Deadbeat Universe: A Textbook On Cosmology, Gravitation, Time, Relativity And Quantum Physics, 2nd edition

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i

The Deadbeat Universe

ii

iii

The Deadbeat Universe by

Lars Wåhlin

Colutron Research Boulder, Colorado

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Copyright © Colutron Research 1997 ISBN 0-933407-03-3 Second Edition Revised 2003

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Preface

We always thought of ourselves as being at the center of the Universe and at rest. It was not until very recently that Copernicus explained how our Earth is orbiting the Sun and that the Sun, not the Earth, is at the center of our solar system. Today’s theories such as Einstein’s special and general relativity, still believe that we and our galaxy, are at the center of the Universe. In fact, relativity with its “cosmological principle” claims that any observer on any galaxy in the Universe can consider him or herself at the center and at rest. In other words, everything is relative and there is no preferred or absolute point in space to relate our location or frame of reference to, thus the term “relativity”. This view creates certain problems. Imagine how difficult it was for astronomers before Copernicus’ time, to set up mathematical equations for planetary orbits with the Earth at the center and at rest and how difficult it is today to deal with a Universe that has more than one center in which we are motionless and at rest. It is understandable why we tend to believe that we are at rest since the star studded sky seems motionless relative to us and we have no feeling or conception of velocity or acceleration. For example, we cannot feel that we are hurling through space around the Sun fifty times faster than a rifle bullet at an orbital velocity of 30 km/second (18 miles/second). Nor do we feel our velocity around the center of our galaxy which is ten times higher or our velocity relative to the rest of the Universe which is still another thousand times higher and which equals c, the velocity of light. There is no doubt that a clever mathematician can construct mathematical equations that will describe planetary orbits with the Earth at the center or equations that will work for a Universe where we assume ourselves to be at rest and everything else moving relative to us. To build a good conceptual theory on such equations is difficult if

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not impossible. The problem is that by accepting the theory of relativity we deal only with relative motion and denounce the existence of absolute motion. Why not accept both? For example, if we are part of a large system in which everything is moving about in an organized fashion, then there will obviously exist both relative motion between bodies as well as absolute motion, with respect to a common center of the whole system. The fact that we are moving at a velocity of c with respect to the rest of the Universe and still subject to a minute cosmic acceleration of a 0 towards its center, is what this work is based on. One can compare our galaxy and the rest of the Universe to a swarm of bees in which all members are moving relative to a common average point. In our Universe, where all matter is subject to a mutual gravitational attraction, such a point is the center of mass of the system or the point to which everything is attracted. Knowing our absolute speed c and gravitational acceleration a 0 in such a system makes it possible to create exact mathematical solutions which can pinpoint parameters such as mass, size, age and temperature of the Universe to mention a few. To date there are no such solutions obtainable for distances beyond the solar system. Most equations in this book are based on the harmonic motion of the Universe and will accurately describe the contracting-expanding Universe. The outcome implies that we are part of a “deadbeat” or a one cycle Universe that is in a state of contraction and the most compelling evidence for this type of cosmology are the equations describing atomic orbits in Chapter 6 section 6.3. It will also be shown that the observed cosmic 2.76 K microwave temperature is a direct result of collective or thermolized radiation from all stars and matter in the Universe. The lifetime of our galaxy is about 8 million, million years and the cosmic model described promotes both evolution and continuous creation (Chapter 7 section 7.4). It is not the intention of this book to reject Einstein’s work since many of his basic equations and discoveries are used throughout. It is merely to point out that the conceptual explanation of his relativity needs to be changed and that further progress can be made if we add

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the idea of absolute and relative motion, as well as absolute and relative energy. It is not the first time a great theory has to be modified. For example, Isaac Newton, the father of modern physics, had his theories modified by Einstein himself and Einstein’s model of a static Universe had to be altered by contemporary science to a dynamic expanding-contracting Universe. We are still far from a perfect theory that will explain everything. The field of natural science is like a labyrinth where progress is made in small steps and where each step usually ends up at a dead end and considerable time passes before a new path can be found. There are two ways to derive a scientific theory. One is by logical reasoning where a theory has to be both conceptually and mathematically sound. The other is by mathematical modeling where equations are structured to fit observations and where conceptual explanations are often missing or misleading. The theory of electric current in solid conductors is one example, where mathematical reasoning requires current to flow from positive to negative, when in reality the opposite is true. One of my favorite subjects is mathematics. I believe mathematics to be a wonderful manmade tool and there is no doubt that mathematical physics has had much success, but I also think that page after page of abstract Picasso mathematics might scare off many potential new scientists. I therefore like to add that it is important to remember that the laws of mathematics must obey the laws of physics and not the other way around. Mother Nature does not know of numbers or digits. She behaves more like an analogue computer rather than a digital computer. This book is written for anyone intrigued by the subject of basic physics and cosmology, and even though it contains numerous equations, only a limited knowledge of algebra and trigonometry is required. In fact, I believe most of the equations can be skipped since numerical solutions are already provided and the purpose of the equations is merely to prove a point, or to describe a scientific statement in rigid mathematical terms. L.W. Boulder Colorado, 1997

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Table of Content Page CHAPTER 1

HISTORICAL BACKGROUND

1

1.1

Early Developments

1

1.2

Later Developments

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1.3

Present status quo

14

1.4

Problems

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CHAPTER 2

THE HARMONIC UNIVERSE

21

2.1

Harmonic Motion

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2.2

The Contraction

25

2.3

Gravitation

29

2.4

Energy and Time

29

2.5

The motion of the Universe

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CHAPTER 3

VELOCITY, ENERGY AND ACCELERATION

35

3.1

Velocity-Energy relationship

35

3.2

Inward acceleration

42

CHAPTER 4 COSMIC DISTANCE AND MASS

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4.1

Our position x0 relative to the center

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4.2

Total mass within x0

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4.3

Mass density

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4.4

Potential energy of matter

54

4.5

Mass and Energy

55

x

page 4.6

The Ether

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4.7

The bending of light by gravity

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CHAPTER 5 RADIATION AND TEMPERATURE

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5.1

Period (Hubble’s time) and frequency of the Universe

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5.2

Angular frequency (Hubble’s parameter)

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5.3

Force constant

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5.4

Radiation

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5.5

Temperature

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5.6

The origin of Planck’s constant h

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CHAPTER 6 ATOMIC ORBITS AND PHOTONS

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6.1

Mass and Radiation

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6.2

Quantum of action and Plank’s constant

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6.3

Particle waves and fixed atomic orbits

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6.4

The Photon

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6.5

The velocity of light

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CHAPTER 7 RED-SHIFTS AND ENERGY BALANCE

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7.1

Red-shifts

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7.2

Energy density of radiation

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7.3

Energy density of matter

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7.4

Continuous creation

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CHAPTER 8 LARGE NUMBER HYPOTHESES AND THE VIRIAL THEOREM

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8.1

Large number ratios

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8.2

. × 1085 Eddington’s Magic Number N = 17507

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page 8.3

Ampère’s equation

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8.4

The Virial theorem and Cosmology

113

8.5

Conclusion

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CHAPTER 9 SUMMARY

121

9.1

Relative and absolute motion

121

9.2

The true Universe?

130

9.3

Building blocks of Nature

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APPENDIX A

Constants and Measures

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APPENDIX B

The Hydrogen Atom

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APPENDIX C

The Problem with E = mc2

153

REFERENCES

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INDEX

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CHAPTER 1 HISTORICAL BACKGROUND Early developments Later developments Present status quo Problems

The ancient Greeks were probably first on record to practice science. They believed everything in our Universe was made up of four single entities; Earth, fire, water and air. Many thousand years have passed and today there are still only four known building blocks of nature namely mass, electric charge, time and length denoted by the symbols m, q, t and l. It is amazing to think that all secrets of our Universe can be unlocked by finding the right combinations of these four symbols. For example, we know that velocity is length per time or l/t while acceleration equals velocity per time v/t or l/t2. Force, which is mass times acceleration, is written as ml/t2 and energy becomes ml2/t2. In this way we should be able to describe any process in nature, whether it involves the smallest atom or the Universe as a whole.

1.1 Early developments Looking at the stars many of us have asked, "Where does everything come from and how long has it been here? Does the Universe have boundaries and how long will it last?" There are no obvious answers to these questions, but it has not discouraged us to search for clues. Even if the over-all picture of the Universe has improved since ancient time, it is still too early to classify Cosmology as an exact science, because despite all information obtained so far there are no exact numbers or exact mathematical solutions at hand which can describe the precise nature of our Universe.

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THE DEADBEAT UNIVERSE

It is interesting to note that some of the basic ideas of today are in fact rediscoveries from the past. For example, we learn from early records that Thales, 580 BC, believed the Moon to be illuminated by the Sun. About the same time Anaximander, 611?-547? BC, thought the Earth was round instead of flat. His contemporary, Anaximines, who at first agreed that the Earth is round and later changed his mind, was first to distinguish between planets and stars. In the fourth century BC Heraclides of Pontus amazingly suggests that the planets Venus and Mercury circle the Sun (Helios) rather than the Earth and that the motion of the stars could be explained by the rotation of the Earth around its axis once in every twenty-four hours. However, a colleague of his, the great philosopher Aristotle, rejected Heraclides’ rotational idea arguing that if the Earth was spinning around its axis then all heavenly bodies, including the planets, would appear to move around us at the same speed. But since the planets move with different velocities it would prove that the Earth is standing still and the planets, including all other heavenly bodies, are moving around us at their own chosen velocities. In fact Aristotle felt that all heavenly bodies were falling in towards the Earth's center which he also believed to be the center of the Universe (this is the first notion of a collapsing universe). In Aristotle’s own words: "As evidence that all heavenly bodies move towards the center of the Earth, we see that weights falling towards the Earth do not fall in parallel lines but always at the same angles to it. Therefore, they are moving towards the same center, namely that of the Earth. It is therefore clear that the Earth must be the center and immobile. From these considerations it is obvious that the Earth does not move, neither does it lie anywhere but at the center of the Universe." The belief that we are at the center of the Universe is shared by many theoreticians even today who have adopted the theory of relativity and its cosmological principle. Shortly after Aristotle’s and Heraclides’ deaths, Aristarchus of Samos extended Heraclides heliocentric idea so that all planets, including the Earth, were moving around the Sun just as we know it today


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(Lovell, (1981)). Heraclides and Aristarchus heliocentric theories were short-lived, mainly due to religious opposition, but were rediscovered nearly two thousand years later by Copernicus.

Fig. 1. The Earth as center of the Universe.

That the Earth is round was deduced in early time from the fact that new stars and constellations will rise above the horizon as one travels north or south. Also, while traveling north or south stars straight above move off at certain angles making it possible to calculate the radius and circumference of the Earth by triangulation. Land disappearing beyond the horizon at sea also gave a clue to the spherical shape of Earth. In one of his essays, Aristotle wrote, "Mathematicians who tried to calculate the circumference of Earth put it at four hundred thousand stades" which is about 74,000 kilometers. It is believed that this written passage stimulated Columbus to undertake his famous new

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world voyage. Later, in the third and second century BC, the circumference of Earth was calculated more accurately by Archimedes who arrived at a value of 55,500 km; Eratosthenes and Hipparchus obtained 46,600 km; Posidonius 44,380 km (today's value is 40,000 km, Munitz (1962)). Eratosthenes is best known for his method of measuring the length and angles of shadows cast by vertical poles at different positions along the Earth's surface. By triangulation he then found the radius and circumference of the Earth. All the above discoveries might not seem very impressive today. We take for granted that the Earth is round and that we belong to one of the planets that encircles the Sun. But two thousand years ago such discoveries were giant leaps in science. To find the first puzzle pieces of our physical world could be compared to the difficulty one would have to imagine a new color never seen before. It is true that many difficult problems have simple answers, but once there is an answer there is no longer a problem and often, once solved, little credit goes to the problem solver. An example of this is the early American township which had posted a $20,000 award to whomever could devise the means or method for removing a large boulder which had rolled down and blocked main street (dynamite could not be used because of nearby buildings). There were many unsuccessful attempts until a bright person appeared who claimed he had a workable solution. When he revealed his idea, "bury it", the towns people felt that such a simple answer was not worth the $20,000 previously offered. Even today solutions and answers to scientific problems do not come easy, but the right answers usually turn out to be simple ones. It is easy to speculate, however, and often numerous and different theories appear about the same subject. This is especially true in the field of cosmology were exact measurements and exact mathematical solutions are not yet available thus making it difficult to rule out even the most exotic ideas.


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In the 14th century Cardinal Nicholas of Cusa tried to break away from Aristotle's theory that for nearly 2,000 years held our Earth as the center of the Universe, a belief which was cherished by the church. Cardinal Nicholas of Cusa thought that the Earth was a moving star like all other stars and that the Universe was infinite in size, because God would not have created anything smaller. The Cardinal's ideas were criticized as being mystical and unscientific because in his infinite Universe, he claimed, each and all bodies would be at the center at the same time. Each body would also be at the periphery and in the interior at the same time. The reason for this is that in an infinite Universe everything can be said to be at the center since there is no limit to its radius. It is interesting to note that modern cosmology follows the same line of thought, namely that any observer on any galaxy in the Universe can consider himself to be at the center of the Universe. This is called the cosmological principle, see page 17. Awkward situations arise when infinity and zero are incorporated into physical phenomena. For example, a point source of energy with zero radius will contain an infinite amount of energy just as a boundless Universe with an infinite number of centers would have. Absurd questions can be asked such as; "What happens if an infinite force strikes an immobile object? What is the probability that another world like ours exists in an infinite Universe?" The answer is that the probability is at least 1:1 and the probability that an infinite number of other worlds exist just like ours with a person like oneself reading the same book etc., is also 1:1. It is the author's opinion that infinity has no real meaning in physics. Nevertheless, many cosmological theories still incorporate infinity. Giordano Bruno was burned at the stake in the year 1600 for supporting the cardinal of Cusa's idea that the Sun and Earth are in motion like stars. Giordano Bruno wrote his first publication on an infinite Universe while residing in England from 1583 to 1585. It is possible that he was influenced by Thomas Digges' treatise Perfit De-

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scription of the Caelestiall Orbes, which was first published in London 1576. Thomas Digges treatise is a translation of Copernicus' work into the English language with some of his own additions. His most important addition was that he believed the Copernican Universe must be infinite (Hoskin, (1997)). Nicolaus Copernicus himself saw his scientific work De revolutionibus orbium coelestium published as he was dying in 1543.

Fig. 2. The planetary system by Copernicus.

Copernicus arrived at the same idea as Aristarchus two thousand years earlier that the planets, including Earth, are orbiting the Sun. The work of Copernicus provides a picture of the solar system as it is described today. His orbital system has served as a model for many theories to follow, such as those involving the motion of stars and gal-


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axies and theories dealing with the smallest atom to the structure of the entire Universe. At first there seemed to be some minor problems with Copernicus' theory because planetary orbits did not appear to be perfectly circular. The problems were solved in a most elegant way by Kepler, who discovered that orbits can be elliptical and during a period of 10 years, from the year 1609 to1619, Kepler established three laws of orbital motion that still stand: 1. Planets move in elliptical orbits around the Sun. One focal point of the ellipse coincides with the center of the Sun. 2 The radial vectors which connect the Sun with each planet sweep out the same area at the same time. 3. The cube of the average distance between each planet and the Sun is proportional to the square of their periods. Kepler's laws are laws of harmonic motion. Kepler has been criticized for having an extraordinary or mystical belief in the harmonics of the world. For example, he tried in vain to find a periodic relationship between the planetary orbits similar to that of the harmonic overtones in music. i.e. Kepler had the idea that orbits might be quantized which in fact is a possibility still open to question. That orbits can be quantized was later proven by quantum physics which describes the organized orbits of electrons in atoms. Acceleration: The year 1590 was a very important year in physics because it was then that Galileo Galilei discovered and measured acceleration. He found that test bodies dropped from a height were falling with increasing speed toward the Earth's surface. Each second the velocity increased by 9.8 meters per second, so that in three seconds, for example, the velocity had tripled to 29.4 meters per second. In mathematical terms the Earth’s acceleration can be expressed as g = 9.8 m/s 2 , where g is the acceleration due to the Earth's gravitational

attraction.

Although acceleration is an everyday occurrence, it was

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never before thought of as a separate event or physical property. It was merely considered a motion. Why is acceleration so important? Because nothing will happen without acceleration. It is not possible to go to work at the office, for example, without having to accelerate and decelerate. Acceleration, which is the same as change in velocity, leads to a change in energy. Acceleration causes bodies to fall toward the center of the Earth with increasing speed due to the Earth’s gravitational field and it generates radiation when electrons accelerate into faster and closer orbits in atoms where the electrons are attracted by the electric field of the atomic nucleus. In fact the whole Universe is in a constant state of acceleration which is evident from the fact that galaxies are receding from each other with velocities that increase with distance (since the distance between galaxies increases with time then velocity must also increase with time which is acceleration). Galileo found the key to a completely new branch in physics called "dynamics". It must also be mentioned that Galileo was one of the first to use the telescope, a Dutch invention, for astronomical observations. Through the use of the telescope it became clear to Galileo that the old view held by the church that the Earth is the center of creation was wrong and that we are in fact orbiting the Sun. Galileo eventually landed in jail or house arrest for supporting the heliocentric doctrine and was ordered to decant. Mass and inertia: Another important observation made at the time, which relates to acceleration, was the concept of inertia. Inertia, which is matter's resistance to acceleration, was an idea invoked by the Frenchman Renè Descartes in 1644. Descartes concluded that a material body in motion will keep its velocity in a straight line unless deflected by another body. The more massive a body is, the more it will resist deflection and consequent acceleration. Mass should not be confused with weight. For example, we know that a bowling ball weighs much less on the Moon than on Earth. Some might think that one should be able to roll a ball faster down a bowling lane on the Moon than on Earth but that is not so. To accelerate the bowling ball to a


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given velocity on the Moon will take the same effort as on Earth because the resistance to acceleration, or inertia of mass, remains the same. Inertia of mass (or simply mass) as we know it today is one of the fundamental entities of nature. The two most basic ingredients in physics, "inertia of mass" and "acceleration", had been discovered but their relationship was not yet fully understood. It was Isaac Newton who put it all in the right perspective when he showed that the product of inertial mass and acceleration is force (force = mass × acceleration or F = ml / t 2 ).

1.2 Later developments

Force: Galileo was born in 1564 the year when Michelangelo died. Galileo died the year 1642, when Isaac Newton was born. The concept of force was not new in Newton's time. Aristotle was aware of the gravitational force that pulled everything towards the center of the Earth including the planets and stars. Kepler realized that by placing the Sun in the center instead of the Earth, the Sun must have an attractive force superior to that of Earth. Attractive and repulsive forces were known to the Greeks who discovered that by rubbing amber (electron), fragments of paper etc. became attracted and sometimes repelled by the amber due to some mysterious force. In the year 1600 William Gilbert published his work On the Magnet which dealt with the repulsive and attractive forces of magnetism and in which the Earth for the first time is being described as a large magnet, which is why we today can talk about the Earth’s magnetic north and south poles. It is believed that Kepler thought magnetism might be the force that caused the planetary orbits to be elliptical. He also visualized the attractive force of the Sun to fall off in intensity with distance. In 1635, seven years before Isaac Newton's birth, Robert Hooke was born on the Isle of Wight. Hooke is said to be first to arrive at the idea of universal gravitation when in 1674, he published his work on the

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Earth's and the planets' motion around the Sun. Hooke was convinced that the force holding the planets in their orbits around the Sun was the same as the gravitational force which pulls a stone towards the center of the Earth. He also maintained that the gravitational force of the Sun decreased with distance. Hooke is also credited with the discovery that the planets Jupiter and Mars are rotating around their axis and that double stars exist. He was first to observe the phenomenon of star aberration and that tails of comets always point away from the Sun. Star aberration is explained as follows: when the Earth is moving around the Sun the positions of stars seem to shift in the direction of our motion. The light rays from the stars could be compared to that of rain falling on the windshield of a moving car. The rain appears to hit the windshield in steep angles although it is falling straight down. The same thing happens to light rays and as the Earth swings around the Sun and starts to move in the opposite direction the position of the stars shift the other way. Force was more or less an intuitive concept until Newton formulated it into a mathematical law of physics and today the unit of force bears his name (if one kilogram is being accelerated so that its velocity increases by 1 m/s every second it will be subject to a force of one newton). Newton's law of universal gravitation marked the beginning of a new era in astronomy and physics. Newton's law states that the gravitational force between two masses m1 and m2 separated F = Gm1m2 / r 2 , where G is the Universal by a distance r is gravitational constant. Newton pictured gravitational force as action over distance in a stationary medium which can be called the ether since at the time, and even now, it is hard to visualize a force acting over a medium of nothingness. Newton's action over a distance can be thought of as field lines of force interacting between bodies where the intensity of the force is proportional to the number of field lines that perpendicularly cut through a unit surface area (F = number of field lines per square meter). There are certain interesting questions connected with a fixed


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ether or absolute space, because it puts the Earth and all the stars in specific positions relative to absolute space. What would happen for example, if all matter in the Universe was removed except for our Earth, could we still say that it rotates and if so, relative to what? Would we still be able to register a centrifugal force at the equator? Since the Earth and stars are moving through space at different velocities, should light waves not reach us at different velocities depending on which direction they come from? Relativity: The above questions have been pondered by many and several new ideas evolved, most noteworthy is the theory of relativity. The first on record to present such a theory was Bishop George Berkeley in 1705. From his writings in The principle of Human knowledge we read: "If every place is relative then every motion is relative, and as motion cannot be understood without the determination of its direction which in its turn cannot be understood except in relation to our or some other body, up, down, right, left. All directions and places are based on relations and it is necessary to separate a stationary body distinctly from a moving one. Let us imagine two globes, and that besides them nothing else material exists, then the motion in a circle of these two globes round their common center cannot be imagined. But suppose that the heaven of fixed stars was suddenly created and we shall be in a position to imagine the motion of the globes by their relative position to the different parts of the heaven." In 1893 Ernst Mach, perhaps not knowing about Berkeley's writings, formulated a physical principle along the same lines which is called “Mach’s Principle”. Mach questioned the nature of inertia (resistance to acceleration) and especially motions that give rise to centrifugal forces. Mach statement reads as follows:

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"For me only relative motion exists. When a body rotates relative to the fixed stars centrifugal forces are produced. When it rotates relatively to some different body but not relative to the fixed stars, no centrifugal forces are produced. I have no objection to calling the first "rotation" as long as it be remembered that nothing is meant except relative rotation with respect to the fixed stars." This is called the “Mach Principle”. The stars, of course, are not fixed but move with extreme velocities relative to us. The vast distance between us and the stars make them appear stationary in the same manner that fast going ships at sea seem nearly stationary at far distances. If the motion of all stars in the Universe is governed by Newton's law of universal gravitation then this must imply that all things must move about a common center, the center of mass of the Universe, which of course brings us back to a Newtonian absolute space. The concept of an absolute space, the constancy of the speed of light and the argument that an infinite Universe would create an infinite gravitational force led to severe conflicts at the end of the19th century and beginning of the 20th century. The search for an absolute space or ether was culminated by the Michelson-Morley experiments which started in 1887 and which showed that the speed of light relative to the Earth is constant in all directions thus disregarding the Earth's orbital motion through a possible ether (see page 121). The constancy of the speed of light at Earth and the inference of Lorentz and Pointcaré that no velocity can exceed the speed of light led Einstein to formulate a different kind of relativity which he named Special and General Theories of Relativity and which forms the basis for scientific thinking of today. In his Special Theory of Relativity Einstein (1905) deals with the constancy of the speed of light in purely mathematical terms and he also formulated the following postulates: 1. The laws of physics take the same form in all inertial frames.


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2. In any given inertial frame, the velocity of light c, is the same whether the light be emitted by a body at rest or by a body in uniform motion. The second postulate simply rejects the existence of an ether, and in Einstein's General Theory of Relativity (1915) the nonexistence of absolute space and ether again brings us back to the Cardinal of Cusa’s infinite Universe where observers anywhere can consider themselves to be at its center. In Einstein's Universe, which has no reference point or common center of mass, inertial forces such as centrifugal forces for example, are generated even in the absence of the fixed stars, in contrast to the earlier relativity theories of Berkeley and Mach. A spinning Earth would, in a mysterious way, generate centrifugal force at the equator even if all other matter in the Universe was removed. This reverts back to Berkeley's argument, "How then can we say that the Earth is spinning and relative to what?" The problem that an infinite Universe must generate an infinite gravitational force field was avoided in Einstein's General Relatively when he proposed that the Universe is bounded but yet infinite. This is explained by introducing a curvature on space allowing the Universe to somehow curve back on itself. Einstein's curvature of space can perhaps be explained as follows: It is an established fact that light rays, which are massless, bend inward as they pass near massive gravitational bodies, such as the Sun. One reason for this is that time slows down with increasing gravitational tension. This means that all physical processes slow down including the propagation of light. Light rays will therefore travel slower as they encounter an increase in gravity. When a beam of light grazes the surface of a gravitational body it will bend. A light ray, if it could travel forever would therefore never leave the Universe since it must bend and eventually curve back on itself due to the immense gravitational field of the Universe. Einstein reasoned that the curvature of space is caused by gravitational fields in which both time

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and speed of light change to form a combined space time dimension in which not just light rays, but everything curves, such as measuring rods, world lines, etc. The infinite but bounded Universe has often been pictured as follows: If two-dimensional beings which can only conceive two dimensions and therefore are unaware of a third dimension, were living on a spherical planet and were traveling in a straight line, they would never find an end to their world since they would move in an infinite number of circles, which would make them believe they are living in an infinite but bounded world. Also, triangles drawn on a spherical surface would never total 180 degrees because of their curved world lines. Einstein’s relativity is a mathematical model, in contrast to a conceptual theory, describing changes in rate of time as well as bent world lines by applying geometry. It can be compared to one explaining the stock market using bar graphs and pie charts, but how it exactly works is still a mystery.

1.3 Recent status quo

The General theory of Relativity, which is based on geometry and curved world lines was, in its earlier stages, describing a static Universe where the overall gravitational force was counteracted by some imaginary repulsive force in order to prevent the Universe from gravitational collapse. The repulsive force was later replaced by the expansion process in the Big Bang theory where the Universe is believed to have originated from a primeval explosion that started from a singular point. The discovery that light waves from surrounding galaxies become more and more red-shifted the farther away they are, led Edwin Hubble to believe that the Universe must be expanding in all directions. The red-shift is interpreted as a Doppler shift where the wavelength of light waves become stretched out towards the red side of the optical


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spectrum as the source of light is receding from us. A popular belief today is that the Universe is expanding and that space, according to General Relativity, curves in on itself in such a fashion that the original center, the center of the primeval explosion, occupies the periphery of the Universe and that the periphery of the Universe is at the center of the expansion. This is not exactly a conceptually clear picture, but if we allow ourselves to deviate slightly from our standard way of thinking we can perhaps picture such an inside-out world. Many cosmologists of today would be offended if we ask them to point their telescopes to the point in heaven where the primeval explosion took place. It would be explained to us that when we look at far away galaxies we are also looking back in time to the beginning when the Universe was born. Therefore, since far away galaxies can be observed in all directions around us, one can conclude that the birth place of the Universe must also be all around us at the periphery and that we, 10 billion years later, are still at the center of the expansion. Energy: Two very important energy relations were established in the beginning of the 20th century. They are Einstein's (1906) energymass equation E = mc 2 and Plank's (1900) constant of radiation h which is energy divided by frequency h = E / ν . At first it was very difficult to justify how energy stored in mass equals mc 2 and not 12 mc 2

according to conventional Newtonian physics and why it is impossible for a mass to reach a velocity of c since its energy E then would reach infinity. There is no problem with the mathematics, but theoretically it is awkward. The same is true of Max Planck's discovery that energy of radiation divided by its frequency is a constant. The problem is that Plank's constant h has the dimensions of energy multiplied by time (Et) or momentum p ( p = mass × velocity ) multiplied by a certain length x. Momentum multiplied by length px = h has no meaning and was believed by many physicists to violate the laws of conservation of energy, because whether a particle with constant velocity and

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momentum traveled one or two meters ( px , p2x , etc.) does not change its energy or state of affairs, but according to Planck’s discovery it will. Heisenberg found a different meaning of Planck’s constant by showing that it can be written as 12 h = ΔpΔx which means that any change in momentum is inversely proportional to a change in distance and that the product of these variables equals 12 h. The intriguing fact is that if we were to determine the momentum of a particle to the highest precision (that is with smallest possible error Δ p) then it will be difficult to pinpoint the particle’s position, because the smaller we make Δ p the larger Δ x becomes (the precision in determining its position). The opposite is true when we try to fix a particle's position to a high degree of precision because then the value of its momentum becomes uncertain. Heisenberg's Uncertainty Principle plays a powerful role in atomic physics and is also believed to have important cosmological consequences, especially in the early Big Bang creation of the Universe. Both Einstein’s energy relation E = mc 2 and Planck’s constant h were great discoveries in modern science but are still not yet well understood. The accidental discovery by Penzias and Wilson (1965) that cosmos is filled with a uniform microwave radiation did establish the fact that the average temperature of the Universe is only about 2.76 degrees Celsius above absolute zero. The microwave radiation is believed to be the remnant radiation from the hot Big Bang explosion. However, the background radiation, which has a blackbody distribution just like a baking oven, can also be explained by the combined heat or scattered radiation from all cosmic heat sources such as stars, galaxies, etc.

1.4 Problems

There are many problems associated with our understanding of the Universe. Most serious is the fact that we don’t have any exact mathematical solutions for how big it is, its mass or mass density or


17

how much energy it contains and what the nature of time is to mention a few. We should, in the author’s opinion, have enough good data collected from astronomical measurement to be able to piece together good a working theory. Presently the most commonly accepted cosmological picture is that of an infinite but expanding Universe. Cosmic red-shifts are interpreted as galaxies receding from us in all directions with speeds that increase with distance and we are believed to be at the center of the expansion and at relative rest. The very unlikely fact that only we should be so privileged as to occupy the center of the Universe, is remedied by resorting to an infinitive Universe. We have seen that General Relativity has adopted Cardinal Nicholas of Cusa's idea that "in an infinite Universe observers on any galaxy will be at the center the same time". The mathematical reasoning is that an infinite radius must have an infinite number of centers and that in an infinite Universe all bodies can be thought of as occupying both the center and the periphery at the same time, which contradicts common sense for most of us.. The theories of relativity do not only allow us to be at the center and periphery at the same time but it also allows us to say that we are either moving with near light velocity with respect to distant galaxies or that distant galaxies move with velocities approaching that of light relative to us. This dual character of nature in relativity theory is called the Cosmological Principle. To add to an already confusing picture we also have to deal with the fact mentioned earlier that when looking at the most distant galaxies, we also look back in time to the birth place of our Universe. Since all distant galaxies appear at the horizon all around us, we must be surrounded by the point where the creation took place. Great efforts have been made to determine whether the expansion is slowing down or not. If so, it would indicate that we are still bound by the laws of mutual gravitation where everything could conceivably collapse back to a singular point. This would be analogous to a stone thrown straight up in the air which,

18


according to the laws of falling bodies, will slow down to a stand-still and accelerate back towards the Earth’s center again. It is the author’s opinion that most of the difficulties mentioned above can be untangled if we simply invert our present picture of the Universe by turning things around. For example, if Hubble had carefully considered the recession of galaxies according to the laws of falling bodies, he could as well have concluded that the Universe is in a state of infall or contraction rather than expansion. Using the example of the stone thrown up in the air and falling back, one can picture galaxies instead accelerating with increased velocity as they fall toward the center of the Universe due to its mutual gravitational attraction. Since galaxies falling ahead of ours have reached a higher speed they will appear to pull away from us and if we look back in the opposite direction we find ourselves moving away from galaxies that are still in an early state of fall. Had the Universe been expanding, a blue-shift would have occurred since we would be gaining on galaxies ahead of us that are slowing down, while galaxies behind are trying to catch up with us. We can also argue, that if all galaxies rush away towards the periphery and origin of our Universe, as claimed by the modern cosmologists, we must in fact be collapsing back to the central point from where the Big Bang creation took place. If what has been discussed in the two last sections appears to be confusing that is because it is. Cosmology as a science is still in its cradle and a solid scientific path to follow has not yet been found. Most of our ideas about the Universe are developed from light spectra and optical observations performed at our remote point of reference here on Earth. What makes it so difficult is that we cannot go out there and check our ideas. The purpose of this work is to take important pieces of knowledge, on the subject of cosmology, and try to piece them together into a possible working theory based on harmonic motion (Wåhlin (1981,1985)). Using simple laws of harmonic motion can in fact, produce simple and understandable equations that yield exact solutions in, until now, the elusive science of cosmology. The text that


19

follows deals with an expanding-collapsing Universe, which obeys the laws of harmonic motion, and should rightfully be dedicated to Kepler who was unfairly criticized for being obsessed by his belief in the harmonics of the worlds. Page 20 shows the front page of Kepler’s work The harmonics of the worlds published the year 1619.

20


Title page of Kepler’s work.

CHAPTER 2 THE HARMONIC UNIVERSE Harmonic motion The contraction Gravitation Energy and Time The motion of the Universe

My God is a Goddess. She is the ruler of creation and evolution and Her laws are divine manifestations that cannot be broken by man. She does not care about our convictions or what we here on Earth believe is right or wrong or what we do or not do. After all we are not alone, there are more heavenly bodies in the Universe than sand pebbles on all our beaches and our presence here on Earth is less than one tick on the cosmological clock and our time might already be running out. My Goddess often seems cruel and hard but Her realm and Her creations are most beautiful and aesthetic. I have spent a lifetime studying Her laws and have learned how to live with them in harmony. She has taught me the art of living to be honest and fair and I am thankful for having been a small part of Her world. My Goddess is Mother Nature.

2.1 Harmonic motion A periodic motion such as the swing of a pendulum or the circular movement of a planet in its orbit must obey the laws of harmonic motion. The name "harmonic" refers to mathematical functions that involve sines and cosines and periodic motions can be accurately described by such functions. A "simple" harmonic motion is usually associated with a to-and-fro movement of a mass about a central point. A typical example of such a motion is the piston in an engine or a

22


weight at the end of a spring, see Fig. 3. A simple harmonic motion is characterized by the fact that the acceleration of the mass is zero as it passes through the central point, x = 0 , of the oscillation and increases on either side of x = 0 with distance. The velocity of the mass on the other hand, is highest at the center and zero at the maximum amplitude or turn around point where x = A . A typical example of a simple harmonic motion involving gravity would be to imagine a hole drilled through the Earth from pole to pole. If we drop a stone through the hole it would accelerate and increase its velocity of fall until the acceleration reaches zero as it passes through the Earth's center. The velocity on the other hand, starts at zero and reaches a maximum as it goes through the Earth's center after which deceleration sets in bringing the stone to a stop as it reaches the other pole.

In a uniform circular motion, such as a planet orbiting the Sun, both acceleration and velocity remain constant. Here, the inward gravitational attraction is equal to the outward centrifugal force created by the rotation. The frequency of oscillation is the number of turns or cycles per second completed by the oscillator and is usually expressed as Hertz (Hz). Instead of using turns per second (each turn equals 360 D of rotation) we can also use angles per second based on the radian system where one radian = 360 o / 2π = 57.3 o standard degrees. Radians per second ( 57.3 o /s) is generally denoted by the symbol ω and is referred to as angular frequency or angular velocity. The name “radian” stems from the fact that the circumference of a circle equals 2π r = 6.28 radii and the distance of one radius along the circumference equals 57.3 o . The relationship between a simple harmonic motion and a uniform circular motion can be demonstrated by connecting a piston to a flywheel as shown in Fig. 3b, where the handle on the flywheel, follows a perfect circular motion in unison with the to-and-fro motion of the piston. A circular motion can also appear as simple harmonic


23

motion, as demonstrated in Fig. 3c, where the flywheel handle appears to move up and down in a to-and-fro motion when viewed sideways.

Fig. 3a. The to-and-fro motion of a weight at the end of a spring performing simple harmonic motions. b) A flywheel and a piston demonstrating the relationship between a circular harmonic motion and a simple harmonic motion. c) The apparent simple harmonic motion of a flywheel handle viewed edge on.

A harmonic motion could theoretically go on for ever without energy being added, assuming that there is no friction slowing it down. When friction sets in, the stored energy of the harmonic motion will dissipate into heat or radiation, which eventually brings the harmonic motion to a standstill. Friction, or loss of energy, has a damping effect on oscillations so that the flywheel in Fig. 3b and 3c will gradually slow down

24


and change frequency to a value that approaches zero. In the case of the spring and weight oscillator shown in Fig. 3a, the frequency remains nearly the same, but the amplitude of the oscillations will diminish to zero as friction sets in. If the energy loss is so large that the oscillation dies out in less than one cycle, the harmonic motion is said to be critically damped. A golf ball dropped from a given height might bounce up and down several times before the damping action, or friction brings it to rest whereas a sandbag, on the other hand, would perform a “deadbeat” since it is critically damped. Harmonic oscillators, such as electrons in atomic orbits or galaxies and planets in motion around gravitational centers of mass, behave in a rather peculiar way when exposed to damping and subsequent loss of stored (potential) energy. Should we be able to slow down a planet in its circular motion around the Sun it would fall into a closer orbit and its orbital velocity and frequency would increase. Alternatively, adding energy to the planet by increasing its orbital velocity would sling it out to a larger orbit, but slower orbital velocity. This seemingly paradoxical behavior, where added energy results in a decrease of velocity and where loss of energy produces an increase in velocity, is not what we usually experience during our daily routine on Earth, where additional energy seems to make everything go faster. However, we now know from experience that the velocity of a manmade satellite increases due to loss of energy by friction as it encounters the Earth's upper atmosphere and spirals closer to Earth. This inverse energyvelocity relationship is not only a characteristic of orbiting planets and manmade satellites, but it also applies to orbital electrons, galaxies and the Universe as a whole. An expanding-contracting Universe can be classified as a simple harmonic oscillator in which the oscillations could go on forever, unless damped by loss of potential energy. A deadbeat, or critically damped Universe, would perform only one cycle of oscillation and all matter contained therein would lose energy and inertial mass to radiation within one cycle while accelerating toward the center, the center of


25

mass of the Universe. We know from practice that matter radiates when subject to acceleration and the evidence so far is that the Universe is losing potential energy, or mass, in the form of heat and light emitted by stars and galaxies. The observed loss of potential energy and mass and subsequent release of radiation, hints to a deadbeat Universe in a phase of collapse.

2.2 The contraction

The view taken by the author is of a critically damped contracting Universe, in which the laws of harmonic motion must be obeyed. The diagram in Fig. 4 shows the Universe as a simple harmonic oscillator in a phase of contraction (a). A comparison of the Universe to the harmonic motion of a weight at the end of a spring is shown in (b) and to a uniform circular motion in (c). Included in the diagram of Fig. 4 (see also page 146) is a list of several basic equations related to the behavior of harmonic motion and which are described in chapters that follow. In a contracting Universe matter must be subject to an attractive force directed toward a central point, x = 0 , the center of mass of the Universe. The inward attraction, which is created by the mutual gravitational field of all matter in the Universe, determines the rate of the cosmic acceleration a 0 . In contrast to the theories of relativity, which allow us to consider ourselves at the center of the Universe and at rest, our present position must fall within a distance between x = 0 and the maximum amplitude x = A . Also, our velocity relative to the center of mass of the Universe must have a value which complies with the mathematical relation v0 = Aω 0 sin 45° where ω 0 is the frequency of the Universe in angular units. If any two parameters could be found which relate to the harmonic motion of the Universe, such as the velocity v0 and acceleration a 0 , for example, it would be possible to excerpt many other features using the

26


equations shown in Fig. 4 and on page 146. These features could then be used in a realistic comparison with scientific facts already known to us.

Fig.4a. The simple harmonic motion of the Universe in a phase of contraction. (b) The Universe compared to a simple harmonic motion of a weight at the end of a spring. (c) The simple harmonic motion of the Universe as related to a uniform circular motion.

By good fortune, exact values of velocity and acceleration are accessible from both astronomical red-shift measurements and from certain actions postulated by quantum mechanics. Red-shifts of distant galaxies show that we are moving with a velocity equal to or very near c, the speed of light, with respect to distant matter. Heisenberg's


27

"Uncertainty Principle", a product of quantum mechanics, proves that matter is subject to a cosmic acceleration (Wåhlin (1981)), because it predicts an ever present change in both momentum and position of matter, which can only be explained by a change in velocity. Change in velocity is the manifestation of acceleration. Once our velocity of fall and rate of acceleration towards the center of mass of the Universe have been established, numerous other parameters can be found, such as frequency of oscillation, distance to the center, total mass, mass density and temperature, see Table 1. In the chapters that follow it will be explained in more detail how the velocity and acceleration are established and how many other properties of the Universe, including those listed in Table 1, can be unveiled.

TABLE 1 1. Velocity: c = 2.997924 × 10 8 m s −1 . 2. Inward acceleration: a0 = 12 =ω12 / me c = 7.62247 × 10 −12 m s −2 .

3. Our position relative to the center:

x 0 = c 2 / a0 = 1.17908 × 10 28 m.

4. Angular frequency: ω 0 = a0 / c = 2.54258 × 10 −20 rad s −1 . 5. Period (Hubble's time): t0 = 2π / ω 0 = 2.47118 × 10 20 s. 6. Total mass of Universe within x 0 : M u = x 02 a0 / G = 1.59486 × 10 55 kg . 7. Mass density: ρ = a0 /( 43 π Gx 0 ) = 2.32273 × 10 −30 kg m −3 . 8. Potential energy of the Universe within x 0 :

E = M uc 2 .

9. Radiation: Lu = M uc 2 / t0 = M u a0c / 2π = 12 hM u / me = 5.80044 × 1051 w.

10. Temperature: T = [M u c 2 /(t0 4π x 02σ ]

1/ 4

= [a0 c 2 / Gt0 4πσ ]

1/4

= 2.766° K .

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THE DEADBEAT UNIVERSE 1 2

11. Red-shifts: z ≈ (2a0 Δx ) / c. (non-relativistic). 12. Energy density of radiation: U r = 6πT 4σ / c = 2.08756 × 10 -13 J m −3 . 13. Energy density of matter: U m = ρc 2 = 2.08756 × 10 −13 J m −3 . 14. Planck's constant: 12 h / 2π = 12 = = me a0 c / ω12 = 5.2729 × 10 −35 Jω1−1 . 15. Gravitational const.: G = (q / me ) 4 hμ 02 /(16πc ) = 6.6445 × 10 −11 Nm 2 kg -2 . q / me = electron' s charge to mass ratio, ω1 = 2π s −1 , h = hs −2 ,

σ = Stephan - Boltzmann' s costant. The picture that emerges from the above physical and mathematical expressions, is of a Universe that is about 100 times larger than current estimates, see (3) Table 1. This means that we can only observe a small fraction of the Universe from our vantage point in space at x 0 . The small circle around x 0 in Fig. 4a, outlines the horizon of our most powerful telescopes of today. A one cycle (critically damped) Universe dissipates all its potential energy to radiation (8) over one period of oscillation (5), which means that all matter will vanish before reaching the center. This would be analogous to an electron dissipating its potential energy to radiation while falling towards the atomic nucleus. The calculated rate of radiation (9) matches the amount of radiation observed in our Universe and the same equations allow us to determine the rate of radiation from individual material bodies in the Universe. For example, the radiant energy from our Sun equals the Sun's potential energy divided by the period of the Universe or M sun c 2 / t0 = 7 × 10 26 watts. The total radiation from all stars and galaxies in the Universe contribute to a black-body temperature just like the heating elements in a heating oven produce a specific oven temperature. The temperature of the Universe (10) can therefore be calculated to equal 2.766 degrees Kelvin, using Stefan's law of radiation.


29

An interesting observation is that the cosmic energy density of radiation (12) and the energy density of matter (13) are equal. This suggests that there is an equilibrium between matter and radiation, which favors the idea of continuous creation proposed by Hoyle (1948), and Bondi and Gold (1948), see Chapter 7, sec. 7.4. For example, it is a known fact that matter can be created by radiation which is proven by the process of pair production where photons convert into one electron and one positron. Hoyle, Bondi and Gold championed a steady state Universe in which everything would remain the same, such as mass density, radiation and temperature. The Big Bang theory however, predicts that the Universe expands and therefore gets thinner and cooler with time. The above authors suggested that spontaneous or continuous creation of matter, would remedy these changes, by filling the void left behind by the expansion with new matter, thus keeping the matter density of the Universe constant. Unfortunately, the steady state theory and continuous creation did not get much support.

2.3 Gravitation

Another remarkable feature of a collapsing Universe is that the universal gravitational constant G can be determined from Planck's constant h, because Planck's constant relates to the value of the inward cosmic acceleration a 0 (2). This relationship, shown by (15) in Table 1, is of great importance because it establishes a connection between gravitation and quantum theory. The theoretical value of G (15) is about 0.004 times less than the measured value at the Earth's surface. A possible explanation of this discrepancy is that the theoretical number represents the astronomical free-space value of G and that measurements near the Earth's surface might deviate slightly in accordance with experimental claims which show an increase in G at short range near massive bodies.

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2.4 Energy and Time

The main substance in the Universe is Energy, which appears in the form of inertial mass (potential energy) and electromagnetic radiation (loss of inertial mass). That mass and energy are related was proposed by Hasenöhrl 1905 and Albert Einstein (1906) who concluded that inertial mass m of a body contains an energy of mc 2 . Inertial mass and gravitational mass are commonly believed to be equivalent which might be debatable. The term inertia refers to the fact that mass can be measured by its resistance (inertia) to acceleration. A body which accelerates to where its velocity increases by 1 m/s every second, when subject to a force of one newton, has an inertial mass of one kilogram. As previously mentioned, mass and weight are not quite the same thing, although the measured quantity is the same. Weight is really a measure of the Earth's gravitational force, g × m , on a mass placed on a weighing scale, where g is the Earth's gravitational acceleration. Mass on the other hand, is a measure of resistance to acceleration. For example, if we try to push a 500 kg miners cart on a frictionless rail, it would take quite some time and effort to get it moving because of its large mass, or inertia. However, it takes the same effort to get the cart moving on the Moon as on Earth, even though its weight on the Moon is about 100 kg or five times less, because the inertial mass of the cart is the same on the Moon as on Earth and inertia has nothing to do with weight. Also, a body that has been accelerated from relative rest to a given velocity will experience an increase in inertial mass which is proportional to the amount of energy spent in the accelerating process. Relative rest refers to a mass at rest relative to a stationary point, such as a laboratory on Earth from which the measurements are taken. Adding energy to matter by acceleration not only increases its inertial mass but it also causes time and physical processes to slow down in its


31

immediate environment. This time effect which is called “time dilation” is hardly detectable under normal circumstances, but studies of high energy or rapidly moving atoms show that atomic clocks slow down proportionally to their increase in energy and consequential increase in inertial mass. One example is the increase in lifetime of high velocity high energy cosmic pions. The structure of time is not yet fully understood. There are no physical laws that define time or the difference between past and future. Isaac Newton believed that “absolute mathematical time flows equably without relation to anything external”. However, the fact that time slows down proportionally to increase in energy or inertial mass proves that time and inertia are related. Inertia of mass equals resistance to acceleration, and should inertia become infinite, all masses in the Universe would offer infinite resistance to acceleration, which means that nothing could be moved and all physical processes including time keeping clocks, would stall and time would come to a standstill. On the other hand, if there was no inertia or resistance to acceleration at all, physical processes would occur instantaneously and time would pass infinitely fast. It is obvious that we are somewhere between these limiting cases where time flows at a rate determined by the present inertia of mass. What determines inertia of mass or time in the Universe? It is the cosmic gravitational tension φUniv = GM Univ / RUniv = c 2 (energy per mass or E/m) in the Universe that sets the parameters of inertia and time. E/m can also be derived from Einstein’s formula E = mc 2 by writing φuniv = E / m = c 2 . Time, like distance, is measured in length. For example, the length of one meter is the scientific unit for distance and the length of one second is the scientific unit for time. Unlike the unit of distance, the unit of time can change from place to place in the Universe as the gravitational tension changes. One standard second is therefore referred to as a second measured at our frame of reference at rest here on Earth. The energy per mass (tension) can also change in


32

a frame of reference that is moving relative to us. In general terms the length of time is directly proportional to tension or the energy per mass ( φ = E / M ) whether the energy is gravitational or kinetic (increase in inertial mass due to velocity) or a combination of both. One possible way to describe time in scientific terms, as seen from our reference in space, is to write the equation sφuniv = s

GM univ = t1 , (one standard second) Runiv

(t)

(1)

where s = 1.11265 × 10 −17 second 3 /meter 2 is a constant or s = t1 / c 2 (one standard second divided by c 2 ) and G is the universal gravitational constant. For example, if the mass and radius of the Universe had been equal to that of the Sun then the length of time would be much shorter or the rate of time would flow about 500,000 times faster to what we are used to i.e. one second at the Sun’s surface would correspond to sGM Sun / RSun = 2.095 × 10 −6 standard seconds. In reality, the 2.095 × 10 −6 seconds generated by the Sun’s gravitational tension does in fact add to the standard second ( sGM Univ / RUniv ) produced by the entire Universe causing time to flow slower at the Sun’s surface by a factor of 2.095 × 10 −6 . The proportional slowing of time with kinetic energy or gravitational tension leads to perplexities such as the "Twin Paradox". A twin brother who traveled several years through space in a space ship at high speeds would on his return find his brother older than himself. Clocks and physical processes slow down on board, due to the increase in energy per mass of the space vehicle and everything within it, generated by its speed. During flight the traveling twin would not be aware of a slow-down in time since everything in his environment would change pace concurrently, including clocks. The slow-down of time due to gravity or speed (time dilation) was realized by Einstein already in the beginning of the 20th century and the twin paradox was one of the products of his great insights. Another noteworthy characteristic of the harmonic Universe presented here is that the rate of time must change with time just as inertial mass and energy changes with time. This


33

means that one second gets shorter every second by the amount of 1 s/t0 = 4.0466 × 10 −21 seconds per second, where t0 is the period of the harmonic motion of the Universe. It also means that we are running out of time, not in the sense that seconds are ticking away, but that seconds are getting shorter and shorter. For example, in a persons average life time of 70 years the total time lost is nearly 0.02 seconds. The constant loss of potential energy, inertial mass and time due to the harmonic motion of the Universe is the price we pay for the privilege of life because nothing would really happen if time stood still and if there was no exchange of energy. The rate of loss in time or inertial mass or energy is determined by the constant ν 0 = 4.0466 × 10 −21 s −1 which is the frequency of the one cycle Universe or ν 0 = 1 / t0 .

2.5 The motion of the Universe

The simple harmonic motion of the Universe as depicted in Fig. 4 is possibly an oversimplified view and many other modes of harmonic motion can be considered, such as the spiral motion, for example, that is typical of many galaxies. It is also conceivable that the Universe is shaped like a disk having spiral arms like as our own galaxy. The Universe could perhaps be thought of as a hierarchy of oscillating systems starting from atoms, solar systems, galaxies, clusters and super clusters of galaxies and ultimately to a meta cluster of all galaxies contained within it. This idea was proposed by Charlier (1908) before galaxies were even discovered and later by Kiang and Saslaw (1969). The most striking feature of the contracting Universe is that absolute velocity is inversely proportional to absolute energy according to the diagram in Fig. 4, while we at our frame of reference here on Earth, as mentioned before, are used to relative velocity being directly proportional to the square root of relative energy. This inconsistency will be explained in Chapters 3 and 9.

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The parameters and features presented in the following chapters are derived as we see them from our vantage point x 0 in space and for practical reasons the simple model in Fig. 4. will be used.

CHAPTER 3 VELOCITY, ENERGY AND ACCELERATION Velocity-Energy relationship Inward acceleration

How many of us are aware of the fact that we are racing through space faster than a rifle bullet at a speed of 30 km per second in our orbit around the Sun? We certainly can’t feel it nor do we feel our velocity around the galaxy which is ten times faster. Furthermore, our velocity relative to the rest of the Universe, which approaches c, the speed of light, and which is 1000 times faster yet, cannot be perceived except for the relativistic effects it generates that we observe in particle accelerators and in measurements involving fast atomic orbits.

3.1 Velocity-Energy relationship It is not difficult to understand the meaning of velocity. Velocity is how far something moves in a specified unit of time. For example, a strong person might slam a tennis ball to a speed of 36 m per second which corresponds to an energy of E = 36 Joules if we use Newton's energy relation E = 12 mv 2 (assuming the tennis ball has a mass of

m = 0.055 kg). Difficulties arise when we attempt to add velocities and energies because of their apparent non-linear relationship. Assume that the ball was hit by a tennis player in the forward direction while traveling on a train which is moving with a velocity of 36 m per second. To a stationary observer at ground the ball would, before it was struck, have the same velocity as the train and therefore an initial energy of E = 36 Joules. The additional energy ΔE of 36 Joules imposed by the player's racket makes the ball go 36 m/s relative to the player and the

36


train but 72 m/s with respect to the observer at ground. Even though the player only added another 36 Joules of energy to the ball the stationary observer at ground records an increase of ΔE = 107 Joules according to E = 12 mv 2 . There is nothing wrong with the mathematics. The point is to show that when velocities are added to an already existing velocity, difficulties arise. The observer, which is at rest on Earth, might say that he is registering an “absolute” increase in the velocity and energy ( Eabs ) of the ball and that the energy differs by a substantial amount from the “relative” energy ( ΔE ) registered by someone on the train. What if we take into account the Earth orbital velocity around the Sun? Careful examination will show that a hypothetical observer on the Sun would see an energy difference of the tennis ball that would differ drastically from the observer on Earth but slightly closer not exactly the same energy difference as seen by the player and passengers on the train. The example of the tennis ball tells us that we are dealing with two types of energies, namely absolute energy and relative energy, as well as absolute and relative velocities. The importance of absolute energy and relative energy can be further emphasized by the following example. Consider the absolute amount of energy E abs = 9 × 1011 Joules required to launch a 2000 kg rocket to a velocity of vabs = 30 km/s. Adding another 9 × 1011 Joules to the rocket while in flight, through the rocket’s own propulsion system, will increase the velocity by a factor of two (disregarding the weight loss of spent fuel). However, had the extra 9 × 1011 Joules been added to the absolute energy at the launch site it would only have increased the velocity of the rocket by a factor of 1.4 or more exactly 2 . Why the difference? In the first case relative energy is added on top of an already existing absolute energy i.e. the energy added in flight also includes the extra amount of energy the fuel itself had gained during the launch of the rocket, while in the second case the extra energy is simply increasing the amount of absolute energy at the launch. It is of utmost importance to note that energy can appear in both absolute and relative form and throughout this book absolute and relative energy of

VELOCITY ENERGY ACCELERATION

37

matter, with respect to our frame of reference here on Earth, will be distinguished as follows: 1. Absolute energy = E0 ( E0 = m0c 2 ). 2. Relative energy = ΔE or ∇E , where E 0 is the absolute energy of matter at our frame of reference with respect to the center of mass of the Universe and ΔE is relative energy added to E 0 (gain in energy) and ∇E is relative energy subtracted from E0 (loss of energy) Normally we do not worry about the Earth’s velocity of 30 km/s around the Sun or our velocity with respect to the rest of the Universe. However, at high velocities relative to us, such as found in particle accelerators and atomic orbits, our velocity relative to the rest of the Universe becomes very important (Mach’s principle). Einstein’s theory of relativity unfortunately, pays no attention to the above since it considers us at rest (thus the term rest mass energy E0 = mc 2 ). Einstein’s velocity-energy equation is usually written as

v =c 1−

1

E ⎞ ⎛ ⎟ ⎜1 + mc 2 ⎠ ⎝

2

,

(l/t)

(2)

where v is velocity of a mass m relative to us generated by the energy E. Einstein’s theory cannot deal with certain situations, for example where energy is lost, for the simple reason that velocity cannot be subtracted from rest or zero velocity. Equation (2) is only accurate in cases where rest mass energy is gained, such as in particle accelerators, but fails in cases where rest mass energy is lost, such as in atomic orbits for example, where energy is lost to radiation (see Chapter 6, section 6.3). Einstein’s theory can be said to be up against the same difficulties that our ancestors faced when trying to unravel the orbits of planets, believing that our Earth was at the center and at rest. It will be shown by rewriting Einstein’s Equation (2) to Equation (5), (as


38

explained below) we discover that the equations are actually basic formulas of harmonic motion which demands that our frame of reference has to move with an absolute velocity of v0 = c relative to a central point x = 0 , the center of mass of the Universe. Our absolute velocity c relative to the rest of the Universe can be established through red-shift measurements. Since red-shifts of most distant galaxies (which by far outnumber close ones) show recession velocities that equal or approach the speed of light c, we can confidently assume that we are moving with c or very near c relative to the bulk of Universe rather than the Universe moving with c relative to us. Therefore, any relative velocity and energy that we observe or experience here on Earth has to be added (or subtracted) to our absolute velocity of vabs = c and absolute energy E0 . From the above analysis it is possible to construct a mathematical representation that nicely conforms with both Einstein’s equation (2) and the equations of harmonic motion shown in Fig. 4, Chapter 2. and on page 146. The diagram in Fig. 4a demonstrates how we and our neighboring galaxies are accelerating toward the point x = 0 , the center of mass of the system and marked along the x-axis is our current position x = x 0 . The circle surrounding x = x 0 represents the present horizon or limit of our largest telescopes (note that we can only see a very small fraction of the Universe). Also shown is the point of maximum amplitude where x = A , from which matter starts falling towards the center. At the maximum amplitude A, where potential or absolute energy of matter is largest, the velocity is zero. Since Fig 4 indicates a loss in potential energy as matter falls with increased speed towards the center x = 0 , where the absolute velocity has reached vabs = 2c , we have an inverse velocity-energy relationship. At our position x 0 absolute velocity and absolute energy of matter is c and E 0 respectively. An increase or decrease in E0 by ΔE or ∇E can be written in the form

vabs = c

E0 , E 0 + ΔE

and

vabs = 2c − c

E 0 − ∇E E0

(l/t)

(3)


39

Equation (3), which yields absolute velocities at different points along the x-axis is derived from the trigonometric functions shown in Fig. 4. Velocities produced relative to our frame of reference, cannot be added or subtracted linearly to our velocity c as Equation (3) might suggest and as was explained by the examples of the tennis ball and rocket, see pages 35-36. Velocities generated relative to us by ΔE or ∇E must be added or subtracted to our absolute velocity c by vector summation. For energy gained, the velocity relative to x 0 becomes 2 Δv = c 2 − vabs ,

(l/t)

(4a)

(l/t)

(4b)

and in cases where energy is lost 2 ∇v = vabs − c2 .

For example, the velocity of a particle that has gained energy in a particle accelerator by a the amount of ΔE is 2

⎛ E0 ⎞ ⎟⎟ , Δv = c − ⎜⎜ c E + Δ E ⎝ 0 ⎠

(l/t)

2

(5)

which gives the same result as Einstein’s Equation (2). When ΔE is much smaller than E 0 , Equation (5) can be reduced to Δv ≈ c 2

2ΔE , E0

(l/t)

(6)

and replacing E 0 with Einstein's E 0 = m0 c 2 leads to Newton's familiar v≈

2E or E ≈ 12 m0v 2 , m0

( ml 2 / t 2 )

(7)

where m0 is mass of matter at relative rest with respect to x 0 . Although the above Equation (5) produces the same result as Einstein's Equation (2) from his Special Theory of Relativity, it contradicts the concept of relative rest because it postulates that our frame of

40


reference must be moving with an absolute velocity of c relative to a common center, the center of mass of the Universe. The inverse energyvelocity relationship and the vectorial addition of velocities are also demonstrated by the diagrams in Fig. 5a and b, where velocities are shown as a function of distance from the center of mass of a gravitating system. Fig. 5a shows the Universe as whole and Fig. 5b illustrates the relationship between velocity and orbital radius of our Earth and the planet Mars.

Fig. 5a. Diagram showing the relationship between energy, velocity and time along the x-axis of the Universe. b. Diagram showing the relationship between velocity and orbital radius in our planetary system. (Radius and velocity in Earth units).

In the diagram of Fig. 5b it can be seen that Mars, which is at a higher gravitational potential energy level than Earth, has a lower orbital velocity. The difference in orbital velocity between Mars and Earth is Δv . If Earth for some reason should move up to Mars' orbit it would need an extra boost of energy in the form of additional orbital velocity. The amount of additional or relative velocity required to sling 2 2 − vmars (see Equation the Earth out to Mars’ orbit equals Δv = vearth


(4a))

41

and not Δv = vearth − vmars as might be expected. Fig. 5b is based

on Newtonian physics and compared to the collapsing Universe we find that the same vectorial summation of velocities apply. As previously stated, it appears that from our limited view of the Universe, we and all galaxies surrounding us, are hurling toward a distant point, too far to observe. This point is the center of mass of the Galaxies ahead of ours, which have system at x = 0 in Fig. 4. accelerated farther and are falling faster towards x = 0 , seem to speed away from us. By the same token we appear to pull away from galaxies that are behind us and still in an early state of fall. For an observer at x 0 it creates the illusion that the whole observable Universe is pulling apart, or expanding, while in reality it is collapsing. Galaxies at the other side of the center, are they not coming towards us? The answer is that we cannot see that far. Our present technology allows us to see only about one percent of the distance to the center of the Universe which means that only about one millionth of the total volume of the Universe is observable to us x 0 assuming a Universe with spherical shape. It should be noted that from our frame of reference at x 0 the maximum velocity of other astronomical objects is c, the speed of light. This is because we are already traveling at c so when looking back to the maximum amplitude A, where matter is at zero velocity, the observed velocity difference must be c. To reach a point at A or zero velocity would require an infinite amount of energy according to Equation (5). Looking towards the center x = 0 where the absolute velocity of matter is 2c , according to the simple model in Fig. 4, the velocity relative to us again appears as c according to the vector sum ∇v = 2c 2 − c 2 = c .

(l/t)

(8)

Note that Equation (8) refers to velocities of Galaxies ahead of ours, which are closer to the center of mass x = 0 , that have lost potential


42

energy and are falling faster toward the center. Relative velocities that are produced by loss of potential energy are denoted ∇v . Another example of velocities produced by loss of potential energy are velocities of electrons captured in atomic orbits where potential energy also is lost to radiation. Velocities relative to us generated by loss of energy are obtained by 2 ∇v = c 2 − (2c 2 − vabs ) ,

(l/t)

(9)

(l/t)

(10)

or 2

⎛ E − ∇E ⎞ ⎟⎟ , ∇v = c − ⎜⎜ c 0 E ⎝ ⎠ 0 2

where ∇E is potential energy lost relative to our frame of reference at x 0 . Equation (10) differs drastically from Equation (5) and Einstein’s Equation (2) but its validity is clearly demonstrated in Chapter 6, section 6.3, which deals with the subject of atomic orbits.

3.2 Inward acceleration

Both Hubble's law and the Uncertainty Principle specify a change in velocity Δv with a simultaneous change in distance or position Δx , and the only mechanism that can cause such changes is acceleration. Slipher (1917) discovered that light from galaxies are red-shifted and concluded that most galaxies must recede from us with high velocities. For example, the Coma cluster of galaxies, 100 Mps = 3.09 × 10 24 m or 3.26 × 10 8 light years away seems to recede with a velocity of 6.86 × 10 6 m/s. The discovery by Slipher of cosmic red-shifts inspired Hubble (1929) to complete more measurements which led him to formulate a law of recession H = Δv / Δx where Δv is the velocity of recession relative from us at Earth and Δx the distance of separation between us and other galaxies. Hubble argued that if red-shifts of distant galaxies


43

are caused by the Doppler effect, where light waves stretch and get longer or redder as galaxies recede, then the red-shifts would indicate how fast individual galaxies separate from each other. The fact that galactic red-shifts are observed in all directions and increase with distance motivated Hubble and many others to believe that the Universe is expanding in a linear fashion as described by Hubble's law H = Δv / Δx . It is unfortunate that Hubble's law was misinterpreted as an expansion law when in reality it has the physical dimensions of angular frequency. The problem is, that in a scenario where everything is expanding, by change in velocity with distance, the expansion law must include acceleration. Instead of writing Hubble’s law as a change in velocity per distance it could perhaps have been better stated as a change in velocity per light-year or H = Δv / Δt , (acceleration) where Δt is light years. Slipher's and Hubble's discoveries clearly show that galaxies separate with velocities that increase with distance, but since distances between galaxies also change with time their velocities of separation must also change with time. From basic physics we know that any change in velocity with time equals acceleration. The expansion scenario therefore, implies that all galaxies and all matter contained within them, are subject to a cosmic acceleration which will be identified as a 0 (Wåhlin (1981)). The rate of a 0 , which is very small, can be derived from Hubble's parameter and using the above velocity and distance values for the Coma cluster we obtain a value of

Δv ( Δv)2 a0 = = ≈ 7.62 × 10 −12 ms −2 . Δt 2Δx

(l / t 2 )

(11)

It is understandable why Hubble and many other astronomers have failed to recognize the presence of acceleration in their observations, since the star filled sky always seems motionless to an observer, like a still picture, and there is no apparent change in velocity or position over


44

any period of time making it practically impossible to visually detect any acceleration. If our Galaxy, including all matter contained within it, is subject to a cosmic acceleration a 0 , it should be possible in some way to detect a small change in velocity and displacement of particles in our laboratory caused by such an acceleration. In fact, such changes are predicted by the Uncertainty Principle. In the laboratory therefore, we should be able to determine the rate of acceleration very accurately from the Uncertainty Principle which postulates that an electron must change velocity momentum Δp and position Δx according to the following equation ΔpΔx = 12 = ,

(ml 2 / t )

(12)

where = is Planck's constant divided by 2π or = = h /(2π ) . Planck’s constant was discovered by Max Planck (1900) who found that electrons in atoms oscillate and radiate energy at fixed frequencies of ω where the energy per ω equals = . The energy of an electron, which is the smallest quantized form of matter, can therefore be expressed in frequency by way of Planck’s constant or Ee = ω e = . This remarkable property proves that a particle such as the electron is oscillating and has a wavelength. Planck’s wavelength for an electron is λe = 2π=c / Ee which is also called the electron’s Compton wavelength. Since Δp and Δx in Equation (12) are inversely proportional and their product equals a constant (Planck’s constant) we can also write

pmax x min = 12 = , where pmax is the electron’s maximum momentum

(ml 2 / t )

(13)

and x min the

electron’s corresponding minimum displacement. Choosing the maximum or rest mass momentum of an electron pmax = me c (from Einstein's relation E e = me c 2 at our reference point x 0 in space) permits us to obtain the associated displacement x min as determined by = . Since


45

= is in units of angular frequency ω1 = 2π / s or one radian per second we can write x min = ao / ω12 and Equation (13) becomes

me ca0

ω

2 1

Solving for a 0

= 12 = .

(ml 2 / t )

(14)

yields an acceleration of 7.622479 × 10 −12 ms −2 which

is the same value as that obtained from Hubble’s law Equation (11). However, a different approach, which avoids the unconventional term pmax = me c , is to write the Uncertainty Principle as

ΔEe Δt = 2π h ,

(ml 2 / t )

(15)

and at small non-relativistic velocities 1 2

me (Δv ) Δt ≈ 4π 2 12 = . 2

(ml 2 / t )

(16)

Velocity of matter can only change as a function of time and without violating the uncertainty principle we can divide both sides by ( Δt )2 and write 1 2

me (Δv ) 4π 2 12 = ≈ , Δt (Δt )2 2

since 4π 2 /( Δt )2 = ω12

(Δv ) Δt

2

≈

(power)

(ml 2 / t 3 )

(17)

we can substitute and change Equation (17) to

ω 12 = me

.

(l 2 / t 3 )

(18)

When we observe red-shifts from the Coma cluster, 3.26 × 10 8 lightyears away, we are also looking back in time. Inserting a value of Δt = 3.26 × 10 8 light-years (1.028 × 1016 s) in Equation (18) above gives which again agrees with a velocity difference of Δv = 6.86 × 10 6 ms −1 observations and the value used in Equation (11). The above results lead to the conclusion that both Hubble's observations and the Uncertainty Principle are a direct consequence of the cosmic acceleration a 0 . One major problem when using Hubble's law is that Hubble's constant H has not yet been precisely determined (van den

46


Bergh (1981)). Hubble's constant also seems to vary with celestial latitude (de Vaucouleurs (1978)) and it was never clear from observations if Δv increases linearly with Δx (Soneira (1979)) or follows a quadratic relationship as shown here (see Chapter 7 sec. 7.1) and which has been suggested by other investigators (Karachentsev (1967), Ozernoy (1969), de Vaucouleurs (1971) and Segal (1980)). The Coma cluster, for which over 800 red-shift measurements are available, is perhaps one of the best candidates for determining the cosmic acceleration a 0 . Also, the quadratic and linear relationships both predict about the same red-shift for the Coma cluster. A large scale quadratic function might appear linear on a small local scale which is probably why most astronomers came to believe that the expansion is linear. The rate of a 0 turns out to be equal to the classical electrons surface acceleration due to its gravitational field or a0 = Gme / re2 , where re = q 2 μ0 / (4πme ) is the electron's electromagnetic radius and μ 0 the permeability constant. This relationship, which may not be a coincidence, will be discussed in Chapter 8. Once the velocity and rate of acceleration are known, numerous other parameters of the harmonic Universe can be obtained. Most important is the loss in potential energy and inertia of mass of matter as it accelerates towards the cosmic center x = 0 . We know from laboratory experiments that acceleration or deceleration causes matter to radiate energy. The heating of a bullet stopped by a brick wall proves this point. High velocity electrons radiate x-rays when stopped by metal targets (anodes) in x-ray tubes where the slowing or deceleration of high velocity electrons creates electromagnetic radiation referred to as bremsstrahlung (braking radiation). All the above cases show that loss of potential energy results in radiation. In the collapsing Universe matter loses potential energy at the rate of ma0 c / 2π (watts) due to the steady cosmic acceleration a 0 . An electron, therefore, continuously radiates a minute amount of power equal to Le = me a0c / 2π = 12 h (watts) where h = h / s 2 is Planck’s constant expressed in power. The term 2π distributes the power over


one cycle and

1 2

47

h is also the observed amount of power radiated in the

electron’s so called zero point energy state. As mentioned before, it is the cosmic acceleration a0 and subsequent change in potential energy of matter that makes the "world go around". If there was no acceleration present there would be no change in energy and nothing would ever happen. It must be remembered that inertial mass is also a measure of energy (Einstein’s E = mc 2 ) so as matter loses potential energy it also loses inertial mass. When inertial mass, or simply mass, converts into radiation it produces electromagnetic waves which again can revert back to mass through collision processes. When a gun is fired the burning powder in the cartridge loses mass to heat and electromagnetic radiation, which in turn will add mass to the bullet as it is propelled to a higher velocity and a higher energy state. So far the terms acceleration and deceleration have been used quite loosely. Why is it that in a collapsing Universe matter loses energy as it accelerates and gains velocity while here on Earth we seem to observe the opposite that energy increases when matter accelerates and gains velocity? From our view point at Earth, it requires additional energy to accelerate a projectile to a given velocity and this energy becomes lost when the projectile decelerates back to a stand-still again. The discrepancy has to do with the paradoxical behavior of harmonic motion where the question of acceleration and deceleration depends on the frame of reference. An astronaut in an earth orbiting spacecraft has to fire booster rockets and add energy to the space craft in order to accelerate out to a higher orbit. An observer on Earth however, would see the astronaut decelerate since the spacecraft in reality moved from a lower and faster orbit to a higher but slower orbit. The opposite is true when retrorockets are fired in order to slow down the spacecraft for reentry because now the earthbound observer will see the astronaut accelerate and speed up as the spacecraft spirals nearer Earth in ever closer but faster orbits. To summarize, an outside observer finds the spacecraft decrease in velocity as a function of added energy and vice versa, while the opposite seems to occur for the astronaut traveling with


48

the spaceship. To an outside cosmic observer therefore, matter in a collapsing Universe will lose energy as it accelerates and gains velocity and conversely slow down when it gains energy. There appears to be a constant interchange between potential energy and radiation in nature as a result of acceleration and deceleration. We know that inertial mass can convert to radiation and likewise, radiation can change back to inertial mass as matter absorbs radiation. One interesting question is: do we live in a Universe where just as much mass is being created as is being radiated? i.e. is there an equilibrium or perfect balance between energy of matter and energy of radiation in the Universe? In the chapters that follow we will find evidence in favor of a matter-radiation balanced Universe. Another interesting effect, first discovered by Zwicky in 1937, is that galaxies and cluster of galaxies seem to have more rotational speed than can be accounted for by their own visible gravitational mass. This missing mass problem can possible be explained by the tidal force and velocity that the cosmic acceleration a 0 will generate over the large cosmic distances occupied by galaxies. The amount of total rotational velocity with the added tidal effect of a0 can be as much as v=

(GM

gal

/ Rgal ) + (4a0 Rgal )

where Rgal is the radius of a galaxy and M gal its visible mass within Rgal . In simple terms, Hubble’s expansion not only increases the velocity between galaxies it also increases the velocity of stars within the galaxies themselves.

CHAPTER 4 COSMIC DISTANCE AND MASS Our position x0 relative to the center Total mass within x0 Mass density Potential energy of matter Mass and Energy The Ether The bending of light by gravity

Of nature’s four building blocks, mass (m), charge (q) time (t) and length (l), length is perhaps easiest to come to terms with. Length cubed, or length × width × height, is volume or space. Space, however, would be meaningless unless filled with something. Space contains gravitational tension produced by all gravitating matter in the Universe. The cosmic gravitational tension in our galactic neighborhood is φ 0 = GM univ / Runiv = c 2 . The gravitational tension is the conveyor of force fields and

electromagnetic waves. The gravitational tension is the Ether.

4.1 Our position x0 relative to the center We see that early and recent theoretical models of the Universe tend to place us in a central position from which the distance to its outside boundaries determines the size of the Universe. With the discovery of galactic red-shifts and recession velocities it was believed that the size of the Universe could not go beyond a point where the recession velocity

50


reached its limit of c, the speed of light. Therefore, it was thought that if the recession velocity increases linearly with distance at a Hubble rate of H = 75 km/Mpc, for instance, then the maximum distance of recession had to be c/H which equals about 4000 Mpc or 1.23 × 10 26 m . This distance or radius turns out to be too small for our Universe and it gives rise to several difficult problems. One such problem occurs when we try to fit the total mass ( M u = c 3 /(GH ) = 1.67 × 10 53 kg) of the Universe into a volume limited by a 4000 Mpc radius since it generates ten thousand times higher mass density per unit volume than is observed, thus creating an apparent missing mass problem. The −30 −3 observed mass density in the Universe is ρ obs ≈ 2 × 10 kg m . The

calculated mass density from Hubble's constant is

qH2 ρ= 4 ≈ 2 × 10 −26 kg m −3 , 3 πG

( m / l3 )

(19)

where q is a dimensionless so called deceleration parameter which could equal 1 or less. G is the universal gravitational constant. The 2 dimensionless deceleration parameter q = a0 x 0 / c is a term invented by cosmologists who favor the “Big Bang” scenario and who need an answer to the question "is the Universe expanding forever or will matter slow down to zero and start to contract again"? In a Universe that obeys the laws of harmonic motion on the other hand, the deceleration parameter has to equal 1 ( q = 1) and can be discarded. The discrepancy in mass density from Equation (19) as compared to observed values is caused by Hubble's constant which holds that the recession velocity is linear with cosmic distance rather than numerically proportional to the square root of distance. A discrepancy in mass calculations of individual galaxies or cluster of galaxies can also appear if Hubble's linear constant is used as a distance indicator. As was mentioned before, Hubble's constant is not really a constant because recession velocities relative to us are not proportional to distance. The problem is that Hubble's relation H = v/x, has the dimensions of angular frequency and is really a measure of the apparent relative change in angular frequency between us and other astronomical objects

COSMIC DISTANCE AND MASS

51

in the Universe just as the planets in our solar system have different periods or angular frequencies around the Sun. The true meaning of Hubble’s constant is therefore angular frequency and by dividing our absolute velocity c with the with the cosmic radius x 0 we obtain the angular frequency of the Universe, H = c/ x 0 = 2π / t0 rad/s, where t0 is

the period of the Universe. A precise way in which to determine the size of the Universe is to calculate the distance between our position x 0 and the center of mass of the Universe at x = 0 using standard equation of harmonic motion. Having established the rate of acceleration a 0 from Planck's constant (Chapter 3, section 3.2) and our velocity in relation to the center of mass of the Universe to equal c, will allow us to exactly determine the distance to the center of mass of the Universe from c2 x0 = = 1.17908 × 10 28 m . a0

(l)

(20)

This is about one hundred times larger than the distance previously obtained by Hubble’s law and it is also about hundred times farther than can be probed by our largest telescopes, which implies that only a small fraction of the Universe is visible to us. It must be emphasized that the parameters c, a 0 and x 0 are derived as they appear to us from our limited field of observation. For this reason, it is impossible to determine the maximum radius or amplitude A of the Universe, but we may assign it a hypothetical distance of A = 1.667 × 1028 m , by assuming that our position x 0 is at an angular displacement of α = 45 0 (see Fig. 4c Chapter 2, section 2.2). The value obtained for x 0 in Equation (20) is in fact in agreement with estimates performed by Narlikar and Burbidge (1981). The basis for Narlikar's and Burbidge's estimates is Equation (19). Instead of thinking of mass density in its usual form consisting of gravitational matter such as stars and galaxies etc. Narlikar and Burbidge argued that the radiation temperature T of the ever prevailing 2.7 0 Kelvin background radiation, discovered by Penzias and Wilson (1965), must have an “equivalent mass” density from which the size of the Universe can be determined.


52

Knowing the energy per unit volume of the T = 2.70 K radiation from U r = 6π T 4σ / c and converting this to equivalent mass per unit volume ρ r = U r / c 2 one can use Equation (19) to solve for the absolute value of H. Inserting the value of ρ r in Equation (19) yields H = c / x = 1.5 × 10 −20 rad/s. Solving for x allowed Narlikar et. al. to obtain a distance of x = c / H = 1.9 × 10 28 m which is very close to the result of Equation (20). Since the distance derived by Narlikar et al. is also about 100 times larger than popular belief they suggested that our Universe could be a small island Universe together with several others of the same size making up a super Universe from which the 2.7 0 K would have originated. The question is however, can one assume that the equivalent mass of the 2.7 0 K radiation is gravitational in nature so that Equation (19) would apply? Perhaps it would have been better to estimate the size of the Universe from its average brightness expressed in brightness per unit volume ( A ) and then determine how large the Universe has to be in order to produce a blackbody temperature of T = 2.7 0 K. This is accomplished simply by writing an equation which contains the product of ( A ) and the spherical volume of the Universe divided by its surface area

A 43 π x 03 T = , 4π x 02σ 4

(none)

(21)

and then solve for x 0

3T 4σ x0 = = 1.2 × 10 28 m, A

(l)

(22)

where σ = 5.670 × 10 −8 is Stephan-Boltzmann's constant. The − 34 −3 A = ρ obs ( LSun / M Sun ) = 8 × 10 w m is brightness per unit volume obtained by multiplying the observed mass density of the Universe with the luminosity to mass ratio of the Sun, i.e. if the average star in a galaxy has the same luminosity and mass as our Sun then the luminosity to mass ratio of matter in the Universe must be the same as the ratio Lsun / M sun .


53

The above method, which is similar to that used to estimate the size of a star from its brightness and temperature, yields the same radius as Equation (20) and very close to the size predicted by Narlikar and Burbidge. It appears obvious that if the observed mass density and the observed radiation density, have equal energy density and occupy the same volume within the Universe, the Universe is neither matter or radiation dominated, but in a state of equilibrium. The fact that Narlikar and Burbidge used the equivalent mass density of the 2.7 0 K radiation to determine Hubble's constant is significant because it suggests that the energy density of radiation equals the energy density of matter in the Universe and therefore eliminates the missing mass problem. Secondly, Narlikar's and Burbidge's results show that on a large scale Hubble’s constant is not a constant but varies over large cosmic distances. This is consistent with the belief of several investigators including the author, that Hubble’s velocity to distance relation is not linear, but follows a quadratic relationship (see sections 3.2 and 7.1).

4.2 Total mass of the Universe within x0

Since the inward force and acceleration are gravitational in nature we can easily determine the total mass of the Universe within our radius x 0 from a0 x 02 Mu = = 1.59486 × 10 55 kg . G

(m)

(23)

The virial theorem GM u = x 0 c 2 is obeyed (see Chapter 8), which means that standard laws of gravitation apply, the same laws that govern orbital motion of planets and satellites in our solar system. The number of stars in the Universe, within the radius x 0 , is about 8 × 10 24

assuming the average star having the same mass as our Sun. It is believed that the average galaxy contains about 1.4 × 1011 stars which leaves us with a total of 6 × 1013 galaxies within x 0 . The


54

number of galaxies and stars outside x 0 is difficult to estimate which means that the total mass of the Universe is unknown and cannot be obtained by the cosmological model presented here. Only the mass within x 0 can be determined.

4.3 Mass density

The mean mass density of the Universe within x 0 by the volume within x 0 or

ρ=

a0 −30 −3 = 2 . 32273 × 10 kg m . 4 3 πGx 0

is M u divided

( m / l3 )

(24)

This is very close to the observed mass density in our part of the Universe (Sandage 1995). Since we can only see about one millionth of the Universe it is not possible to tell whether the mass density is the same everywhere or if it varies with the distance from the center of the Universe. The above mass density corresponds to about 2.5 electrons or positrons per cubic meter.

4.4 Potential energy of matter

Potential energy or absolute energy is, from a cosmological point of view, the total energy that is potentially available and stored in the mass of matter. Potential energy of matter varies along the x-axis (radius) of the Universe and its magnitude at our position x 0 is E 0 = φ 0m0 = m0 c 2 ,

(ml 2 / t 2 )

(25)

where φ 0 = GM univ / x 0 is the cosmic gravitational tension and E 0 and m0 the energy and inertial mass of matter at relative rest in our frame of reference at x 0 .


55

From the laws of harmonic motion, which involves mathematical functions such as sine and cosine (see Fig. 4. Chapter 2), we may find the potential energy of matter along the x-axis by the use of trigonometric functions as follows; for values between 0 − 45 0 E=

E0 2

sin α

,

(ml 2 / t 2 )

(26)

(ml 2 / t 2 )

(27)

and from 45 0 − 90 0

E = 2 E0 cosα ,

where E 0 is the potential energy or absolute energy of matter at x 0 .

4.5 Mass and Energy

We have so far only dealt with energy and velocity but not space. Before assigning certain properties to space it must first be clarified what is meant by energy and field. Energy of matter is confined to mass and comes essentially in three forms: Gm02 1. Self energy of matter Es = joules r 2. Potential or absolute energy of matter E 0 = 3. Radiation L = ∇m c 2ν watts,

L=

(ml 2 / t 2 ) GM um0 joules Ru

2(∇m)2 c 4 w/photon h

(ml 2 / t 2 )

(ml 2 / t 3 )

where h is Planck’s constant, ∇m the inertial mass lost and converted to radiation and ν is the frequency of the radiation. Self energy of matter is the energy stored in a body’s own gravitational field and potential energy is the energy of a mass stored in an external gravitational field. The self energy of the Earth's own gravitational

56


2 field equals ( E s = GM earth / Rearth ) and is about 14 times less than the

Earth’s potential energy stored in the Sun's gravitational field ( E p = GM sun M earth / Rsun−earth ) or 1.5 × 10 9 times less than its absolute potential energy E0 stored in the gravitational field generated by the entire Universe ( E 0 = GM univ M earth / Runiv = M earth c 2 ). Any change in potential energy of a mass involves radiation. Loss of potential energy produces radiation while an increase in potential energy occurs when matter absorbs radiation. Mass is also related to charge squared since mass can be converted to charge and vice versa by the relation m = q 2 μ 0 /( 4π r ) , where r is the radius of the body and μ 0 the permeability constant. Energy of mass or charge is not confined within the mass or charge itself, but resides in its external field. What is field? Field is a term often used casually and should really be divided into two categories namely, energy fields and force fields For example, is the Sun's gravitational field at the Earth's surface greater than that of the Earth’s own gravitational field? The answer is both yes and no, because if one refers to the energy field or energy per unit mass, which is gravitational tension ( φSun = GM Sun / RSun−Earth ), then the Sun dominates the Earth’s own field ( φ Earth = GM Earth / REarth ) at the Earth’s surface by a factor of 14 (see Fig. 6). If we refer to the force field on the other hand, or force per unit mass which is acceleration ( a = GM / R 2 ), then the Earth’s own field at the Earth’s surface is greater than that of the Sun by a factor of 1600. The force-field is in fact created by the gradient of tension. Gravitational tension is equal to gravitational potential and acceleration is the same as potential gradient. Gravitational potential should not be confused with gravitational potential energy. Gravitational potential or tension, from a cosmological point of view, decreases with distance from the gravitating source and reaches zero at infinity. For earthbound observers, it has often been customary to assign gravitational potential a negative sign. This stems from the observation that it requires energy to move a body away from a gravitating source such as the Earth. Problems occur however, if we make gravitational potential and


57

gravitational energy mathematically negative (anti-gravity). A student may try his or her calculator to determine the orbital velocity of an earthbound satellite from v = − φ Earth . To avoid confusion, gravitational potential will throughout this book, be referred to as gravitational tension and together with gravitational energy will carry a positive sign.

Fig. 6. The gravitational tension of the Sun, Earth and Jupiter superimposed on the gravitational tension φ univ = c 2 of the Universe.

4.6 The Ether

The Sun's gravitational tension is 14 times larger at the Earth's surface than that of the Earth's own gravitational tension. However, when we realize that the gravitational tension generated by all matter in the Universe ( φUniv = GM Univ / RUniv = c 2 ) is 1.5 billion times more one might wonder what impact, such a huge gravitational tension, will have on us here on Earth. In fact, the immense cosmic gravitational tension, in which the Earth and the entire solar system are immersed, serves as a medium, or ether, for the propagation of photons or ripple of electromagnetic waves. It is the magnitude of the cosmic gravitational tension that determines inertia of mass and, therefore, the rate of time


58

(see Chapter 2, section 2.4) and the speed at which electromagnetic waves propagate through space. The cosmic gravitational tension is also the conveyor of gravitational force fields, since gravitational force is simply the gradient of gravitational tension. Fig. 6 shows the magnitude of the cosmic gravitational tension as compared to that generated by the Sun, Earth and Jupiter. The gravitational tension contributed by the Sun and the planets are in reality very small bumps on top of the cosmic tension. Summarized in Table 2 are the major components of gravitational and electric fields such as self energy, potential energy, energy between two bodies, tension, force and acceleration. FIELD COMPONENT Self energy

Gm 2 r

E

Potential energy E 0

Energy between two bodies

Tension φ

Force between two bodies F

Acceleration

GRAVITATIONAL

a

Table 2.

ELECTRIC

q2

4πε 0 r

GM Univm0 = m0 c 2 RUniv

GM Univ q 2 μ 0 RUniv 4π r

Gm1m2 r

q1q2 4π ε 0 r

Gm r

q

4πε 0 r

Gm1m2 r2

q1q2 4πε 0 r 2

Gm r2 Different components of gravitational and electric fields.

The velocity of electromagnetic waves through the cosmic gravitational tension (ether) is inversely proportional to the magnitude of the gravitational tension just as absolute velocity is inversely proportional to absolute energy (Chapter 3). The additional increase in


59

the cosmic gravitational tension near individual gravitating bodies, such as the Sun for example, will therefore, have the effect of slowing the free space value of c, the speed of light, to an amount determined by 2

⎛ φuniv ⎞ c ⎟⎟ ≅ , v = c⎜⎜ 2 Gm φ φ + ⎝ univ sun ⎠ 1+ rc 2

(l/t)

(28)

where φ sun is the gravitational tension of the Sun at the radial distance r from the center of the Sun and m the mass within r. The slowdown in velocity causes the wavelength of photons entering the gravitational field to shrink and become blue-shifted by 2

zblue

⎛ φuniv ⎞ Δv Δλ 2Gm ⎟⎟ ≅ = = = 1 − ⎜⎜ . (none) λ c rc 2 ⎝ φuniv + φsun ⎠

(29)

By the same token photons or rays of light leaving a gravitational field become red-shifted by the same amount, which is an effect that has been observed for light waves emanating from the Sun’s surface. The change in velocity of light, entering and leaving a gravitational field, can cause rays of light to bend. The bending of light in gravitational fields is of historical interest since its discovery and prediction by several scientists including Einstein, has had some very important historical implications.

4.7 The bending of light by gravity

The experimental proof that light bends near a gravitational body and the attempts to explain this phenomenon is very interesting, because it led to the impetuous acceptance of Einstein’s relativity theories which to some extent has guided us down the wrong path. Newton himself queried the possibility of light being bent by gravity but it was not before the beginning of the 19th century that a German


60

astronomer, Johann Georg von Soldner (1804), presented calculations based on Newton’s corpuscular theory that light weighs and bends like high speed projectiles in a gravitational field, which produced a value of 0.87 arc second bending angle for light grazing the Sun. Fifteen years earlier Sir Henry Cavendish made the same calculations, but his results were never published (Will, (1993)). Over hundred years later, in the year of 1911, Einstein published his first paper on the bending of light (Einstein, (1911)) based on the equivalence principle which holds that anything, regardless of mass, accelerates equally in a gravitational field. Einstein obtained a deflection angle of

α=

2GmSun (radians) = 0.87 arc second , c 2 rSun

(none)

(30)

which is the Newtonian value and the same angle previously published by Soldner. It appears that Einstein and the scientific community at the time were unaware of von Soldner’s paper and it was not until the year 1921 when Soldner’s work was rediscovered. About four years after his first paper in 1911, Einstein had developed the General Theory of Relativity (1915) which prompted him to modify the above Newtonian value by adding an effect of curved space thus increasing the bending angle of Equation (30) by a factor of two or

α=

4GmSun = (radians) , c 2 rSun

(none)

(31)

resulting in a new value of 1.75 arc seconds (1.7505395 arc second). The new result was published by Einstein on November 18, (1915) and was experimentally verified by Crommelin and Eddington during the solar eclipse expeditions of May 29, 1919. When the sunlight was blocked by the Moon, starlight grazing the Sun’s surface became visible and the bending angle of the starlight was revealed. While the expeditions were being planned, Eddington wrote: “The present eclipse expeditions may for the first time demonstrate the weight of light (i.e.


61

Newton’s value) and they may also confirm the added effect of Einstein’s weird theory of noon-Euclidean space, or they may lead to a result of yet more far-reaching consequences of no-deflection” (Eddington, (1919)). The verification of Einstein’s additional bending angle of 0.87” due to curved space led to instant fame and blind acceptance of his relativity theories. Einstein’s relativity theories, although they produce the right numbers, have often been criticized for being weird since they are conceptually difficult and often impossible to understand. One serious problem with Einstein’s bending of light theory is the assumption that light will accelerate and fall closer to a gravitating body according to the equivalence principle, which would account for the first 0.87 arc second of bend, while the rest, or 0.87 arc second, is due to the obscure warping of space. The main objection is that since light is mass less it does not accelerate but, on the contrary, it decelerates or slows down in a gravitational field in accordance with observations. However, the introduction of warped or curved space is what formed the basis for Einstein’s General relativity theory which can be classified as mathematical model based on geometry. The bending of light in gravitational fields is better explained by Snell’s law of refraction (see Fig. 7) which is a law that was experimentally established by Willebrod Snell and theoretically by René Descartes over three hundred years ago. Snell’s law is based on the discovery that when light enters a medium which retards its velocity of propagation, such as a piece of glass or a gravitational field, it will bend at an angle determined by the combination of its change in velocity and angle of incidence. The advantage of using Snell’s law is that it eliminates both the idea that light weighs and the notion of curved space. Snell’s law (or in France, René Descartes’ law) can be written as n=

c sin α ' = v sin α ' '

and

Δv Δ(sin α ) = , c sin α '

(none)

(32)


62

where n is the index of refraction, α ' the angle of incidence, α ' ' the angle of refraction and c the incoming speed of light and v the retarded speed of light in the refracting medium. Light is a wave and will bend when it enters the refracting medium at an angle and if the velocity of propagation from one medium to the other varies. The diagram in Fig. 7 shows a train of waves entering a glass prism at an angle and how each wave-front breaks at the surface and changes direction due the change in velocity and consequent shrinkage in wavelength caused by a traffic jam effect in the slower refracting medium. Observe that the wave length λ ' ' of the beam inside the prism is shorter than λ '

Fig. 7. A beam of light entering and exiting a glass prism showing the wavelength in the glass being shorter because of the slower velocity of propagation.

outside the prism. It is in fact the change in wavelength that causes the light beam to bend. Snell’s law can therefore be rewritten as n=

λ ' sin α ' = and λ ' ' sin α ' '

Δλ Δ(sin α ) = , λ' sin α '

(none)

(33)

Note that the beam of light bends twice, once at the entrance and once at the exit of the diffracting medium. Fig. 8 shows a beam of light entering the Sun’s gravitational equipotentials and their associated


63

angles of incidence. The angles of incidence range from 0°−90° producing a mean incident angle of α ' = 45° . The change in wavelength Δλ , divided by the original wavelength λ ' of light grazing the Sun’s surface, from Equation (29), is 2

⎛ φuniv ⎞ Δλ ⎟⎟ . = 1 − ⎜⎜ + λ' φ φ ⎝ univ sun ⎠

(none)

(34)

Fig. 8. A beam of light entering the gravitational field of the Sun showing the Sun’s different gravitational equipotentials and corresponding angles of incidence.

The optical bending of a light beam grazing the Sun’s surface therefore according to Snell’s law α bend = 2(α '−α ' ' ) or

is

α bend = 2[α '−sin −1 (sin α '− sin α ' ( Δλ / λ ' ))] = 1.75053023 arc second , (35)

64


where α ' = 45D is the mean angle of incidence. The factor of “2” is necessary since light has to pass through two refractive indices, one at the entrance and one at the exit of the gravitational field, produced by the Sun. The solar gravitational deflection of electromagnetic waves has been accurately measured during the last decade for both light and radio waves. One of the latest measurements, which was reported by Lebach et al. (1995), and which claims a precision of 0.1% agrees with Equation (35). In fact, Equation 35 yields about 1 × 10 −5 arc second less bending angle for the Sun than Einstein’s Equation (31). The velocity of light is constant in a gravitational equilibrium where the gravitational tension is constant and where no potential gradient or acceleration exists. For example, electromagnetic waves produced at Earth travel with a constant velocity relative to Earth regardless of our orbital velocity around the Sun, because the Sun's acceleration at Earth is canceled by the centrifugal acceleration produced by the Earth's circular orbit, i.e. the Earth will experience a uniform gravitational tension along its orbit around the Sun thus ensuring a uniform gravitational tension in its path which might be thought of as ether dragging. The Earth's own change in gravitational tension with altitude (acceleration), however, will cause a small change in the velocity of propagation with altitude. Light from extra terrestrial objects such as stars and galaxies that intersect the Earth in its orbit, will be subject to aberration. Light from these objects will be tilted just as rain will appear more and more tilted on a windshield screen when traveling at an increasing velocity. The bending due to aberration can be calculated using Snell’s law where n = c / vearth . Equation (28) shows that the velocity of electromagnetic waves cannot slow to zero no matter how strong the gravitational tension is, thus most probably ruling out the existence of so called black holes.

CHAPTER 5 RADIATION AND TEMPERATURE Period (Hubble’s time) and frequency of the Universe Angular frequency (Hubble’s parameter) Force constant Radiation Temperature The origin of Planck’s constant h

In his book, The Harmony of the Worlds, Kepler presents his third law, known as the harmonic law, which describes how our planets are bound to the Sun in organized orbits where the square of the period of an orbit is directly proportional to the cube of the orbital radius. The discovery of the harmonic law, which was made before Newton’s laws of motion, led Kepler to believe that practically everything in our Universe had to be governed by harmonics and he even wrote music scores representing the harmonic motions of the planets. Although Kepler has been criticized throughout time for his extraordinary beliefs in harmonics he was not far off the mark. Half the energy in the Universe, which is confined to radiation, is nothing else than spectra of frequencies and waves in perfect harmonic relationships. The physical laws that rule atoms, planets, galaxies and clusters of galaxies are harmonic laws of motion and there is nothing wrong in assuming that the whole Universe is a large harmonic oscillator.

5.1 Period (Hubble’s time) and frequency of the Universe The period or time it takes for the Universe to complete one cycle is simply

t0 =

2π x 0 = 2.47118 × 10 20 s . c

(t)

(36a)


66

which equals a frequency of

ν0 =

1 = 4.046667 × 10 − 21 Hz . t0

( t −1 )

(36b)

Using the simple harmonic oscillator model in Fig. 4, Chapter 2 and assuming that our position is at an angular displacement of 45 Ο will leave us with about 6 × 1019 s before reaching the central point x = 0 , where all matter, now at x 0 , will have been dissipated into radiation. The rate at which matter radiates energy is therefore mc 2 / t0 . In the following text t0 is defined as Hubble's time and is the period of our oscillating Universe as seen from our reference point x 0 .

5.2 Angular frequency (Hubble's parameter)

Since harmonic motions are cyclic in nature we can divide each cycle into a circular 360 D rotation and state that one cycle per second is the same as an angular frequency or angular velocity of 360° per second. In practice, angular frequency or velocity is usually expressed in radians per second where rad/s = 360 Ο / 2π s = 57.3Ο per second , see Fig.

9.

Fig. 9. The diagram illustrates a 360Ο per second rotation as compared to 1 radian or 57.3° per second rotation.

67

RADIATION AND TEMPERATURE

The angular frequency or angular velocity of our Universe, assuming a simple harmonic motion, is therefore,

ω0 =

a0 = 2.54258 × 10 −20 rad s −1 . x0

( t −1 )

(37)

(t −1 )

(38)

The angular frequency or velocity is also given by

ω 0=

c = 2.54258 × 10 −20 rad s −1 , x0

for a circular or spiraling motion around its center. Birch (1982), who studied polarization of distant radio sources, discovered a large scale harmonic motion of the Universe at an angular frequency of the order of ω 0 ≈ 10 −20 rad s −1 . This is very close to the above calculated value. If the Universe follows a purely simple harmonic motion, where all matter is falling in straight lines toward the center of mass of the system, then the angular frequency will stay constant at any distance x from the center. If matter spirals in towards the center one can expect the angular frequency to change with x just as electrons and planetary orbits change angular frequency as a function of their orbital radii. We are, unfortunately, at the present time only able to see a very small portion of our Universe which makes it difficult to decide what kind of harmonic motion we are part of. Although the feeling of the author is that we may be part of a large spiraling meta galaxy, we can still use equations of simple harmonic motions to unveil many unknown properties of the Universe.

5.3 Force constant

When dealing with a simple harmonic motion it is often practical to use the mathematical term k = ma/x where k is called the force constant.


68

In the oscillating Universe the force constant is k=

M u a0 = 1.031040 × 1016 kg s −2 /(4π 2 ) . x0

(m / t2 )

(39)

The force on matter in the Universe can also be written as F = kx and is directed toward the equilibrium point x = 0 . The force constant can also be expressed as k = M uω 02 , and has the dimensions of angular energy. In the large scale Universe, gravity is responsible for the force constant k.

5.4 Radiation

It is quite obvious that if matter loses all its potential energy to radiation over one period of oscillation, such as in a critically damped collapsing Universe, the rate of energy radiated by matter should be equal to the potential energy of matter E 0 divided by the period t0 of the cosmic oscillation. As a result, the rate at which matter radiates energy, as seen from our reference point x 0 in space, must equal L=

E0 = ν 0 E 0 (watts), t0

(ml 2 / t 3 )

(40)

where L is the total luminosity or flux of radiation produced by matter in the Universe. From observations within the visible region of our Universe one can see that the ratio of mass to luminosity remains fairly constant over many orders of magnitude. Dividing the mass of the Sun by its flux of radiation produces about the same mass to luminosity ratio M/L as when we divide the mass of a galaxy with its flux of radiation or luminosity. The same ratio appears when we divide the total observable mass of the Universe with its total flux of radiation. Matter in the Universe will therefore, radiate energy as a result of the inward acceleration, just as atomic electrons radiate when falling closer to the nucleus. Also, from the theory of electromagnetism it has been established that matter radiates energy while accelerating

69


and in a critically damped oscillator, such as the collapsing Universe, the energy radiated due to the cosmic acceleration a 0 is L=

ma0 c = E 0ν 0 (watts), 2π

(ml 2 / t 3 )

(41)

which is identical to Equation (40). The luminosity of the whole Universe within our radius x 0 is therefore

M u c 2 M u a0 c = = 5.80044 × 10 51 (watts). (ml 2 / t 3 ) (42) Lu = 2π t0 The diagram in Fig. 10 shows power radiated as a function of mass

Fig. 10. Luminosity L as a function of mass M in the Universe.


70

for various matter in the Universe. Most of the data were obtained from Allen (1973) and Huchra (1977) and the solid line represents calculated values using Equations (41) and (42) ranging from the smallest quantum of matter, the electron, to the entire Universe. It is interesting to note that an electron, according to the diagram in Fig. 10, will radiate Le =

me a0c = E0ν 0 = 12 h 2π

(watts),

(ml 2 / t 3 )

(43)

where h is Planck’s constant expressed in power or h = h/s 2 , E 0 is the electron’s rest mass energy and ν 0 the fundamental frequency of the Universe. The equation implies that an electron, even at relative rest, has a zero-point radiation state and a specific temperature which will be discussed in the next section. Since matter is quantized and the electron being the smallest quanta of matter, it means that radiation has to be quantized as well. Equation (43) proves this fact because all symbols in the equation are constants including Le . Equation (43) can also be written as Le

α1

= 12 = ,

(ml 2 / t )

where α1 = 1 rad/s 2 , is unit angular acceleration, and

(44)

= = h /( 2π ) is

Planck’s constant defined as power per unit angular acceleration (the origin of Planck’s constant is discussed in section 5.6 at the end of this chapter). The fact that radiation is quantized and only appears in small power pulses rather than a continuous flow of energy allows us to write the radiation formula as follows: M L = 12 h (watts). me

(ml 2 / t 3 )

(45)

For example, the Sun must radiate Ls = 12 hM s / me = 7.23 × 10 26 watts of which L = ( Ls e ) − p= 3.826 × 10 26 watts escapes unrestricted as pure radiation and where e is the emissivity of the Sun’s surface and p the


71

amount of radiation converted kinetic energy that propels the solar wind. The emissivity e is the ratio of a body’s specific radiation leaving its surface as compared to the specific radiation produced inside the body and varies for different surface materials, but can never be greater than one. The emissivity e for the Sun is not well known. There is also a considerable amount of radiant power lost in the collision of solar photons with matter particles at and near the solar surface. This gives rise to the exterior solar wind which consists of high velocity particles ranging from electrons and ionized hydrogen to some of the heavier elements. The Sun’s radiation equals a mass loss of about L / c 2 = 4.25 × 10 9 kg/s which does not include the mass swept away by the solar wind. The amount of mass removed by the solar wind is comparable to the amount of solar mass lost to radiation. If radiation is generated by the acceleration a0 due to the collapse of our Universe, what part do nuclear transformations play in the heating of stars? Nuclear transformations are most probably the result of the extreme heat in stars rather than the cause of it, and judging from the Sun's neutrino flux, less than one-third of the solar energy is involved in nuclear reactions. Lanzerotti et al. (1981) point out that nuclear mechanisms in the Sun are not clearly related to the solar power output since they found no correlation between solar activity and solar neutrino flux. It should be mentioned that the existence of the elusive neutrino might not yet be an established fact. See Bagge, (1985). Estimated M/L ratios for stars in our own galaxy, based on double stars, do not agree with the radiation mechanism presented here since they do not fall along the straight line in Fig. 10, but are believed to follow the relation M 3 / L . This discrepancy can perhaps be explained by the fact that most observed double stars are in a high state of collapse or acceleration towards their own common centers of mass and therefore lose more energy to radiation than would be expected if they were only subject to the cosmic acceleration a 0 . It must also be remembered that estimates of M/L ratios based on double stars are not at all conclusive, since they make up only a few percent of the total star


72

population in our galaxy, and does not necessarily represent the true M/L ratio for all the rest of the stars in our galaxy.

5.5 Temperature

Temperature is interesting because it has no physical dimensions, yet its effect can be felt and measured. Temperature actually relates to the intensity of radiation or energy emitted per second per square meter, which is the same as power per unit surface area. For example, the Earth, which appears as a disk to the Sun with a radius of REarth , 2 blocks πREarth of the Sun’s radiation and, at our distance from the Sun, receives about L / A = 1371 watts per square meter, where A is unit 2 surface area. Since the Earth’s spherical surface area is 4πREarth or four 2 times larger than the above πREarth area of received radiation the Earth will, as it rotates on its axis, in reality collect an average flux of four times less or 343 watts per square meter which, according to Stefan’s law, corresponds to an average temperature of T = ( L / σ ) = 279° 1 4

Kelvin or 5.7° Celsius, where σ = 5.670 × 10 −8 (Stefan-Boltzmann's constant). We know that the Earth is in a temperature equilibrium and radiates as much energy as it receives, namely 343 watts per square meter, which means that the global average temperature generated by the Sun’s radiation is 5.7° Celsius in spite of any greenhouse effect. From the total radiant power emitted by the Universe (Equation (42)) one can calculate its temperature from Stefan's law ⎛ Lu Tu = ⎜ ⎜ 4π x 2 σ 0 ⎝

1/4

⎞ ⎟ ⎟ ⎠

= 2.766° K ,

(none)

(46)

(none)

(47)

or 1/4

⎛ a0 c 2 ⎞ ⎟ Tu = ⎜ ⎜ 4π G σ t ⎟ 0 ⎠ ⎝

= 2.766° K ,

73


Equations (46) and (47) suggests that the 2.766° K blackbody temperature is the product of scattered or thermolized radiation from discrete sources such as galaxies and stars etc. and from the dipole anisotropy of the observed radiation we can determine the direction of our motion in the Universe, which appears to be towards 10.4hR.A. and -18 dec. on the celestial sphere (Smoot et al (1977)). The dipole anisotropy is caused by the movement of our Galaxy relative to the 2.766° K background radiation so that in the forward direction Doppler shifts make the background radiation appear slightly hotter than in the direction we come from. From these minute Doppler shifts we obtain an apparent drift velocity of about 500 km/s relative to a point from which a 2.766° K photon last scattered. The dipole anisotropy might be partly caused by a 0 , the amount of acceleration of our galaxy towards the center of the Universe. One can, therefore, calculate the distance to the photons last point of scatter, which equals the photons mean-free path in inter-galactic space, from v2 l = ≈ 1.64 × 10 22 m , 2a0

(l)

(48)

where l is the mean-free path and v is our velocity relative to the point from where the photon was last scattered. One very interesting observation is that the black-body temperature of the Universe is equal to the black-body temperature of an electron using Equation (46) or ⎛ Lu Te = ⎜ ⎜ 4π x 2 σ 0 ⎝

1/4

⎞ ⎟ ⎟ ⎠

⎛ ma c = ⎜ e2 02 ⎜ 8π r σ e ⎝

Are we allowed to speculate? Universe?

1/4

⎞ ⎟ ⎟ ⎠

= 2.766° K . (none)

(49)

Could an electron be just another

There are several cosmological theories based on thermodynamics which involve the interaction between matter and radiation in the Universe. Most noteworthy is perhaps the oscillating cosmological

74


model offered by P.T. Landsberg et al. (1992). The model describes a Universe that goes through a numerous amount of expansions and contractions in which heat and matter exchange place.

5.6 The Origin of Planck’s Constant

When Max Planck discovered one of nature’s most mysterious constants h, quantum physics was born. The problem was, and still is, that Planck’s constant has the physical dimensions of

h = E / ν , (energy per unit frequency)

(ml 2 / t )

(50)

which is difficult to comprehend since it indicates that energy comes in the form of frequency which at the time could not be explained by classical physics. Max Planck himself nearly abandoned his theory after years of frustration trying to solve this mystery. Modern researchers have no problem with Planck’s constant and do not question its origin since they simply believe it is a constant of nature and therefore needs no explanation. This attitude caused a split between classical physics, which demands a conceptual explanation to all physical phenomena, and modern quantum physics which is satisfied as long as the mathematical equations work out. It does not seem right, however, that there should be more than one kind of physics and a conceptual explanation of Planck’s constant along the guidelines of classical physics would certainly bridge the gap between classical and modern thought. The harmonic Universe does in fact offer a reasonable answer to the question as why energy comes in steps of frequency. The harmonic motion of the Universe, provides the fundamental frequency from which all other frequencies are harmonic overtones. In other words: since the whole Universe oscillates at a fundamental frequency of ν 0 , then all matter contained within it will oscillate at the same frequency, or at any harmonic of ν 0 just as overtones on a violin string are multiples of the string’s own fundamental frequency, see Fig.

75


11. The fundamental frequency of matter in the Universe Equation (36b) is ν 0 = 4.04665 × 10 −21 Hz .

from

Fig. 11. A vibrating violin string showing the fundamental frequency and the first and second overtones

Only harmonics such as 2ν 0 , 3ν 0 , 4ν 0

etc. of the fundamental

frequency can exist which means that an electron having a frequency of 1 Hz (which according to Planck’s discovery corresponds to an energy of E = 1Hz × h = 6.6 × 10 −34 Joules) will oscillate at approximately the 247 × 1018 th harmonic of the fundamental frequency ν 0 . To trace the origin of Planck’s constant h we need to start from Equations (41) and (43) which show that matter is subject to a constant change in energy (power) while partaking in the fundamental frequency of the oscillating Universe. An electron will therefore,


76

radiate a fixed amount of power (Equation 43) generated by the fundamental frequency ν 0 which is simply

Le = E0ν 0 = 12 h = 3.313 × 10 −34 w,

(ml 2 / t 3 )

(51)

where E 0 is the electron’s rest mass energy. Energy or power cannot change instantaneously but must change as a function of time. To change the frequency of an electron from its fundamental frequency ν 0 to its first harmonic 2ν 0 will therefore involve power Le and time Δt Le ΔE = = E0 2ν 02 = 2.681334 × 10 −54 w s −1 . 2 Δt ( Δt )

(ml 2 / t 4 )

(52)

Since a change in energy ΔE is directly proportional to a change in frequency (see Equation (50) we can write ΔE Δν , ∝ 2 ( Δt ) ( Δt )2

(53)

where Δν = ν 0 represent the change in frequency required to step up to the next harmonic. Since the two terms above are proportional to each other then dividing one by the other will equal a constant or ΔE /( Δt )2 = 6.626075 × 10 −34 w s −1 /Hz s −2 = h, 2 Δν /( Δt )

(ml

2

/ t ) (54)

where h is Planck’s constant. The above reduces to h=

ΔE / Δt Power per angular acceleration , Δν / Δt

(ml / t ) 2

(55)

which further reduces to h = E / ν Energy per Hertz .

(ml

2

/ t ) (56)

To sum up, the reason why energy appears in steps of the fundamental frequency ν 0 is explained by the fact that the whole Universe is oscillating and all matter within the Universe will share its fundamental frequency, which causes energy of matter to change in


77

steps unison with changes in the fundamental frequency ν 0 . When Max Planck discovered the relationship described by Equation (56), which is the reduced form of Equations (54) and (55), he could not possibly know its meaning since it can be seen that most information of its origin is lost in the reduction process. Max Planck was not able to determine the smallest quantum of energy because he had no idea what the lowest possible frequency (fundamental frequency) was. The smallest quantum of energy is E r = ν 0 h = 2.6813 × 10 −54 J. The smallest amount of energy in the Universe is the electron’s gravitational self energy E g = Gme2 / re (see Chapter 4, section 4.6). If we divide E g by E r we obtain another one of natures mysterious ratios

α=

Eg Er

= 7.29735 × 10 −3 The finestructure constant. (none)

78


CHAPTER 6 ATOMIC ORBITS AND PHOTONS Mass and Radiation Quantum of action and Planck's constant Particle waves and fixed atomic orbits The Photon The velocity of light

Only a few hundred years ago Copernicus explained that the Earth is not at rest nor at the center of the Universe, but is moving around the Sun at a tremendous velocity. Copernicus’ discovery led to simple calculations which can be used not only for our own solar system but also for atomic orbits and the motion of stars and galaxies. It is understandable however, that our Earth for so long was believed to be at the center of heaven and at rest since we can’t feel ourselves hurling through space and from our point of view it certainly looks like all heavenly bodies are moving around us. Today we smile at our ancestors belief not realizing that our text books still teach us a similar doctrine. Modern relativity and cosmology has no problem in placing our reference frame at the center of the Universe and at rest. The harmonic Universe on the other hand, demands that we move with an absolute velocity of c relative to a cosmic center of mass. This opens up a new avenue in physics that present relativity is unable to treat, namely that velocities created by loss of energy differs in magnitude from velocities generated by gain in energy. The harmonic Universe offers answers to many unsolved phenomena. One such phenomenon is the capture of electrons in atomic orbits where energy is lost to radiation.

6.1 Mass and Radiation It has been stated that mass and radiation are the two basic essentials in Nature. Both represent energy in different forms. Inertial mass equals potential or stored energy whereas radiation equals power or transformation of stored energy to radiation. We have also seen that the increase in energy, ΔE , of a bullet fired from a rifle increases the


80

inertial mass of the bullet by Δm = ΔE / c 2 . Energy is transferred by radiation from the burn of the powder which itself loses at least as much mass as is gained by the bullet. When the bullet is stopped by a target the inertial mass gained converts back to radiation which heats both the bullet and target. Transfer of energy cannot occur instantaneously but must take place over a certain length of time, Δt . The transfer of radiant energy is measured in power L = ΔE / Δt

(watts). Power is carried by individual photons which are massless wavelets or power pulses traveling with the speed of light. Each photon or wavelet is half a wavelength in size and has a frequency of 1 ν = 2 L / E , where E is the energy of the photon.

6.2 Quantum of action and Planck’s constant

Chapter 5 section 5.6 describes how energy appears in frequency. It was explained by the fact that we are part of an oscillating Universe in which all matter is subject to the same fundamental frequency ν 0 or ω 0 . Energy can therefore only appear in steps of its fundamental frequency just as frequencies of standing waves on a vibrating string only occur in steps set by the fundamental frequency of the string. The simple harmonic motion of the Universe can also be compared to the harmonic motion of a pendulum in a gravitational field. The fundamental angular frequency of a pendulum's to-and-fro motion is approximately

ω0 =

g , x

( t −1 )

(57)

where g is the gravitational acceleration and x the length of the pendulum. However, during its to-and-fro motion the pendulum also sweeps through numerous amounts of secondary frequencies of

ω // =

v , x

( t −1 )

(58)

ATOMIC ORBITS AND PHOTONS

81

where v is the linear velocity of the pendulum that varies between zero and maximum as the pendulum moves from its highest to its the lowest position. The same is true for our Universe where the frequency of the expanding-contracting motion (to-and fro-motion) is ω0 =

a0 , x0

( t −1 )

(59)

( t −1 )

(60)

which corresponds to a circular motion of

ω0// =

v0 . x0

For matter at relative rest with respect to our frame of reference at x 0 , the linear velocity v0 = c and circular frequency ω 0'' = ω 0 . The fact that matter at relative rest in our frame of reference has an angular frequency of ω 0 while hurling through space at a velocity of c, is certainly not noticeable to us. It becomes noticeable, however, as soon as its relative rest is disturbed by a change in the absolute values of v0 since it creates a measurable change in the fundamental frequency ω 0 which in turn changes the rest mass energy E0 . The change of E 0 as a function of change in frequency can easily be determined since the frequency change Δν of an electron that has gained energy of ΔE relative to our frame of reference as derived from Equation (54) and (55) Chapter 5, section 5.6 is

ΔE / Δt ΔE = = h, Δν / Δt Δν

(ml 2 / t )

(61)

(ml 2 / t )

(62)

or in power per unit angular acceleration ΔE / Δt Le = = =, Δω / Δt α1

where Le is the power needed to change the electron’s angular α1 = 2π / s 2 (angular acceleration) and frequency by = = 1.054495 × 10 −34 w s 2 is Planck's constant expressed in power per


82

unit angular acceleration. The angular frequency of an electron that has gained an additional energy of ΔE is therefore

Δω =

ΔE , (radians per second) =

( t −1 )

(63)

( t −1 )

(64)

and converting to standard frequency in cycles per second

Δν =

ΔE , (Hertz) h

where Δν is frequency in Hz and h = 6.625590 × 10 −34 J/Hz is Planck's constant expressed in energy per Hz. Since frequency appears only in even multiples of the fundamental frequency ω 0 or ν 0 the energy of E = ω 0 = or an electron becomes quantized in multiple steps of E = ν 0h .

6.3 Particle waves and fixed orbits

It has been described how energy appears in steps of frequency, a phenomenon that was empirically discovered by Max Planck in the early part of the 20th century. Planck's discovery was limited to the electromagnetic radiation of photons generated by atomic oscillators. Each time an electron is captured in a lower orbit of an atom, it loses potential energy which is emitted as electromagnetic radiation with a frequency of ν = ∇E / h , where ∇E is the energy loss involved in the capture process. Since electromagnetic radiation, or photons, propagate through space with c, the velocity of light, they can also be assigned specific wavelengths of λ = c / ν . In 1924 Loius de Broglie (1924) made the bold suggestion that energetic particles, such as electrons, moving with a velocity of v, might also be assigned wavelengths of

λ=

h , (de Broglie) mv

(l)

(65)


83

a prediction which was closely verified experimentally by Davisson and Germer (1927) and by Sir G.P. Thomson. It was perhaps unfortunate that de Broglie used the particle’s momentum p = mv in his theoretical treatment of particle wavelengths since Planck's constant relates to energy and not momentum. The problem with de Broglie's equation, is that it is a non-relativistic equation and it refers to a particle having a full wavelength when in reality particles, just like photons, only appear in half a wavelength. The fact still remains that de Broglie’s theory of moving particles having the same wave character as photons is one of the greatest achievements in quantum theory. The wave character of moving particles has to do with the fact that energy appears in frequency, see Equation 64, and if we divide the velocity of a moving particle by its acquired frequency we obtain its wavelength. The correct wave equation for particles having gained energy is thus 1 2

λ=

1 2

hΔv 1 ( 2 wavelength) , ΔE

(l)

(66)

where ΔE is the particle’s gain in rest mass energy and Δv the acquired velocity of the particle relative to our frame of reference. There is one more twist to the particle wavelength story which has not been considered before, namely that a particle can have two different wavelengths depending on whether the amount of potential energy is lost or gained. This is because the harmonic Universe model predicts different relative velocities for a particle depending on whether it has lost or gained energy. An example of velocities produced by gain in potential energy are particles accelerated in particle accelerators and an example of velocities resulting from loss of potential energy is the capture of electrons in atomic orbits where energy is lost to radiation. The dual energy-velocity relationship predicted by the harmonic or


84

collapsing Universe model (see Chapter 3, section 3.1 Equations (5) and (10)) is here summarized by Equations (67) and (68) and compared to velocity equations based on Newton's law of motion (69) and Einstein's relativity theory (70) 2

⎛ E0 ⎞ ⎟⎟ , Collapsing Univ.(Energy gained) Δv = c − ⎜⎜ c E + Δ E ⎝ 0 ⎠ 2

(67)

2

⎛ E − ∇E ⎞ ⎟⎟ , Collapsing Univ. (Energy lost) ∇v = c 2 − ⎜⎜ c 0 E ⎝ ⎠ 0

Δv or ∇v =

2

E , m

Δv or ∇v = c 1 −

Newton

(Energy lost or gained)

(68)

(69)

1 , Einstein (Energy lost or gained) (70) (1 + [ E / E0 ])2

where Δv and ∇v are changes in the velocity of a particle that has gained or lost energy relative to an observer at rest in our frame of reference. E 0 is the particle’s so-called rest mass energy or the absolute energy of matter at relative rest in our frame of reference ( E 0 = m0 c 2 ). ΔE and ∇E represent gain or loss in energy relative to our frame of reference. Both Equation (67) and Einstein's Equation (70) yield the same results for particles which have gained energy while Newton's non-relativistic Equation (69) will deviate more and more as velocity increases. In atomic orbits where velocities are produced by loss of energy only Equation (68) works while both Einstein's and Newton's equations fall short. The reason for this is that neither Newton or Einstein treated velocities produced by loss of energy any differently from velocities caused by gain in energy in their computations. This discrepancy becomes apparent when we employ the above equations in atomic orbit calculations and the result serves as irrefutable proof that our frame of reference is moving with a velocity of c, a fact that cannot be


85

ignored. For example, the circumference 2πr of the innermost orbit that an electron can occupy in an atom is equal to half the electron’s particle wavelength in accordance with de Broglie’s wave theory 1 2

and replacing

Zq 2 , λe = 4ε 0 ∇E 1 2

1 2

( l)

(71)

λ with Equation (66) yields

h∇v Zq 2 = , ∇E 4ε 0 ∇E

(l)

(72)

where Z is the atomic number; ∇v the sum of the electron's and nucleus' orbital velocities around their common center of mass; ∇E the sum of their orbital energies; q the elementary charge; and ε 0 the permittivity constant. Solving for the electron's closest orbit in oneelectron atoms (neutral hydrogen, singly ionized helium, doubly ionized lithium and so on) using the different velocity Equations (68), (69), (70) will result in 2 ⎡ ⎛ Zq 2 ⎞ ⎤ mn ⎟⎟ ⎥ × ∇E e = E 0 ⎢1 − 1 − ⎜⎜ , Collapsing Universe ( ) 2 ε h c m + m ⎢ ⎥ ⎝ 0 ⎠ ⎦ n e ⎣

(73)

m (Zq 2 ) mn Ee = e 2 2 × , (mn + me ) h ε0 8

(74)

2

⎡⎛ 1 E e = E0 ⎢⎜ ⎢⎜ 1 − (Zq 2 /(2ε hc ))2 0 ⎣⎝

Newton

⎞ ⎤ mn ⎟ − 1⎥ × . Einstein ⎟ ⎥ (mn + me ) ⎠ ⎦

(75)

The term mn / (mn + me ) , where mn and me are masses of the atomic nucleus and electron respectively, reduces the orbital energy to that of the electron only. The graphs in Fig. 12 show deviation in calculated values of orbital energies using the above equations as compared to measured values published in the Handbook of Physics and Chemistry. Both Einstein's theory of relativity and Newton's mechanics break down

86


since they do not consider the motion of our frame of reference relative to the rest of the Universe and, thus, do not distinguish between energy gained or energy lost. Equation (73) which is based on the harmonic Universe shows no serious deviations. A small Compton red-shift of Δλ ≈ 5.3 × 10 −14 m was discovered in the published values and has been deducted. Much work has been done trying to apply Einstein's relativity theory in calculating atomic orbits, which has led to extremely complicated equations involving a multitude of correction factors such as the Dirac-Fock correction (Desclaux, (1981)); Self energy correction (Mohr (1981)); Uehling vacuum polarization correction (Uehling, (1935)); higher order vacuum polarization correction (Källen (1955)); and nuclear size correction.

Fig. 12. Calculated ground state energies (highest ionization potential) for oneelectron atoms as a function of atomic number Z compared to available measured data. Deviation of calculated values from measured values are shown in %. The three curves are based on Einstein's theory of relativity; Newton's law of motion; and the velocity equation postulated by the harmonic or collapsing Universe.


87

In the lower Z-range between 1 - 20, these corrections produce results very close to Equation (73) but deviate drastically at higher atomic numbers. To summarize, the results show that it is possible to calculate atomic orbits without any of the above correction factors if we take into consideration that our local frame of reference is traveling with the velocity of c and that the orbital velocities of electrons in atoms are the result of loss of rest mass energy. This, in the author’s opinion, supports the collapsing Universe theory which is based on the laws of harmonic motion and which conform with Kepler's (1619) notion of a harmonic Universe presented in his much criticized work The Harmonics of the Worlds.

6.4 The photon

Photons are hard to envision. We know that a photon is a wave and needs a medium to propagate through, just as acoustic waves or ripples on the surface of a pond. We also know that a photon is not a continuous train of waves but only half a wavelength long, which gives it a pulse or particle appearance. The medium through which photons propagate is the universal gravitational tension φuniv which permeates all cosmic space. As explained previously the universal gravitational tension is generated by all matter in the Universe and is determined by the expression φuniv = GM univ / Runiv = c 2 (Mach’s Principle). One can perhaps describe a photon as a massless power pulse or wavelet which propagates through space at the speed of light. The energy E of a photon can be expressed in temperature such as T = E / 32 k , where k = 1.380 × 10 −23 J/T is Boltzmann's constant and T the photon’s temperature in degrees Kelvin. One can also determine the power per unit area delivered by a photon from Stephan’s law of radiation L / A = σT 4 , where σ = 5.67 × 10 −8 watts/m 2T 4 is StephanBoltzmann's constant and L the photon’s power of radiation. From the above information one can obtain a reasonably good idea of the physical


88

shape and size of a photon. For example, since the photons frequency and wave length are given by

ν=

E h

and

λ=

hc , E

(76)

and the total power of a photon is

L = 2νE , (watts)

(ml 2 / t 3 )

(77)

then the cross sectional area of the photon must equal L ( 32 k )4 A= or 1 = ( 12 λ ) 2 . 2 4 σT σ 2 hE

(l 2 )

(77b)

(l)

(78)

Since the length of the photon equals l=

Ec 1 = 2λ, L

we can roughly assume that a photon takes up a space in the shape of a cube with each side measuring 12 λ , see Fig. 13.

Fig. 13. The photon.


89

Below are some typical characteristics of a photon: 1 2

λ=

Ur =

ch ch = , half a wavelength 2E 3kT E

( λ) 1 2

3

=

81(kT ) = , energy/m3 3 3 2c h 4

kT

3 2

( λ)

3

1 2

2E 2 Ec 9(kT ) , power (watts) L= =1 = 2 h λ h 2

(l)

(79)

(m /(t l))

(80)

2

2

L / A = σT = 4

4

L

( λ) 1 2

81(kT ) = . 2c 2h 3

(ml 2 / t 3 ) (81)

2

watts/m 2

(m / t 3 )

(82)

6.5 The velocity of light

Although Maxwell successfully formulated his theory on electromagnetic fields and established that the velocity of electromagnetic waves equals c, the velocity of light, the question remained: what kind of universal medium or ether are electromagnetic waves propagating through? Great efforts have been spent in the search of a stationary ether, and numerous experiments have been performed trying to detect the Earth's movement through such a medium. Since no change in the velocity of electromagnetic waves at the Earth’s surface could be found in any direction, regardless of the Earth's motion through space, many abandoned the concept of an ether and claimed that electromagnetic waves ought to exist in empty space and therefore do not need a propagating medium. However, laws of physics require a propagating medium for any type of wave and the physical properties of such a medium must be


90

φ=

p

ρ

(tension),

( l2 / t2 )

where p is the pressure in newtons/square meter and

(83)

ρ the mass

density (inertia) per cubic meter. The pressure p of the medium is equivalent to energy density in joules per cubic meter. It is obvious to most of us that space is not empty, since it contains not only the gravitational tension of our Earth, Sun and planets, but the gravitational tension of the whole Universe. As mentioned before, the gravitational tension of the Universe at our vantage point x 0 in space, which equals φuniv = GM univ / Runiv = c 2 , by far exceeds the gravitational tension generated by our Earth and solar system (see Chapter 4, section 4.6). However, gravitational tension of individual gravitating bodies will add to the cosmic gravitational tension φuniv and cause the propagation of electromagnetic waves to slow down in their immediate vicinity determined by 2

⎛ φuniv ⎞ c ⎟⎟ ≅ v = c⎜⎜ 2 ⎝ φuniv + Δφ ⎠ 1 + 2Gm /( rc )

(l/t)

(84)

where Δφ = Gm / r is the gravitational tension of a body at the radius r and m is the mass of the body within r. The velocity of light at the surface of the Sun for example, is about 2

⎛ ⎞ φuniv ⎟⎟ = 2.997911 × 10 8 m/s v = c⎜⎜ ⎝ φuniv + (Gmsun / rsun ) ⎠

(l/t)

(85)

or 4 × 10 −4 % slower than the free space value of c.

It is important to note that it is not the gravitational acceleration but the gravitational tension φ that determines the velocity of light. Gravitational acceleration is the gradient of the gravitational tension φ and its magnitude at a distance r of the gravitating body is a = φ /r =

Gm Newton/kg r2

(l / t 2 )

(86)


It turns out that the Sun’s gravitational acceleration

91

φsun / rsun in the

vicinity of Earth is over a thousand times weaker than the Earth’s own surface acceleration φ earth / rearth . However, the gravitational tension of the Sun φSun at Earth, is fourteen times larger than the Earth’s own gravitational tension φearth measured at the Earth’s surface. We find that light slows down near a gravitating body because of the increased gravitational tension. Gravitational tension also determines the rate of time (see Chapter 2, section 2.4) so that any increase in gravitational tension has the effect of slowing all physical processes, including clocks as well as the speed of light. The fractional slow-down of light, however, is twice that of the slow-down of clocks. Gravity can bend or retard electromagnetic waves but Equations (84) shows that it is impossible to completely stop the propagation of an electromagnetic wave no matter how strong the gravitational field is. It is doubtful therefore, that gravitating masses exist from which light cannot escape such as “black holes”, championed by many astronomers, unless they possess an infinite amount gravitational tension, which is unrealistic.

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CHAPTER 7 RED-SHIFTS AND ENERGY BALANCE Red-shifts Energy density of radiation Energy density of matter Continuous creation

Religion teaches us that matter in all its forms, including ourselves, is created and did not evolve from natural processes. Natural science, on the other hand, claims the opposite, that our Universe has evolved to the state we see it today. There should be no conflict, because both claims are true.

7.1 Red-shifts Red-shifts are shifts in spectral lines towards a lower frequency or longer wavelength. Red-shift of wavelengths are usually expressed by the dimensionless ratio z

z=

λ − λ0 λ Δλ = −1 = , λ0 λ0 λ0

where λ0 is the original wavelength and λ

(none)

(87)

the longer red-shifted

wavelength.

In astronomy there are at least three major types of red-shifts which can be classified as follows: 1) Doppler red-shifts 2) Gravitational red-shifts


94

3) Time dilation red-shifts Most common are Doppler shifts that are purely velocity related. The spectral red-shift, z, of light from a galaxy that is moving away from us with a velocity of Δv equals c 2 − Δv 2 −1. z= c − Δv

(none)

(88)

Astronomical bodies, such as stars and planets that are bound by gravitational fields, exhibit gravitational red-shifts which relate to the slow-down of physical processes and the difference in gravitational tension that a photon has to cross as it climbs the potential well of a gravitational field. For example, photons created at the surface of the Sun are subject to a gravitational red-shift that is determined by the difference in gravitational energy ΔE or tension Δφ between the Sun and Earth and is equal to

z=

GM sun GM sun − c 2 Rsun c 2 RAU

=

ΔE , E

(none)

(89)

where Rsun and RAU are the radii of the Sun and of the Earth's orbit around the Sun respectively. Here ΔE is equivalent to the difference in energy of a photon that has crossed the difference in gravitational tension between the Sun and Earth and E is the original photon energy at the Sun’s surface. It should be pointed out that in reality the frequency of the photon, as seen from an outside observer, does not change crossing the potential well. The difference in clock rate between the Sun and Earth is what causes a difference in frequency reading at each location which we interpret as change in photon energy. Time dilation at high speeds creates a red-shift of spectral lines proportional to the increase in inertial mass and clock time and due to the increase in potential energy. Time dilation of matter can also be interpreted as an increase in tension Δφ ≈ Δ(v 2 ) caused by motion.

RED-SHIFTS AND ENERGY BALANCE

95

Time dilation is therefore, detected on fast moving bodies where not only time but all physical processes are observed to slow down. As physical processes slow down, such as atomic oscillators, the frequency of radiation also slows down, causing wavelengths to shift toward the red giving rise to a red-shift of

c2 Δt ⎛⎜ z= = t1 ⎜⎝ c 2 − Δv 2

⎞ ⎟ − 1 = ΔE , ⎟ E0 ⎠

(none)

where t1 is the rate of time at our frame of reference and Δt

(90) is the

increase in length of time due to the time dilation effect. The nature and magnitude of gravitational red-shifts and red-shifts caused by time dilation are but small when compared to the Doppler effect, see Fig. 14.

Fi g. 14. Velocity, gravity and time dilation red-shifts as a function of energy

96


Observed one-way red-shifts could also be a combination of all three processes mentioned, especially when astronomical objects are involved. Astronomers normally use Doppler type red-shifts to determine velocities of stars and galaxies relative to our frame of reference and in some cases possibly disregarding the time dilation effect and red-shifts caused by high gravitational fields. As mentioned earlier the discovery that light from distant galaxies become more and more red-shifted with increasing distance, and thus receding from us, caused astronomers to draw the conclusion that our Universe is expanding. The expansion was assumed to have started from a primeval explosion "the Big Bang" believed to have originated 1/H ≈ 18 thousand million years ago. The assumption that we are expanding was based purely on intuition and not scientific deduction. To exemplify how recession velocities can be misunderstood, consider several massive bodies falling from a given height. Observers on one body will see their neighbors recede at velocities that increase with distance along the line of fall. For example, imagine several persons on a misty day jumping off the edge of Grand Canyon at intervals of one second. If the visibility allowed, a person somewhere in the middle would see the nearest person who jumped ahead, pull away with a velocity of about 10 m/s and the person before him moving away with double the speed of 20 m/s. The reason of course, is that any one ahead has fallen, or accelerated longer, and reached a higher speed. Looking back in the other direction, however, one would see oneself pull away from the nearest person with a velocity of 10 m/s and 20 m/s from the next individual, etc. If the bottom of the canyon and its walls were obscured by the mist no one would be aware of falling and as far as can be seen, everybody appears to recede from each other, while in fact everybody is in free fall toward a common center, the center of mass of the gravitating Earth. The example of falling bodies is analogous to galaxies falling and accelerating towards the center of mass in a collapsing Universe. The increase in velocity makes galaxies ahead of ours seem to speed away,


97

and for the same reason galaxies behind us will appear to recede, because we are traveling faster toward the center than they are. From our limited view of the Universe this creates an illusion of expansion. Recession velocities as a function of distance are greatest when measured along the x-axis (see Chapter 2, Fig. 4) and decrease with increasing angles. Since we are falling along the x-axis (seemingly towards the Virgo cluster), which lines up closely with our galactic polar axis, then stars along the galactic plane obscure other galaxies to considerable galactic latitudes (“the zone of avoidance”). In the two windows available, one above the northern galactic pole and one above the southern galactic pole, one can expect the velocity/distance relationship to vary by a considerable amount, which also seems to be the case and which could perhaps explain the directional dependency of Hubble's parameter (de Vaucouleurs, (1978)) and the great scatter in its experimental value. Having established a physical reason for red-shifts of receding galaxies leads to equations based on the laws of harmonic motion that enables us to calculate red-shifts as function of distance. Relative recession velocities along the x-axis, as seen from our position at x 0 , can be estimated

from Δv ≅ 2a0 R

or

more precise by the

trigonometric functions shown in Fig. 4, Chapter 2 and on page 146. Exact recession velocities and red-shifts along the x-axis as a function of distance Δx or R as seen from our reference point x 0 in space are therefore looking back toward A 2

⎧ ⎡ Δx ⎞⎤ ⎫ ⎛ 1 Δv = c − ⎨ Aω 0 sin ⎢cos −1 ⎜ + ⎟⎥ ⎬ , ⎝ 2 A ⎠⎦ ⎭ ⎣ ⎩ 2

(towards A) (l/t)

(91)

and looking forward toward x = 0 2

⎧ ⎡ ⎛ 1 ∇x ⎞ ⎤ ⎫ 2 ∇v = ⎨ Aω 0 sin ⎢cos −1 ⎜ − ⎟⎥ ⎬ − c , (toward x = 0 ) (l/t) ⎝ 2 A ⎠⎦ ⎭ ⎣ ⎩

(92)

98


where Δv and Δx are velocity and distance respectively in the direction of A while ∇ v and ∇ x symbolize velocity and distance towards x = 0 .

Fig. 15. Hubble diagram showing red-shifts along the x-axis as a function of relative distance. Hubble's linear law is also shown for both 50 km/s and 150 km/s.

Recession velocities and red-shifts as a function of distance Δx , ∇x or R from Equation (91) and (92) are shown in Fig. 15. Hubble's linear velocity distance relationship is indicated by its two extreme values of H=50 km/s Mpc and H=150 km/s Mpc respectively. Recent data obtained by the Ia Supernova Cosmology Project are plotted on the above Hubble diagram. The theoretical light output of a typical Ia supernova is very strong, about 2 × 1038 watts or M = −23.5 in astronomical units, and can be seen very far. The Ia supernovae data show that Hubble’s law is actually a quadratic law ( v 2 / R ≈ 2a0 ) although it might appear linear on a small scale.


99

It also seems reasonable to believe that light from very distant galaxies should not only shift the spectra toward the red side but should also redden due to the fact that light in intergalactic space will scatter and contribute to the 2.766 o Kelvin temperature black body radiation. This type of reddening, which is similar to the reddening of sunlight during sunset or by dust, does not shift the spectral lines. It is conceivable, in the author’s opinion, that most galaxies in the Universe cannot be seen since their scattered light has already melted in with the glow of the black body radiation which perhaps should be recognized as Olbers’ light. Olbers in 1823 pointed out that if the Universe was filled with an infinite amount of stars as bright as the Sun, then every point in the sky would shine as bright as the Sun, even during night. This is known as Olbers’ paradox (Jaki, (1991)). Even if the Universe is not infinite there should still be a glow, but of much less intensity, which of course can be identified as the feeble glow of the 2.766 Ο Kelvin black body radiation. The amount of measured gravitational red-shift or blue-shift depends very much on the position of the observer and can be complicated. For example, for an observer on Earth the relative slow-down in clock rate of atomic oscillators at the Sun’s surface is Δν / ν = GM Sun /( RSun c 2 ) which in itself should cause a red-shift in atomic spectra of z = 2 × 10 −6 . However, the velocity of light at the Sun’s surface is slower by a factor 2 of Δv / c = 1 − (φuniv /(φuniv + φ sun )) which will generate a blue shift of z = 4 × 10 −6 where φuniv = c 2 and

φsun = GM sun / Rsun .

The resultant

solar shift is a blue-shift of z = 2 × 10 −6 . As the photon leaves the Sun’s surface on its voyage to Earth, the velocity of propagation increases stretching the photon’s wavelength to a red-shift of z = 4 × 10 −6 . The net effect is that we on Earth will observe a solar red-shift of z = 2 × 10 −6 . The frequency of a solar photon, as seen by an outside observer, might not change during its voyage to Earth. However, another accepted view is that the decrease in a photons energy and subsequent frequency while climbing the gravitational energy barrier between


100

Sun and Earth is canceled by the increase in clock rate as it reaches the Earth.

7.2 Energy density of radiation

The Temperature T = 2.766° Kelvin of the black-body spectrum is the result of scattered radiation from all discrete sources in the Universe and its energy density can be determined as follows: Consider a point source radiating electromagnetic energy in all directions. Such a 3 source will fill a spherical volume of 43 π (ct1 ) with radiant energy in one second, where ct1 is the radius of the sphere or the distance radiation propagates in t1 = 1s . The amount of radiation produced

and contained within the sphere equals the amount of radiation leaving the sphere in one second or E 2 = t1T 4σ 4π (ct1 ) , 2π

(ml 2 / t 2 )

(93)

where T 4σ is the radiant flux density at the surface and 4π (ct1 )

2

is the surface area of the sphere. If we divide the radiant energy E by 3 the volume 43 π (ct1 ) of the sphere we obtain the density of radiation in terms of potential energy and for T = 2.766° K we have 6πσ T 4 Ur = = 2.087 × 10 −13 Jm −3 , c

(m /(t 2l ))

(94)

(m /(t 2l ))

(95)

Equation (94) can be rewritten as U r = aT 4 ,

where a = 3.565 × 10 −15 Jm −3 K −4 replaces the term 6πσ / c . The constant a is useful for determining the energy density of black body radiation.


101

7.3 Energy density of matter

The energy density of matter in the Universe as seen from our vantage point x 0 is simply U m = ρc 2 = 2.087 × 10 −13 Jm −3 , where ρ

(m /(t 2l )) (96)

is the mass density of the Universe calculated from Equation

(24) which is the same as the observed mass density of the Universe. The striking result is that the energy density of matter is equal to the energy density of the cosmic black body radiation (see Equation (94)).

7.4 Continuous creation

If the energy densities of matter and radiation in the Universe are in exact equilibrium as seen above, then in the author's opinion, they were most probably always in a state of equilibrium. It is very unlikely, considering the large cosmological time scale, that at this instant we should be in the exact middle of a transit between a radiation dominated Universe and a matter dominated Universe. If in fact an equilibrium persists between matter and radiation, it will promote the idea of a Steady State Universe involving continuous creation in which everything in general remains balanced and unchanging. The Steady State Universe was first proposed by Hoyle (1948) and Bondi and Gold (1948). This theory was more or less abandoned because of the difficulty at the time to account for the observed 2.766° K black-body radiation (Narlikar, (1978); Narlikar and Kembhavi, (1980)). If we adopt the view of a Steady State Universe in which matter and radiation are kept in balance, then we must also accept the actuality of continuous creation, since matter must be replenished at the same rate it is dissipated by radiation. The rate at which matter must be created by radiation (e.g. pair production) is about one electron-positron pair per m 3 per 6 × 1012 years. The creation of matter from radiation such as pair production is a known physical phenomenon where light quanta

102


or photons can change into matter in the form of one electron and one positron. The photons must have at least the same combined energy as the potential energy of one electron plus one positron or 2 −13 2me c = 1.637 × 10 J which means that the photons must have energies in the gamma ray range. There is no lack of gamma rays in the Universe. The steady state and continuous creation scenario is consistent with a collapsing Universe where matter is losing all its potential energy to radiation as it falls towards the center of mass of the Universe. The radiation in turn travels out through space to eventually convert back into matter and again become subject to the inward gravitational acceleration. A rigid physical and mathematical theory of continuous creation might eliminate difficult arguments between different schools of thought concerning the origin of matter and subsequent evolutionary processes. Questions such as: was there first light before matter; when did it all start; and how long should it last? are still beyond the author's comprehension.

CHAPTER 8 LARGE NUMBER HYPOTHESES AND THE VIRIAL THEOREM Large number ratios Eddington's Magic Number N = 1.7507 × 10 85 Ampère's equation The Virial theorem and Cosmology Conclusion

A large number such as 1042 or 1085 might not appear monumental. It is easy to write down and compared to infinity it is insignificant. But how many of us remember the inventor of chess who, when offered a reward of his own choice, asked the maharajah for a grain of rice on the first square, two on the next and four on the third and so on until all sixty-four squares on the board had been filled. “Is this all you ask?” said the maharajah not realizing it would take more than 1019 grains of rice to accomplish this. In fact the amount of rice would cover all India with a foot or 30 cm layer of rice. Had he asked for 10 85 rice grains it would have filled a spherical volume 10 billion light-years across.

8.1 Large number ratios Ever since Herman Weyl (1919) come across the large numerical ratio of 4 × 10 42 there has been a great interest in trying to connect it with other number ratios that appear in nature having a similar magnitude. Weyl derived at his large number ratio in a roundabout way by comparing the classical or electromagnetic radius of the electron re = qe2 /( 4πε 0me c 2 ) to the hypothetical radius of a particle with the same

charge qe having an electrostatic energy equal to that of the electron’s gravitational energy. This ratio turns out to be the same as the ratio between the electron’s electrostatic coulomb force Fe to its gravitational force Fg or the energy E0 of the electron’s electrostatic

104


field to the electron’s self energy due to its gravitational field E g . Weyl further speculated that the above large ratio might also be the ratio between the radius of the Universe and the radius of the electron. In fact some years later in 1931 the astronomer John Q. Stewart pointed out that the radius of the Universe from Hubble's law, and Einstein’s relativity theory, divided by the radius of the electron, comes within a factor of one hundred of Weyl's ratio. In the same year Arthur S. Eddington (1931) speculated that the ratio of the electrostatic force to the gravitational force between an electron and a proton equals the square-root of N, the number of particles in the Universe. For reasons unknown Eddington thought that N had to be taken as the number of protons rather than electrons. Up to the present time the general view has been that the large number ratios which appear in nature are more or less a coincidence rather than a serious natural relationship. In the pages that follow, it will be shown that Weyl's notion and Eddington’s speculation are in fact true manifestations of nature because both Weyl's large number ratio and Eddington's magic number fit precisely the harmonic model of our Universe described in this book. An attempt will be made to find the connection between Weyl's and Eddington's large numbers and the mechanical structure of the Universe.

8.2 Eddington’s magic number N = 1.7507 × 10 85

Since the proton is found unstable we will duplicate Eddington's arguments, the only difference being that N will represent the number of electrons and positrons in the Universe and that N will equal the ratio of the electrostatic to gravitational forces, or energies between The two such particles as suggested by Weyl. In Eddington's Expanding Universe (1923) one can follow the reasoning that led to his large number hypothesis and the basic equations underlying his theory are

LARGE NUMBER HYPOTHESES AND VIRIAL THEOREM

GM u = c2 , Ru

(l 2 / t 2 )

105

(97)

and the ratios between the electron’s electrostatic and gravitational forces and energies Fe E0 q 2 /( 4πε 0re ) = = = N, Fg E g Gme2 / re

(none)

(98)

which equals Weyl’s number. In the above equations Ru is the radius of curvature of the Universe; M u its mass; c the maximum speed of recession of distant objects; G the gravitational constant; N the number of particles of mass me in the Universe; q the electron's charge; and re = q 2 /(4πε 0me c 2 ) is the electrostatic or classical radius of an electron (the proton's mass has purposely been replaced with the electron's or positron's mass me ). From Equation (98) it immediately follows that N = 1.7507 × 10 85 and the total mass of the Universe becomes M u = Nme = 1.5948 × 10 55 kg .

(m)

(99)

(l )

(100)

(m / l 3 )

(101)

The radius of curvature from Equation (97) is then Ru =

GM u = re N = 1.17908 × 10 28 m , 2 c

and the mean mass density of the Universe equals

Mu H2 = 4 = 2.3227 × 10 −30 kg m −3 . ρ=4 3 3 π Ru 3 πG

In the above equation, H is Hubble's constant. Hubble's constant, which has the dimensions of angular frequency, relates to the fundamental frequency ω 0 of the Universe or

ω0 =

c = H = 2.5425 × 10 −20 rad s −1 , Ru

(t −1 )

(102)


106

which corresponds to a time epoch or period of t0 =

2π

ω0

= 2.4712 × 10 20 s .

(t)

(103)

Further calculations show that the gravitational acceleration at Ru equals a0 =

GM u Gme = 2 = 7.6225 × 10 −12 m s −2 . 2 Ru re

(l / t 2 ) (104)

The significance of the large number hypotheses becomes clear when we compare the above results with the physical constants and parameters listed in Table 1, Chapter 2, section 2.2. Perhaps the most important feature revealed by the above equations is that the electron's gravitational potential energy at Ru , generated by the rest of the Universe, energy

equals the electron's

rest mass energy and electrostatic

GM ume q2 = = me c 2 . E0 = Ru 4πε 0re

(ml 2 / t 2 )

(105)

We can therefore change expression (98) by replacing the electron's electrostatic energy q 2 /(4πε 0re ) with the potential energy GM ume / Ru and write Weyl's number to equal E 0 GM ume / Ru = = 4.183 × 10 42 , 2 Eg Gme / re

(none)

(106)

from which other natural ratios and the square root of Eddington’s magic number can be obtained Ru = re

Mu M u Ru =3 = N = 4.184 × 10 42 . me me re

(none)

(107)

It is not known whether Weyl or Eddington arrived at their hypotheses by intuition or by considering the substitution described in Equation (107). Another interesting consequence of the large number


107

ratios is the ratio of the electron's gravitational self-energy to its smallest quantized energy (see Chapter 5, section 5.6) due to the fundamental frequency of the Universe equals Gme2 / re 1 = , (finestructure constant) (none)(108) α= ω0= 137.036 where = is Planck's constant h divided by

2π . This is the so-called

fine structure constant which was of special interest to Eddington, From the laws of harmonic motion we can also determine the electron's change in potential energy (radiation) as a function of time ∇E me c 2 me a0 c = = = Le , Δt 2π t0

(ml 2 / t 3 )

(109)

Le = 12 h ,

(ml 2 / t 3 )

(110)

and

where

Le

is the power radiated by an electron generated by the harmonic motion of the Universe and h is Planck’s constant expressed in power. From Equations (104), (109) and (110) follows the ratio

G re2π = . h me2 c

(lt / m2 )

(111)

Since Plank's constant is accurately determined from experiments we can derive G = 6.6445 × 10 −11 as the postulated free space value of the universal gravitational constant. This is about 0.004 times less than the current estimated value obtained at the Earth's surface. There are however, both theoretical (Fujii (1971) and Long (1980)) and experimental (Long et al. (1976) and Stacey et al. (1981)) claims indicating a small departure, at the Earth's surface, from the astronomical or free space value of G, which could explain the above discrepancy. It should be mentioned that the universal gravitational constant G is one of the least well known constants in nature and the

108


method of obtaining G has often been criticized (Kazuaki Kuroda (1995)).

8.3 Ampère's equation

It is difficult to ignore the implications of Weyl’s and Eddington's large number hypotheses. The large number hypotheses has intrigued many theoretical physicists and most noteworthy P.A.M. Dirac (1938). Dirac, who unfortunately did not refer to Weyl’s or Eddington's work, also compared other large numbers with the electron-proton electric to gravitational force ratio. The amazing results of the large number hypotheses is that they seem to offer a link between electromagnetic and gravitational forces. The same is true for Ampère's law of electrodynamics which hints to the fact that c, the velocity of matter due to the harmonic motion of the Universe, is responsible for the electron's Coulomb force and that its gravitational force is generated by the cosmic acceleration a 0 : 1. The force, or glue, that holds the electron together preventing it from flying apart due to its own charge is brought about by the electromagnetic “pinch” effect generated by its velocity of c in space. The magnitude of this force equals q 2 /( 4πε 0 re2 ) = me c 2 / re . 2. The gravitational energy of an electron seems to originate from the “pinch” effect generated by the cosmic acceleration a0 and the magnitude of this energy is me a0re . The pinch effect originates from electromagnetic forces created by a moving charge. The electromagnetic force, which points inward towards the center of a moving charge, tends to squeeze the charge together. The magnitude of this force can be determined by Ampère's equation (1820). This equation was first derived from the observation that a force appears between two parallel cylindrical conductors that carry currents of electric charge. If the currents through both


109

conductors flow in the same direction the force is attractive and if the currents flow in opposite directions the force is repulsive. The diagram in Fig. 16 shows Ampère's experiment consisting of two parallel ia and ib respectively. The force conductors carrying currents between the conductors is given by

F = lib Ba = l

ia ib μ 0 , 2π R

(ml / t 2 )

(112)

where l is the length of the current elements which are separated by the distance R. The magnetic field generated by one current at the site of the other is B. If we consider the current elements as individual charges ( i = q / t ), such as single electrons moving at a velocity of formula becomes

Fe = where

l q a qb μ 0 ⎛ re ⎞ ⋅ ⋅⎜ ⎟, t 2 2π R ⎝ R ⎠

l = 2 re is the length of

c, Ampère's

(ml / t 2 ) (113) the current

element or in this

case

Fig. 16. Two parallel wires which carry current in the same direction attract each other.

the size of an electron and t = 2re / c is the time required for the electron to move the distance l. The term (re / R ) changes Ampère's equation from cylindrical symmetry of conductors to the spherical geometry of


110

single electrons. So far we have discussed the force between conductors or two electrons moving side by side immersed in each others magnetic fields. Since a moving electron is also immersed in its own magnetic field it will be subject to a self-induced pinching force that tends to squeeze it together according to Equation (113) thus counteracting the electron’s own Coulomb force which wants to blow it apart. In fact, Equation (113) can easily be converted to Coulomb’s force equation by replacing t with 2 re /c and μ 0 with 1 /(ε 0 c 2 ) which results in

q e2 (Coulomb’s law). Fe = 4πε 0 R 2

(ml / t 2 ) (114)

There is also a force Fg superimposed on charges that are mutually accelerated which according to Ampère's law (113) can be written as

Fg = 12 a ⋅

q a qb μ 0 ⎛ re ⎞ ⋅ ⎜ ⎟, 2π R ⎝ R ⎠

(ml / t 2 )

(115)

l / t 2 in (113) is replaced by 12 a, were a represents acceleration. Inserting a 0 , the acceleration due to the harmonic motion of the Universe, makes Equation (115) identical Newton's law of gravitation Fg = a0 ⋅

where ma and mb

q a qb μ 0 re Gma mb = (Newton’s law), (ml / t 2 ) (116a) 2 2 4π R R

are the masses of two electrons separated by a

distance R. Equation (116a) might prove that the agency responsible for generating gravitational attraction is the cosmic acceleration a 0 . From the above analysis we find that Ampère's equation (113) predicts that an electron moving with a velocity of c generates a pinch force on itself by being immersed in its own magnetic field B. The force due to this pinch effect is opposite and equal to the electron’s own Coulomb force and thus serves as the force binding the electron together. We have also seen that a similar pinch effect appears when an electron is subject to acceleration as shown by Equation (116a) which can also be written as


qe2 μ 0 Fe = a0 = me a0 . 4π re This leads to the fact that

111

(ml / t 2 ) (116b)

me = qe2 μ 0 /(4π re ) , or that the electron’s

inertia of mass is equivalent to the electron’s inertia of charge (selfinduction). What if two massive objects are accelerated relative to our laboratory, does the gravitational force between them increase in accordance with Equations (116a and 116b)? The answer is probably yes but we must first convert relative acceleration to absolute acceleration similar to that of relative and absolute velocities as described in Chapter 3, section 3.1. The increase in gravitational force between two objects being accelerated by a given acceleration a relative to our frame of reference should (without proof) be equal to

⎡ a 2 ⎤ Gma mb F ≅ ⎢1 + • ⎥ × , R2 ⎣ 2c a0 ⎦

(ml / t 2 )

(117)

where c • = c per second. The gravitational force generated should be attractive for bodies decelerating relative to our frame of reference. In Equations (116a and 116b) only the smallest amount of quantized matter were considered, namely that of the electron and the positron. However, there should be no problem with other particles, charged or neutral, if we assume that all matter is made up of electrons and positrons as suggested by Eddington’s magic number N. Ampère's formula gives only the magnitude of force between moving charges and not the direction as to whether attractive or repulsive. Also, the gravitational force of Equations (116a, 116b and 117) is believed to be of monopole nature and cannot be shielded against. The cosmic structure revealed by the large number ratios is that of an oscillating Universe (expanding-contracting) in which the maximum speed of recession c can be interpreted as our velocity relative to distant matter or to the center of mass of the Universe. The frequency of oscillation is ω 0 = a0 / c and a 0 represents the cosmic acceleration

112


associated with the harmonic motion.

The cosmic acceleration a 0

described in Chapter 3, section 3.2, is in fact evident from the discovery and observation that galaxies separate with velocities that increase with distance of separation. Since the distance of separation increases with time, the velocity must also increase with time. An increase in velocity with time is acceleration. The large number hypotheses are, in the author's opinion, not just a coincidence but a direct result of the laws of harmonic motion where the energy or force between matter is balanced about a common center, the center of mass of the system. For example, if we compare the electron’s gravitational self energy Gme2 / re to the rest mass energy generated by the rest of the Universe GM ume / Ru , the following ratios appear GM ume / Ru Ru = = Gme2 / re re

Mu = N. me

(none)

(118)

(none)

(119)

Since GM u / Ru equals c 2 we can also write

φ c2 E F = 0 = e = u = N, Gme / re E g Fg φ e

where E 0 is the electron’s rest mass energy and E g its gravitational self energy. One important point here is that the electron’s rest mass energy E 0 = me c 2 is equal to the electron’s electrostatic energy E0 = q 2 /(4πε 0 re ) which implies that both energies are of the same origin or one electron would otherwise carry double the energy, which is not the case as shown from electron inhalation studies. Electron inhalation is a phenomenon where a positive charged electron (positron) is absorbed by a negative charged electron and where both particles mass and charge disappear into radiation in the form of two photons, each photon carrying an energy of E0 . The large number ratios can also be derived from the Virial Theorem which in the author’s opinion, again


113

proves that the large number ratios are not a mere coincidence but are simply in compliance with the laws of Nature. 8.4 The Virial Theorem and Cosmology

The Virial Theorem was originally applied to thermodynamics and statistical mechanics. It is now a very important tool in all branches of physics because it provides a means for checking the validity of both experiment and theory. Most cosmological models of today fail the virial test creating a missing mass problem and the purpose here is to test our cosmic model based in toto on the Virial Theorem. The result again leads to a relationship between G, the Universal gravitational constant, and h, Planck’s radiation constant. The Virial Theorem is the creation of Rudolf Julius Emanuel Clausius, 1822-1888, who was first to apply the doctrine of probabilities, in a systematic way, to the kinetic theory of gases and he also introduced the concept of entropy which is another important contribution made by him to science. The Virial Theorem states that the virial of a body or a large assembly of bodies in a confined space is defined to be the sum

∑ = 12 ri Fi ,

(ml 2 / t 2 ) (120)

where ri and Fi are the position and force vectors respectively, acting on the ith particle. In Latin virial means strength or stress, but in scientific terms we can describe it as half the product of the stress due to the attraction or repulsion between two particles in space multiplied by the distance between them, or in the case of more than two particles half the sum of such products taken for the entire system. In simple language, the Virial Theorem tells us that the stress or potential energy inside a given volume is balanced by the kinetic energy of matter residing within that same volume or 1 2

m

U

ρ

= 12 mv 2 ,

(ml 2 / t 2 ) (121)


114

where U = E / V (energy density) is the energy per unit volume, ρ = m / V (mass density) is the amount of matter per unit volume and v is the velocity of a particle or the root mean velocity of particles with The potential energy might be electrostatic, binding mass m. molecules together, or it could be the gravitational energy which holds our solar system or entire Universe together.

The usefulness of the Virial Theorem can be demonstrated in several ways. For example, reducing Equation (121) to U

ρ

= v2 ,

(l 2 / t 2 )

(122)

implies that energy density divided by matter density relates to a fixed velocity squared. If we insert the energy per unit volume as equal to the molecular binding energy in a metal (stress modules or Young's modules), and divide it by the matter density, we can solve for v, the wave velocity of sound in the metal. When applied to the electrostatic field produced by one or n number of electrons and positrons, Equation (122) becomes U

nq 2 V = ⋅ = c2 , ρ 4πε 0 reV nme

(l 2 / t 2 )

(123)

where q and me are the electron's or positron’s charge and mass respectively, and re = q 2 /( 4π ε 0 c 2 ) is the electrostatic or classical radius of an electron and positron. The symbol V in Equation (123) represents the volume in which n electrons or positrons are contained. The result is v = c, the velocity of light, or the speed at which electromagnetic waves and field changes propagate through space. Since U / ρ , which represents energy per mass, is equal to c 2 which in turn equals the cosmic gravitational tension φuniv we can rewrite the Virial Theorem as follows:

φUniv =

E = c2 , m

(l 2 / t 2 )

(124)


115

which leads to Einstein's energy relation E = mc 2 (see Appendix C). The Virial Theorem is used extensively in astrophysics and cosmology and it is in this branch of physics where it is often disobeyed. There is no apparent problem locally, such as within our own solar system. It is at large extra-galactic distances where observation and most present theories drastically deviate from the Virial Theorem and it worsens the larger and farther away these systems are, as was discovered many decades ago by F. Zwicky (1937). Zwicky found that the apparent mass of the Coma cluster is many times larger than the combined mass of all its individual galaxies, thus creating a missing mass problem (see Chapter 3, p48). Most cosmological models suffer the same difficulty since they predict a mean mass density that is orders of magnitude higher than what is being observed from luminosity measurements and galaxy counts. The mean mass density of the Universe is usually obtained from the standard relations Mu qH 2 ρ=4 =4 , 3 π R π G 3 3 u

(m / l 3 ) (125)

where G is the free space value of the gravitational constant and H is Hubble's constant. A typical value of H = 50 km s −1 Mpc −1 will yield a cosmic mass density of 1 × 10 −26 kg m −3 that is much too high compared to the observed ρ ≈ 2 × 10 −30 kg m −3 . The dimensionless constant q is the so called "acceleration parameter" and must equal unity or the Virial Theorem is violated. However, an acceleration parameter considerably less than unity at maximum recession velocities is often suggested as a solution to the mass density problem but proof of its validity is lacking. The "Einstein - de Sitter" model of the Universe, for example, requires that q = 12 , which is still too high to account for the above discrepancy in mass density. If Hubble's constant is defined as H = v / R or H = c / Ru where c is the limiting speed of recession and Ru the radius

116


of curvature of the Universe, then Equation (125) variation of the Virial Theorem and can rewritten as GM u c 2 = = a0 (acceleration) , Ru2 Ru

is simply a

(l / t 2 )

(126)

(l 2 / t 2 )

(127)

or GM u = c2 Ru

(velocity 2 ).

The exact interpretation of Hubble's constant is, therefore, angular velocity and its physical dimensions H = v/R inform us that we are part of a harmonic motion rather than a linearly expanding Universe. If the Virial Theorem is truly universal, it must be valid not only for subatomic particles and planets, but also for galaxies and for the entire Universe. Consequently, it should be possible to construct a cosmological model, based on the Virial Theorem, and assess the result against known physical parameters. A simple approach would be to consider the gravitational interaction between the smallest known quantum of matter, the electron, and the rest of the Universe. In a virial Universe (where M u ∝ Ru2 ) the ratio of the gravitational energy of the whole Universe to the gravitational energy of the smallest known matter (the electron) equals 3 2

3

GM u2 / Ru ⎛ Ru ⎞ ⎛ M u ⎞ Ru M u ⎜ ⎟ = = = , ⎜ ⎟ ⎜m ⎟ Gme2 / r r r m ⎝ ⎠ ⎝ e ⎠ e e

(none) (128)

and the energy ratio of the electron’s attraction to the rest of the Universe and to its gravitational self energy is 1

GM u me / Ru Ru ⎛ M u ⎞ 2 ⎟ = N. = =⎜ 2 re ⎜⎝ me ⎟⎠ Gme / re

(none) (129)

Equations (128) and (129) are easily modified to the Virial theorem


GM u Gme = 2 = a0 , Ru2 re

117

(l / t 2 ) (130)

where a0 = 7.62247 × 10 −12 m s −2 is the rate of the cosmic acceleration or gravitational force per mass in the Universe which also equals the electron’s gravitational surface acceleration. Since GM u / Ru2 = c 2 / Ru it immediately follows that the radius of curvature of the Universe is c2 Ru = = 1.17908 × 10 28 m . a0

(l) (131)

The total mass within the radius Ru equals

Ru2 a0 Mu = = 1.59486 × 10 55 kg , G

(m) (132)

and the mean mass density of the Universe becomes

ρ=

Mu = 2.32273 × 10 −30 kg m −3 , 3 4 Ru 3π

(m / l 3 ) (133)

which agrees with the observed value. The angular frequency of the expansion-contraction is given by H = ω0 =

c = 2.54258 × 10 −20 rad s −1 , R

(t −1 ) (134)

u

which corresponds to an epoch, or period, of t0 =

2π

ω0

= 2.47118 × 10 20 s .

(t) (135)

The potential energy of matter at Ru is E=

GM um = mc 2 , Ru

(ml 2 / t 2 ) (136)

which conforms with Einstein's mass-energy relation E = mc 2 .


118

The above equations again describe an oscillating (expandingcontracting) Universe which is about 100 times larger than current estimates and that is in perfect agreement with the large number hypotheses and the values listed in Table 1, Chapter 2, section 2.2. Whether we are part of a one or a multi-cycle Universe is determined by the rate of energy that is lost to heat or radiation. We have shown from the laws of harmonic motion, that a one cycle “deadbeat” (critically damped) Universe must dissipate all its potential energy over one period of oscillation, and the equivalent rate of radiation is M uc 2 Lu = = 5.80044 × 10 51 watts . t0

(ml 2 / t 3 )

(137)

Consequently, an electron must generate ΔE me c 2 me a0 c = = = Le , Δt 2π t0

(ml 2 / t 3 ) (138)

Le = 12 h ,

(ml 2 / t 3 ) (139)

or

which is the same as obtained by Equation (110). The rate of radiation from Equations (137) matches the amount of radiation found in our Universe and the total amount of radiation from all matter determines the blackbody temperature of the Universe from Stefan's law: 1 4

⎛ M uc ⎞ ⎟⎟ = 2.766° K , T = ⎜⎜ 2 4 R t π σ u 0 ⎠ ⎝ 2

(none) (140)

where σ = 5.6705 × 10 −8 (Stefan-Boltzmann's constant). As can be seen, the Virial Theorem predicts exactly the same large number ratios as anticipated by Weyl, Stewart and Eddington.


119

A one cycle cosmological model based on the Virial Theorem does not comply with a constant value of Hubble's constant H = Δv / ΔR . The problem is, that even if H = Δv / ΔR might appear linear on a small scale, it in fact follows the quadratic function v 2 / R which becomes more and more clear the further in space we probe. In contrast to other cosmological models, which usually contain vague numerology and therefore are difficult to challenge, the virial Universe provides exact solutions and can be experimentally tested. For example, from Equations (130), (138) and (139) it follows that

G re2π = , h me2 c

(lt / m2 )

(141)

and if we replace the electron's dubious radius with the electromagnetic term q 2 μ 0 /( 4π me ) we can write G ( q / m )4 μ 20 = . 16πc h

(lt / m 2 )

(142)

Since Planck's constant h is accurately determined from experiments we again derive G = 6.6445 × 10 −11 as the postulated free space value of the universal gravitational constant.

8.5 Conclusion

The Large Number Hypotheses and the Virial Theorem offer the same exact mathematical solutions as the harmonic Universe described previously. The idea that Ampère's law (see section 8.3) could provide a clue to gravitation is not new. Maxwell spoke of Ampère's discovery as "one of the most brilliant achievements in science. It is perfect in its form and unassailable in accuracy; and it is summed up in a formula from which all phenomena may be deduced" (Whittaker (1951)). It also seems proper to conclude with Eddington's own words: "A large ratio appears when we compare the electric force between a proton and electron with the gravitational force between them. I have long thought

120


that this must be related to the number of particles in the Universe and I expect that the same view has been entertained by others. The above ratio is of the order of N." The large number ratios seem to hint at a relationship where the mass of the Universe increases proportionally to the square of its radius or M u ∝ Ru2 which might favor a non-uniform Universe, unless pancake shaped, as suggested by several investigators. It is difficult at the present time, to detect any non-uniformity in mass density over the entire Universe since the range of our most powerful telescopes at the present time is less than 1% of its radius Ru .

CHAPTER 9 SUMMARY Relative and absolute motion, an overview The true Universe? Building blocks of Nature

Nature is an interplay between mass, charge, time and length. It produces acceleration and velocity generating the most precious commodity of all, namely energy, which can be stored (potential energy) or used (power). Stored energy is the product of mass (m) and tension (φ ). Spent energy or power always involves radiation Charge (q) and mass (m) are related through the constants ε 0 and μ 0 . The change in energy with time (power) is nature’s gift of life. Without change in energy nothing would ever happen.

9.1 Relative and Absolute Motion, an overview Isaac Newton, the ground breaker of modern physics and mechanics, summarized physics in three laws of motion which are still in broad use today. He believed the Earth was orbiting the Sun through a fixed space or ether so that our frame of reference, the Earth, would have an absolute velocity with respect to stationary space. If Earth in its orbit is moving around in stationary space, which acts as a medium for force fields and the propagation of light waves, then space must have some physical properties. It should be possible to perform an experiment that could detect the Earth's motion through stationary space or the so called "ether". Michelson and Morley (1887) were first to attempt such an experiment using sensitive optical interference methods. It was thought that the travel of light waves, from a light source to a mirror and back, would be different along the direction of the Earth's orbital

122


motion through the ether than at right angles to it. The result was that no difference in travel time was detected. Ritz explained the null result by suggesting that c, the speed of light, is always constant with respect to the light source, but other scientists at the time favored the idea that our Earth is dragging the ether along in its motion. This is perhaps closer to the truth since the gravitational tension of all matter in the Universe is what constitutes the ether and serves as a medium for force fields and electromagnetic waves. As long as the gravitational tension is constant the speed of light stays constant. There is no change in the gravitational tension φuniv at the surface of the Earth, regardless of its orbital motion around the Sun, that will change the speed of light in any direction along the Earth’s surface except in the vertical direction where the Earth’s own gravitational tension changes with altitude. The change in the Earth’s gravitational tension with altitude (acceleration g) will change the speed of light, see Appendix C. Fitzgerald (1889) and, independently, Lorenz (1892), on the other hand, believed that the null result could be explained if one assumed that the length of the measuring instrument shortened in the direction of motion. This turned out to be an appealing approach and it had its origin in the discovery, at that time, that matter could not accelerate to exceed the speed of light c. It had been observed that velocity v, ( v = distance per time) did not increase proportionally to the square root of kinetic energy as envisaged by Newton but seemed to shrink at high values, never to reach the speed of light c. This behavior was attributed to the shrinkage of distance and was assumed to affect anything moving at high velocity. The proposed shortening in the length of objects, including instruments and measuring rods in the direction of velocity v relative to the ether, was attributed to a factor β which limits the velocity to the speed of light

β = 1 − (v 2 / c 2 ) .

(none) (128)

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This shortening of length is known as the Lorenz-Fitzgerald contraction and the idea has been widely used in situations where energy and velocity are transferred from coordinates in one frame of reference to another, which are moving with a velocity of v relative to each other. However, instead of changing length of coordinates in moving frames by β it is just as easy to modify the rate at which time flows by β or to change both. This variation of the Lorentz-Fitzgerald idea is exactly what Einstein (1905) had in mind when he introduced his Theory of Special Relativity which led to the concept of space-time

E0 + E = E 0 / β

(ml 2 / t 2 ) (129)

or v = c 1−

1 = c 1− β2 , 2 [1 + ( E / E0 )]

(l/t) (130)

where E is the kinetic energy of a mass m due to its relative velocity v and E0 = mc 2 is the rest mass energy of an object. The interesting but perhaps troublesome outcome of Einstein's theory is that it eliminates the use of length when transforming velocities from one coordinate to another thus abolishing the concept of a fixed space and the existence of an ether. Einstein's velocity equation (130) improved Newton's law of motion to the point where it can precisely describe the behavior of particles with high relative velocities such as found in high energy particle accelerators. In fact, Einstein’s equation is only accurate in situations where matter has gained velocity due to gain in energy. The diagram in Fig. 17 shows how Einstein’s relativistic formula differs from Newton’s law of motion when used to calculate velocities of high energy electrons. Einstein’s energyvelocity formula has been verified numerous times in particle accelerators where matter has gained energy and is perhaps one of the greatest successes in physics of the 20th century. But one problem still remains, namely that neither Newton’s law of motion nor Einstein's

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relativistic equations work satisfactorily for the high relative velocities found in atomic orbits, where velocities are created by loss of rest mass energy. In fact, Newton's laws of motion offers a slightly better fit for the energy-velocity relationship of atomic energy spectra than does

Fig. 17. Velocity of high energy electrons as a function of Energy according to Newton and Einstein.

Einstein's theory, see Fig. 12, Chapter 6. The reason why both Newton's law of motion and Einstein's theory of relativity do not work for atomic orbits has to do with the fact that both theories consider us at rest (thus the term rest mass energy) and therefore ignore the influence of our own motion with respect to the rest of the Universe. Einstein's theory goes as far as to state that everything is relative and that no observer occupies a privileged position in the Universe because there is no absolute space or ether to reference our position or velocity to. Although our Earth, the solar system and our galaxy, are moving relative to other astronomical objects the theory claims that it is just as valid to say that other astronomical objects are moving relative to us and that we, therefore, can consider our frame of reference here on Earth to be at rest. By the

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same token an observer at any other galaxy can consider her or himself to be at the center of the Universe and at rest. Herein lies the snarl with Einstein's relativity theory since it essentially places ourselves as stationary observers in a mathematically centralized position. As previously explained it creates the same difficulty that haunted our predecessors for thousand of years when they firmly believed that our Earth was at the center of heaven and at rest and how an impossible task it was to understand any mathematical equation describing the planets including our Sun orbiting the Earth. It is for this same reason that both the Lorentz-Fitzgerald transformation of velocities and Einstein's relativity theories are difficult to understand and why they have always been a subject of debate. Even though the mathematical equations, in some cases, provide correct numerical answers there is still a small number of investigators who cannot accept a mathematical theory unless it makes physical sense; while many mainstream scientists of today seem to feel that a mathematical equation, which gives a correct numerical answer, constitutes a physical law. Even if Lorenz-Fitzgerald and Einstein's equations give correct answers, it is the interpretation of the physics that is amiss because it does not take into account our absolute motion in the Universe and, therefore, breaks down when applied to atomic and astrophysical orbits where velocities relative to us are generated by loss of potential energy (see Appendix C). To illustrate the importance of absolute motion consider, for example, the Earth's orbital velocity v0 around the Sun using standard Newtonian mechanics v0 = GM sun / Rorb. = 30km/s .

(l/t) (131)

Adding energy in the amount of ΔE to the Earth's orbit would sling the Earth out to a higher orbital radius but slower orbital velocity of v0 − Δv =

2( E orb − ΔE ) M earth

(l/t) (132)


126

where Δv is the change in orbital velocity due to the added energy ΔE . However, should the Earth's orbit on the other hand, experience a loss in potential energy it would fall closer to the Sun, but with an increase in orbital velocity of

v0 + ∇v =

2( E orb + ∇E ) , M earth

(l/t)

(133)

where ∇v is the change in orbital velocity due to ∇E , the loss in energy, see Fig. 18.

Fig 18. Diagram illustrating change in Earth’s orbital velocity and frequency Δν and ∇ν as a function of change in orbital energy ΔE and ∇E respectively.

The diagram in Fig. 19 shows a curve labeled "Energy gained", which is constructed from Equation (132) and a curve labeled "Energy lost" which is constructed from Equation (133). The two curves, "Energy gained" and "Energy lost" demonstrate that for an equal change in energy ΔE = ∇E the velocity Δv does not equal ∇v or 2

ΔE ⎛ Δv ⎞ ≠⎜ ⎟ , ∇E ⎝ ∇v ⎠

(none)

(134)

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which informs us that two different equations must be used depending on whether energy is lost or gained, for the simple reason that our frame of reference, the Earth, is not at rest. The same is true for objects seen from our point of reference relative to the rest of the Universe, a reality which has been neglected and explains why existing theories fall short in accurately predicting velocities of atomic orbits where electrons have lost energy to radiation. Therefore, it cannot be ignored

Fig. 19. Change in the Earth's orbital velocity as a function of change in energy.

that we ourselves are in motion when trying to determine motion of matter relative to us. We have to abandon the doctrine of both Einstein’s relativity and Newton’s concept and accept the fact that we are part of a Universe in which all matter has its own peculiar position and absolute velocity with respect to the center of mass of the system. The intention of this book has been to show that it is possible to construct equations of motion that are both conceptually clear and physically sound, and that will work for both energy gained and energy lost, as well as for absolute and relative velocities. For example, if we use Newton’s Equation (131), that was used to describes the Earth's


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orbit around the Sun, but change the mass and radius to that of the whole Universe we obain

vabs = GM Univ. / RUniv. = c , where

(l/t) (135)

vabs = c can be considered our absolute velocity relative to the

rest of the Universe. At our frame of reference in the Universe, the potential energy of matter equals E0 = mc 2 . The absolute velocity as a function of gain in potential energy is therefore

vabs = c

E0 . E 0 + ΔE

(l/t) (136)

As previously shown relative velocities of bodies in the Universe, that have gained kinetic energy relative to our frame of reference, are obtained from the vector sum 2

Δv = c − v 2

2 abs

⎛ E0 ⎞ ⎟⎟ , = c − ⎜⎜ c E + Δ E 0 ⎠ ⎝ 2

(l/t) (137)

which can be reduced to Einstein's relativistic Equation (130) v = c 1−

1 = c 1− β2 . 2 [1 + ( E / E0 )]

(l/t) (138)

On the other hand, when energy is dissipated relative to our frame of reference, such as when electrons or astronomical bodies are captured in orbits and where potential energy is lost relative to us , the above Equations (137) and (138) are invalid. The correct equation for velocities relative to us that are produced by loss in potential energy is therefore 2

⎛ E − ∇E ⎞ ⎟⎟ . ∇v = c − ⎜⎜ c 0 E0 ⎠ ⎝ 2

(l/t) (139)

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The curves in Fig. 20 show the velocity of an electron relative to our frame of reference as a function of energy gained and energy lost. The straight line represents Newton's law. The curve labeled "energy gained" is constructed from Equation (137) and which also conforms with Einstein’s Equation (138). The curve labeled “energy lost” is constructed from Equation (139) and fits perfectly situations where absolute energy has been lost such as in atomic orbits, see detailed description in Chapter 6, section 6.3. The energy-velocity Equations (138) and (139), identical to Equations (5) and (10) in Chapter 3, which were developed from the cosmic harmonic model pictured in Fig. 4, Chapter 2.

Fig. 20. Change in energy of an electron as a function of its change in velocity.

In summation, consider two observers, one at Earth and one outside the Universe. The outside observer will see our galaxy and Earth fall with an absolute velocity of c toward the center of the Universe. The outside observer will also see our galaxy being accelerated at a rate of

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a 0 toward the center of mass of the system. Contrary to the cosmic observer, the observer on Earth tends to see himself at rest relative to the bulk Universe. Both observers however, will find that potential energy of matter at Earth equals E0 = mc 2 .

9.2 The true Universe ?

What has been described so far is a Universe based on the harmonic model shown in Fig. 4, Chapter 2. This model is basically a mathematical model which appears to works well for a stationary observer here on Earth using standard physical units for energy, time, However, these units are not the same velocity and mass etc. everywhere in the Universe but will change drastically with location. Time flows slower at the Sun’s surface than here on Earth and faster on the planet Pluto. This means that fundamental constants such as the gravitational constant G and Planck’s radiation constant h, which both have physical dimensions involving time, are not the same everywhere and as a result the relationship between energy and velocity can not be the same at different locations in the Universe. The reason for this is that the cosmic gravitational tension φuniv = GM univ / Runiv , which determines the energy of matter, varies with the cosmic radius and can therefore not be the same everywhere. At our position x 0 in the Universe φuniv = c 2 and the potential energy of matter is E0 = m0φuniv . The fact that energy per mass , inertia of mass and consequently time (see section Chapter 2, section 2.4), change proportionally with tension makes it difficult to exactly evaluate physical events at other localities in the Universe, using the same standards for physical constants as here on Earth. Although the harmonic model of the Universe presented in this book, seems to function satisfactorily there are some questionable features

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which need to be addressed. For example, should we not be able to add velocities linearly in the same direction as we are accustomed to rather than by vector summation and is the edge or the end of the 1.6674 × 10 28 m as predicted by the Universe really at exactly harmonic model in Chapter 2? Absolute velocity and potential energy per mass of matter as a function cosmic radius predicted by the harmonic model are shown in Fig. 21. Absolute velocity and energy and absolute radius were obtained from the Equations in Fig. 4, Chapter 2. The diagrams also

Fig. 21. Calculated absolute velocity, observed velocity and energy per mass as a function of Cosmic radius using the standard mathematical model presented in previous chapters.

show the observed relativistic velocities Δv and ∇v as seen from our vantage point here on Earth, and which are also predicted by vector summation see Equations (137) and (139) and Einstein’s Equation (138). However, it does not seem natural that there should be two types of velocities, absolute velocity as predicted by the harmonic model and relativistic velocity according to Equations (137, 138, and 139). In the

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authors opinion velocities should add linearly in the same direction and by vector summation only if they point in different directions. I therefore believe, that in the relativistic energy-velocity Equations (137) and (139) and Einstein’s Equation (138) it is not the velocity but the energy that appears both as absolute and relative. In Chapter 3 section 3.1, it was shown that energies do not always add linearly as demonstrated by the examples of the tennis ball and rocket. Doubling the energy of a rocket at the launching pad does not make the rocket go twice as fast but increasing the energy in flight by two will double the velocity. Changing the diagram in Fig. 21 to reflect the idea that both relative and absolute energy can exist and how it will relate to the observed velocity is accomplished by mathematically replacing the components in Fig.21 and construct a new digram such as Fig. 22.

Fig. 22. The standard cosmic model modified to show both relative and absolute energy as related to observed velocity.

The Diagram in Fig. 22 presents a more sensible view of our Universe where relative energy is energy of matter as seen from our vantage point in space and absolute energy as seen from an observer

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outside the Universe. Both absolute and relative energy will equal E0 at our galactic reference point x 0 in space. The observed velocity at x 0 , or our velocity relative to the rest of the Universe, is c and increases or decreases by Δv or ∇v on either side of x 0 . The increase and decrease in observed velocity as a function of cosmic distance is not linear as suggested by Hubble’s law but follows a quadratic function. The other problem with the harmonic model presented in Fig. 4 is that at maximum amplitude A, the radius is precisely 1.6674 × 10 28 m where the gravitational tension; potential energy of mass; and length of time become infinite. This is purely a mathematical solution which stems from the fact that we only know the amount of mass in our Universe within our radius of x 0 and not how mass is distributed outside x 0 . A close examination of the diagram in Fig. 22, shows that relative energy of matter increases linear with radius up to our galactic position at x 0 . It seems likely then that relative energy should

Fig. 23. A more realistic description of the Universe. Velocity and potential energy of matter shown as a function of cosmic radius seen from our vantage point in space and expressed in Earth units.

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continue to increase in a linear fashion past our position x 0 to its maximum radius or amplitude A as shown by Fig. 23. Allowing the relative energy, or gravitational tension of the Universe, to increase linearly with its radius beyond x 0 means that the mass of the Universe must increase proportional to its radius squared ( M U ∝ RU2 ) which is in exact agreement with the Large Number Hypotheses and the Virial Theorem described in Chapter 8, see page 116 and 120. The increase in inertial mass as a function of radius squared hints to a Universe with a non-uniform mass density, unless pancake shaped, as suggested by several investigators. A pancake shaped or disk shaped Universe could possibly represent the ultimate structure of cosmos completing the hierarchical system from atoms, solar systems, galaxies, cluster of galaxies to a meta galaxy of near infinite size. The diagram in Fig. 23 portrays, in the author’s opinion, such a Universe in a authentic way. It shows our distance from the cosmic center and for comparison a distance of 15,000 Mpc centered around our galaxy. At the present time a distance of 15,000 Mpc is still far beyond the reach of our telescopes. The diagram in Fig. 24 zooms in on a small section of Fig. 23 spanning 400 Mpc along the cosmic radius in each direction of x 0 . Note that the observed velocity of nearby galaxies appear linear with distance but then further deviates with distance. This explains why the illusive “linear” Hubble’s law has never been established. In fact, several astronomers (A. Dressler (1987), (1994), Riess et al. (1996) and Perlmutter et al. (1998) ) have recently discovered that Hubble’s velocity-distance relationship is not linear but changes slightly with distance which they ascribe to a small amount of acceleration caused by some unknown force. The force is of course generated by the gravitational mass of the Universe and the observed acceleration is the cosmic acceleration a 0 (see Fig. 15. Chapter 7)

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Does the radius of the Universe and the gravitational tension and potential energy of matter increase forever? There must be a limit to the size and mass of the Universe otherwise the laws of physics break down. But if there is a limit to the Universe how do we describe empty space beyond the boundaries of our Universe? Empty space contains nothing and we cannot assign properties to nothing or nothingness. This is a difficult subject since it is practically impossible for most of us imagine empty space as nothing or something that does not exist. We seem to understand that we cannot visit or live within the boundaries of a country that does not exist but yet we seem to find it possible to visualize a boundless void outside our Universe where nothing exists, a place filled with an infinite amount of nothing!

Fig. 24. A small section along the radius of the Universe centered around our Galaxy showing the change in velocity, tension, time and potential energy of matter as a function of cosmic radius.

One point of view is that since space outside the Universe is filled with its gravitational field which decreases to zero at infinity, one could

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argue that the Universe together with its gravitational field is infinite and boundless. Alternatively, if the energy of the gravitational field is quantized and divided into a finite amount of small energy/mass packets, like sand pebbles on the beach, then one could expect the number of energy packets to eventually run out before reaching infinity, thus favoring a finite Universe.

9.3 Building Blocks of Nature Length: As mentioned in Chapter 1 the building blocks of nature are mass, charge, length and time or m, q, l and t. Length or distance is probably the one building block that is easiest to understand. Length however, has no physical significance unless joined by any of the other three building blocks. For example, length per time is velocity and mass per length determines the strength of gravitational tension ( φ = Gm / r ). Length × width is surface area and is an important spatial

dimension when dealing with pressure, temperature or radiation. Mass per unit surface area ( m / l 2 ) is often used as a measure of pressure although a more sophisticated term for pressure is newton per unit surface area ( m /(t 2l ) ). The surface temperature of a body is determined by the power radiated per unit surface area of the body or T 4 = L / Aσ

where σ is Stefan-Boltzmann’s constant. Length × width × height is volume or space and is also quite meaningless unless filled with gravitational tension, mass or charge. We can imagine empty space but it is doubtful it can be detected. It was believed that gravity could warp space or bend world lines. I do believe that light rays will bend inside a volume filled with a non-uniform gravitational tension and that measuring sticks can shrink or expand, but I do not believe that the elements of space itself “length × width × height” can change. The fact that light bends near gravitating bodies is, therefore, not due to warped space or bent world lines but is caused by the same effect that bends

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light in glass, namely Snell’s law. A peace of glass does not warp space or bend world lines, see Chapter 4, section 4.7. Mass: If the units of length, width and height do not change what about mass? In scientific terms mass is often referred to as either gravitational mass or inertial mass. Many are of the opinion that both are equal, which is a subject of debate. Let us follow the historical progress that led to the concept of inertial mass which starts with Galileo Galilei 1564-1642. It is said that Galileo obtained his ideas for his famed experiments while attending a church service during which he also observed and timed the swing of a chandelier hanging from the ceiling. One experiment that followed is here described in Galileo’s own words:

I had one ball of lead and one of cork, the lead ball being more than hundred times heavier than the one of cork, and suspended them from two equally long strings, about four or five bracchia in length. Pulling each ball away and releasing them at the same instant from their vertical point of rest, they fell along the circumferences of their circles having the strings as radii swinging back to near the same vertical height of origin and then returned along the same path. This free pendulum motion, which repeated itself more than hundred times, showed clearly that the heavy body kept time with the light body so well that neither in hundred swings, nor in thousand, will the former pass the latter by even an instant, so perfectly do they keep step. The experiment clearly showed that the pitch of a pendulum does not change with mass or weight even though gravity exerts a much stronger force on a heavier weight. The next test was to see whether a heavier weight would fall faster than a lighter weight. Galileo is said to have dropped different weights from the tower of Pisa, see Fig. 25, and found that they reached ground at the same time. In modern terms, two different weights are accelerated at exactly the same rate even if

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the Earth’s pull is stronger on the heavier mass. It is often said that since inertia of mass is the same as resistance to acceleration, then pull by the Earth’s although twice the mass means twice the gravitational field, the resistance to the pull will also double, thus canceling any effect of change in mass leaving the acceleration unchanged. This explanation is not quite right because the Earth’s gravitational field does not care about the mass of an object but bestows the same rate of acceleration on any object immersed in its field. This is simply

Fig. 25. Galileo’s experiment and discovery of acceleration at the tower of Pisa.

explained by the fact that the Earth’s acceleration g is solely determined by the gradient of the Earth’s own gravitational tension or g = φ Earth / REarth and not by the mass of the body being attracted to it. However, the view that the inertial force of a body (the force that resists acceleration) exactly balances the gravitational force that attracts it,

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did stick in the mind of scientists for a long time and has led to some questionable theories. One such theory concerns the “equivalence principle” which goes as far as to state that inertia of mass is the same as gravitational mass, since gravitational force cannot be distinguished from inertial force. The proof given is Einstein’s famous example of an observer inside a windowless elevator. Two conditions are considered, one where the elevator is stationary suspended by a cable in the gravitational field of a gravitating body such as the Earth, and the other where the lift is being pulled by the cable at a steady rate of acceleration far outside any gravitating body. In both cases the observer will feel his feet pressed to the floor and inside his windowless elevator the observer would supposedly not be able to tell whether he is subject to a gravitational force or an inertial force, since they both are considered equivalent. This is not quite right, because for one thing, the lines of force inside the elevator, when subject to a gravitational force are not parallel but always converge to a point which coincides with the center of mass of the gravitating body. When pulled at a steady accelerating rate, the lines of force are always parallel. Also, a steady change in clock rate and a change in the velocity of light will take place in the latter case caused by the steady increase in velocity and tension Δφ ≅ Δ(v 2 ) , see Appendix C. Another problem with the “equivalence principle” occurs when it is applied to the bending of light near gravitating bodies. Here the “equivalence principle”, which assumes that any substance (including photons) will accelerate towards a gravitational center at an equal rate regardless of mass (whether zero or infinite), predicts that a massless beam of light will accelerate and bend in the same manner that the path of a massive projectile will accelerate and bend when passing near a gravitational source. The angle of deflection is determined by Newton’s law of gravitation. In reality, massless light beams, contrary to massive objects, do not accelerate, they slow down and decelerate when entering a gravitational field and the bending of light in gravitational fields is better explained by Snell’s law (Chapter 4, section 4.7).


140

Our concept of mass is not very clear. How is gravitational mass m g related to inertial mass mi ?

In our frame of reference at Earth we

often define the mass of a body by its inertia or resistance to acceleration mi = F/a i.e. if a body accelerates by a = 1 meter per second per second when subject to a force F of one newton (nt) its mass is 1kg. We can, therefore, determine inertial mass from Newton’s laws of motion mi =

2E k , Newton v2

(m) (140)

where Ek is the kinetic energy involved in accelerating the body to a given velocity of v relative to us. There is one problem here, namely that at high relative velocities a relativistic increase in inertia of mass becomes notable. This relativistic mass increase, which was discovered by Einstein, makes Newton’s law obsolete so Equation (140) needs to be changed to mi =

Ek + E0 , c2

(m) (141)

Einstein

where E 0 = m0 c 2 is the mass equivalent energy of the body at rest relative to our frame of reference. Since E 0 and c 2 are constants and E k is a variable it means that mi must increase proportionally with E k . It is also important to remember that the rate of time in a system that has been accelerated changes proportionally with the energy of the system, see for example, Equation (1) and the “twin paradox” Chapter 2 section 2.4. Gravitational mass on the other hand is determined by Newton’s law of gravitation

mg =

Eg R GM

=

Eg

φ

, Newton

(m)

(142)

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where R is the distance of m g from the center of a gravitating mass M and E g is the energy required to move m g from R to infinity. The gravitational tension generated by M at R is φ . Since E g is directly proportional to φ it means that the gravitational mass mg will always remain constant. In contrast to inertial mass gravitational mass does not change with energy. However, the rate of time changes proportionally with gravitational tension. For example, we have seen that the rate of time is slower at the Sun’s surface than here on Earth due to the Sun’s higher gravitational tension. The result is that all physical processes on the Sun are slower, including processes involving acceleration. This slow-down of acceleration can be interpreted as an increase in inertia. This leads to the argument that the relativistic mass increase discussed above is really not an increase in mass but an increase in the length of time thus retarding or slowing down the rate of acceleration which we interpret as increase in inertia of mass. By the same token we can say that gravitational energy increases the length of time which is not reflected in Equation (142). Conclusion: Whether the tension of mass ( E / m ) is raised by an increase in velocity relative to our frame of reference or by an increase in a surrounding gravitational field mass stays constant but inertia will change due to the change in rate of time. Time: From the above it appears that out of the three main building blocks of nature length, mass and time, only time is a variable. A meter is always a meter and a kilogram is always a kilogram and will remain unchanged anywhere in Cosmos but the unit of time, the second, varies at different locations in the Universe. Most interestingly is that length of time is determined by the combination of mass and length (M/R) which is proportional to Tension and Energy. Time therefore must be related to Tension and Energy. The question is: Does tension or energy determine the rate of time or does the rate of time determine the amount of energy?

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APPENDIX A Constants and Measures

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APPENDIX A

Constants and measures 1383230 . × 10 −23 JK −1 λc 2.4263105822 × 10 −12 m a0 7.622470 × 10 −12 ms−2 AU 1.495979 × 1011 m M earth 5.976 × 1024 kg Rearth 6.37103 × 106 m N 17507 . × 1085 qe 1.60217607 × 10 −19 C re 2.8179409238 × 10 −15 m me 9109389754 . × 10 −31 kg E0 8187111216 . × 10 −14 J me / m p 5.4461701311 × 10-4 Elementary charge q 1.60217607 × 10 −19 C Fine-structure constant α 7.29735308 × 10−3 Frequency of Universe ν0 4.046645 × 10 −21 Hz Frequency of Universe ω0 2.542582 × 10 −20 rad / s Gravitational constant G 6.6445 × 10 −11 m3 kg-1s-2 Hubble’s time (period) t0 2.47118 × 1020 s Lightyear ly 9.460530 × 1015 m Mass density of Universe ρ 2.32273 × 10 −30 kg Microwave Backgrond Temperature T 2.766o Kelvin Parsec pc 3.085678 × 1016 m Permeability of vacuum μ0 1.2566370614 × 10 −6 NA -2 Permittivity of vacuum ε0 8.854187817 × 10−12 Fm-1 Planck’s constant h 6.62607554 × 10 −34 Js Proton mass mp 1.6726231 × 10 −27 kg Solar mass M sun 1.989 × 1030 kg Solar radius Rsun 6.9599 × 108 m Speed of light in vacuum c 2.99792458 × 108 ms −1 Stephan-Boltzmann constant (area) σ 5.6705119 × 10 −8 Wm −2 K −4 Stephan-Boltzmann constant (volume) a 3.565 × 10−15 J m − 3 K − 4 Time universal constant s 111265 . × 10 −17 s3 m −2 Universe (distance to center) Runiv 117908 . × 1028 m Universe (mass within Runiv ) M univ 159486 . × 1055 kg Year y 3.155692597 × 107 s Boltzmann constant Compton wavelength Cosmic acceleration Distance Sun-Earth Earth mass Earth radius Eddington’s magic number Electron charge Electron classical radius Electron mass Electron rest mass energy Electron-proton mass ratio

k

145

146


Absolute velocity Acceleration Distance from center of mass Force constant Absolute potential energy of matter Maximum amplitude Our distance to center of mass Angular frequency of Universe

v = Aω 0 sin α a = xω 20 x = a / ω 20 = A cos α k = ω 20 M E = E0 / ( 2 sin α ) (0o − 45o ) E = E0 2 cos α (45o − 90o ) A = 1.667 × 1028 m x 0 = 1179 . × 1028 m ω 0 = 2.543 × 10 −20 rad / s

APPENDIX B The Hydrogen Atom

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APPENDIX B

149

APPENDIX B THE HYDROGEN ATOM

The Hydrogen atom with its naturally fixed orbits has served as a testing ground for many theories. There is no doubt that standard quantum theory, based on Planck’s constant h as a constant of angular momentum together with Einstein’s relativistic velocity theory, has improved the accuracy in determining the position of atomic orbits but they still do not give exact results. Two reasons were given in Chapter 6 section 6.3 namely that Planck’s constant is not compatible with momentum since it is an energy constant and that Einstein’s relativity theory falls short in dealing with velocities created by loss of potential energy such as experienced by electrons in atomic orbits. The simple laws of the Deadbeat Universe, however, provide the right energy-velocity relationship for electrons in atoms, and as will be shown, it puts the orbits in their right places to a precision limited only by the accuracy of the physical constants involved in the calculations.

The Hydrogen Atom Atoms have been thought of as miniature solar systems ever since the beginning of the 20th century. It was Ernest Rutherford (1911) who proved this point of view to be true through scattering experiments but it remained a mystery why the atomic orbits could only exist in certain well defined radii from the central nucleus. A student of Rutherford’s, Niels Bohr, discovered that the well defined orbits seemed to appear in certain steps of the angular momentum h = mevr where h is Planck’s constant, me the electron’s mass, v the orbital velocity and r the radius of the orbit. Soon thereafter, a remarkable discovery was made by Lois de Broglie, which would revolutionize not only atomic physic but particle physics as well. Lois de Broglie showed that moving particles, such as electrons have wave properties similar to that of light rays and that their wavelengths, just like light, could be determined by Planck’s constant and from the particles momentum or λ = h /(mv) . de Broglie


150

also showed that orbits can only exist at atomic radii where the orbital circumference 2π r equals the electron’s particle wavelength in wave lengths of 1 2 λ , 1λ , 1 1 2 λ , 2λ , 2 1 2 λ , 3λ etc. This motivated the Austrian physicist Erwin Schrödinger and others to formulate orbital equations based on wave properties and statistics which is about as far as atomic physics has progressed today. Even if atomic physics has advanced significantly since Rutherford, Bohr and de Broglie together with Einstein’s relativity theory, the equations do not fit measurements too well. The problem is twofold. First, Planck’s constant is an energy constant and not compatible with momentum so de Broglie’s momentum equation λ = h /(mv) has to be replaced by the energy relation

λ = hv / E . This becomes clear if we compare λenergy

1 hv hv h = = ≠ λmomentum = 2 2 = . 1 mv ν E mv 2

v

(l)

(143)

were ν = E / h is the frequency of the particle associated with its energy. 2 1 We know that Newton’s energy relation 2 mv , from which the momentum is deduced, is not accurate at high relativistic velocities and therefore, any quantum theory based on particle momentum is doomed and cannot yield correct answers. The second problem is that we can not use Einstein’s relativistic velocity equations because orbits in atoms are created by loss of potential energy whereas Einstein’s relativity is only accurate for velocities generated by gain in potential energy as explained in Chapter 6 section 6.3. The purpose of this appendix is to show that by applying relativistic velocities generated by loss of energy, together with de Broglie’s wave theory based on energy rather than momentum (see the atomic orbit Equation (73) in Chapter 6 section 6.3), one is able to obtain accurate energies for the different orbits and wave lengths of n in the Hydrogen atom E orbit

⎡ 2 ⎞2 ⎤ ⎛ mn Zq ⎟ ⎥× = E 0 ⎢1 − 1 − ⎜⎜ ⎟ ⎥ (m + m ) , ⎢ n 4 ε h c 0 n e ⎝ ⎠ ⎥ ⎢⎣ ⎦

(144)

APPENDIX B

151

Orbital energies using Equation (144) are listed in Table 3 and also shown in the energy diagram of Fig. 26. Orbit

n=Wave length

Energy Joules

1

1

2

1

5.446719526 × 10 −19

3

1 12

2.42075986 9 × 10 −19

4

2

1.361676403 × 10 −19

5

2 12

8.714731931 × 10 −20

6

3

6.051891492 × 10 −20

7

3 12

4.446290966 × 10 −20

8

4

3.404188964 × 10 −20

2

2.178709658 × 10 −18

Table 3. The first eight orbits in Hydrogen and their energies.

Fig. 26. Energy diagram of the Hydrogen atom as predicted by the “Deadbeat Universe”.

An electron can fall from any orbit down to the ground state orbit, or orbit number 1, as shown by the arrows in Fig. 26. The orbiting


152

electron can also fall between different orbits which is not shown here. As the electron falls in to a closer orbit, potential energy is lost to radiation. The amount of energy escaped, in the form of radiation equals the energy difference between the orbits. In Fig. 26 the wavelength of the radiation liberated is listed next to each arrow. The spectrum of wavelengths listed is that of the Lyman series. One interesting phenomenon is the electron’s spin or rotation around its own axis. The spin will shift the orbital energies causing each orbit to split up into small fractions. This energy fragmentation can be seen in the emitted wavelengths and is called finestructure splitting. The effect of the electron’s spin is shown in Fig. 26 by the double line in orbit number 2 and is caused by tidal forces similar to that of the Earth’s tidal influence on the Moon. For example, consider the Moon being captured in its 28 day orbit around the Earth. At first the Moon would not rotate around its own axis and would show the same face relative to the fixed stars but it would appear to rotate once every 28 days relative to Earth. However, tidal forces between Earth and the Moon will soon bring this relative rotation to a stop so that the Moon now will show the same face to the Earth at all times but rotate once every 28 day around its axis relative to the fixed stars. The energy consumed in generating the Moon’s rotation or spin around its axis relative to the fixed stars will cause the Moon’s orbit to fall slightly closer to the Earth. An electron in orbit around the atomic nucleus will suffer the same fate and will eventually rotate with the same frequency around its axis as around its orbit. The energy ratio between the two rotations will therefore equal the ratio of the orbital radius to that of the electron’s radius R / re where R is the distance between the electron and the nucleus. The maximum loss of energy due to the electron’s spin in orbit 2, for example, in Hydrogen is therefore 2

E spin

2 n8π ε 0 E orb re ⎡ mn + me ⎤ nre E= = 7.2550833 × 10 −24 J , = ×⎢ ⎥ 2 R q ⎣ mn ⎦

(145)

where E orb is the electron’s energy in orbit 2 and n=1 is the wave number or wave length of orbit 2. E is the is combined orbital energy of both the electron and nucleus. The electron’s electromagnetic radius is re = q /( 4πε 0me c 2 ) and q is the elementary charge.

APPENDIX C

APPENDIX C The Problem with E = mc

2

154


APPENDIX C

155

APPENDIX C 2 THE PROBLEM WITH E = mc

In 1905 Einstein concluded that the energy of a mass m is equivalent to mass × velocity of light squared. According to the literature (Pais (1982)) Einstein’s proof is as follows: “If a body gives off the energy L in radiation, its mass diminishes by L / c 2 .” This links energy with the velocity of light or the propagation of electromagnetic waves, and the conclusion drawn was that energy of mass must be related to the speed of light so that L = mc 2 . Today, nearly 100 years later, the equation has seen no change except that the symbol for energy L is usually replaced by E. There has never been any doubt about the accuracy of the equation as long as the speed of light remains constant. However, recent advances in space science and satellite technology suggest that there are changes in both the speed of light and the rate of time, which poses the question: is energy of mass really proportional to the speed of light squared and how can we test the validity of Einstein’s proof? This essay provides such a test, and the result is quite contradictory since it will show that energy of mass in Einstein’s equation is not proportional to the velocity of light squared, but is proportional to a universal gravitational potential or gravitational tension φ univ .

The Problem with E = mc 2 Einstein’s equation E = mc 2 suggests that energy stored in mass is proportional to the velocity of light squared. In fact, the opposite is true. Energy stored in mass is inversely proportional to the velocity of light squared. For example, if energy of mass is proportional to the velocity of light squared, then energy of mass will be less in situations where velocity of light is slower, e.g. at the surface of the Sun or near a

156


massive black hole where light is believed to slow down to zero preventing it from escaping the black hole’s strong gravitational field. This is an example of how mathematics can sometimes give a correct numerical answer but a misleading physical picture. The problem is that c 2 in Einstein’s E = mc 2 has always been associated with the speed of light, when in reality c 2 in Einstein’s equation has a different meaning than the speed of light. Although the numerical value and the physical dimensions of c 2 are the same as those of the speed of light squared, its function is quite different. For example, the distance between New York and Albany has a totally different meaning from the same distance squared, which describes surface area. Another example is the amount of potential energy per unit mass required to lift a mass to a certain height h above the Earth’s surface under the influence of the Earth’s gravitational acceleration g

E / m = gh (meter/second) 2 ,

( l 2 / t 2 ) (146)

where the answer is velocity squared, even though the physical process described does not involve velocity. In reality, c 2 in Einstein’s equation and velocity squared in the above equation represent gravitational potential or gravitational tension φ . Einstein’s energy equation should therefore be written as

E = mφuniv , where φuniv = GM univ / Runiv ≡ c 2 .

(147)

Here φuniv is the cosmic gravitational tension, or the amount of Energy per unit mass E/m generated by the Universe (Wåhlin ( 2002)) at Runiv , the distance to Earth from the center of mass of the Universe, and M univ is the mass of the Universe within Runiv . G is the universal gravitational constant. Note that φuniv includes both the Earth’s and the Sun’s gravitational tension a location on Earth. The Sun’s gravitational tension at Earth is 100 million times weaker than φuniv and the Earth’s own gravitational tension at its surface is an additional 14 times less. The cosmic gravitational tension φuniv

APPENDIX C

157

determines the rate of time. The gravitational tension φuniv also establishes the speed of light and therefore acts as a propagation medium for electromagnetic waves which means that any change in the cosmic gravitational tension changes the speed of light. The ratio of change in time and rate of clocks and physical processes is

φuniv

and

φuniv + Δφ

φuniv

φuniv − ∇φ

,

(148)

where Δφ represents an increase in gravitational tension and ∇φ a decrease in gravitational tension relative to an observer. For example, when a clock progresses one second on Earth the same clock on the Sun’s surface would read tsun = tearth

φuniv

φuniv + Δφ

= 0.99999788 sec ,

(149)

where Δφ = Gmsun / rsun is the gravitational tension of the Sun added to φuniv and neglecting the influence of the Earth’s gravitational tension. The slowdown of solar time has been measured by Snider (1972). A reduction in gravitational tension, on the other hand, will speed up clocks or physical processes and the propagation of light. A clock raised to any height above ground would therefore run faster because of the decrease in the Earth’s gravitational tension ∇φ with altitude. The rate of a clock at height h above ground relative to a clock on the Earth’s surface is theight = tearth

φuniv

φuniv − ∇φ

,

(150)

which has been verified by many experiments (Pound and Rebka (1960)) including the Mössbauer effect involving very sensitive measurements of atomic frequency spectra. However, the change in the propagation of light caused by changes in the cosmic gravitational tension is twofold. First, light is subject to the same slowdown in time as observed for clocks and physical processes described above. Second,


158

it is also subject to the change in the tension of the propagating medium. The result is that the speed of light is affected twice, i.e. vlight

⎛ φuniv ⎞ ⎟⎟ = c⎜⎜ φ φ + Δ ⎠ ⎝ univ

2

2

and vlight

⎛ φuniv ⎞ ⎟⎟ . = c⎜⎜ φ φ − ∇ ⎠ ⎝ univ

(151)

These equations are in agreement with experimental data. One such experiment involved the timing of radar waves bouncing off Venus (Shapiro (1971)) while crossing the gravitational field of the Sun. The change in velocity and time by gravitational fields is often referred to as gravitational red shifts when generated by Δφ and blue shifts when produced by ∇φ . The velocity of light remains constant whenever the cosmic gravitational tension φuniv is constant. Since gravitational tension φ represents energy per unit mass, there are other processes that can increase or decrease the energy of mass, e.g. kinetic energy. A fast-moving jet aircraft will generate a level of energy per unit mass of v 2 , where v is velocity of the jet. This will raise the gravitational tension of the jet aircraft by 12 v 2 , (nonrelativistic) thus slowing clock s and physical processes accordingly, so that the clock rate onboard the jet would be t jet = tearth

φuniv

φuniv + 12 v 2

. (non-relativistic)

(152)

This is true only if the change in gravitational tension due to the jet’s altitude is not considered. Experiments involving clocks on board jet aircraft (Hafele (1972)) verify Equation (152). Also, an increase in the half-life of decaying cosmic particles has been observed, which has been attributed to the high velocities of the particles as they enter the Earth’s atmosphere. The time change due to velocity is known as time dilation.

APPENDIX C

159

Orbiting satellites are subject to both time dilation and gravitational blue shifts, which compete against each other, as shown in the following equation tsat = tearth

φuniv

φuniv . (non-relativistic) − ∇φ + 12 v 2

(153)

Both ∇φ and v 2 can vary due to the position of the satellite relative to the Sun’s, Moon’s and Earth’s gravitational tensions. The orbital velocity v of a satellite is determined relative to the fixed stars. Clocks on a GPS satellite (Ashby (2003)) orbiting at an altitude of 26500 km will run faster by about 4.92 × 10 −10 second per second and clocks onboard the space station, which orbits at the much lower altitude of 380 km above Earth, will run slower by 2.88 × 10 −10 second per second compared with clocks on the Earth’s surface, ignoring the effect on clocks on Earth caused by the Earth’s own rotational velocity. For relativistic velocities we need to replace the Newtonian 12 v 2 with

φrel = φuniv

φuniv

φuniv − 12 v 2

− φuniv

(154)

Another interesting consequence of Equation (153) is Einstein’s twin paradox, where a twin traveling in a space ship will age more slowly than the twin remaining on Earth, so that on the return from the voyage the returning twin will be younger than the one on Earth. Here ∇φ in Equation (153) can be replaced by the square of the escape 2 velocity ( vesc = Gmearth / rearth ) required for the space ship to leave the Earth's gravitational field and v 2 equals the square of the velocity of the space ship when traveling through space.

Conclusions

Replacing velocity of light squared in Einstein’s equation E = mc 2 with φuniv has several implications, two of which are mentioned below.


160

The slow-down of light in strong gravitational fields according to Equation (151) offers a simple explanation for the bending of light near gravitating bodies such as the Sun. The index of refraction n according to Snell’s law or René Decartes’ law is n=

Δv Δ(sin α ) c sin α ' = and = c sin α ' v sin α "

(155)

where c and v represent the speed of the incoming and retarded light respectively and α ' and α " are the angles of incidence and refraction respectively. Since the mean incident angle of light penetrating the Sun’s gravitational equal-potentials is 45° then the total angle of refraction becomes

α bend = 2{α '− sin −1 [sin α '−Δ(sin α ]} = 1.75053023 arc second

(156)

for light grazing the Sun’s surface. The factor “2” is necessary since light has to pass through two refractive indices, one at the entrance to and one at the exit from the gravitational field. The outcome of Equation (151) also casts serious doubt on the existence of black holes, since the equation shows that light cannot slow to zero in order to be prevented from leaving a black hole’s gravitational field no matter how strong the gravitational field.

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INDEX

INDEX A

Acceleration, 1,7 Acceleration, cosmic, 26,27,42,145 Allen, 70 Ampère's equation, 107 Anaximander, 2 Anaximines, 2 Angular frequency, 66 Archimedes, 4 Aristarchus, 2,6 Aristotle, 2,3 Ashby, 159 B Background radiation, 51 Bagge, 71 Bending of light, 13,59,139,159 Berkeley, George (Bishop), 11,13 Big Bang, 14,16,18,29,50,96 Birch, 67 Black holes, 64,91 Black-body radiation, 16,99,100 Black-body temperature, 72,99,28 Blue shift, 18,59,99 Boltzmann’s constant, 87,145 Bondi, 29,101 Bremsstrahlung, 46 Burbidge, 51,53 C Cavendish, Sir Henry, 59 Centrifugal force, 11,13,22 Charlier, 33 Clausius, 113 Collapsing Universe, 2,18,29,38, 41,46,48,68,83,86,96,101 Columbus, 4. Coma cluster, 43,46,115 Compton red-shift, 86, Compton wavelength, 44,145 Continuos creation, 29,101 Copernicus, 3,5,6,7,79 Cosmological Principle, 2,5,17 Crommelin, 60 Curvature of space, 13,105,115,117

166

D

Davisson, 83 Deceleation parameter, 50 de Broglie, Loius, 82,83,150 de Vaucouleurs, 46,97 Descartes, 8,61,160 Desclaux, 86 Digges, Thomas, 5 Dirac, P.A.M., 86,108 Dirac-Fock correction, 86 Doppler shift, 14,43,73,93-96 E Ether, 10,12,57,89,121 Eddington’s magic number, 104, 106,145 Eddington, 60,104,106,107,108, 118,119 Einstein, Albert, 12-16,2932,37,39,47,59,84-86,104, 115,122,138 Electron, 9 Electron's electromagnetic radius, 46,103,145 Emissivity, 71 Energy self enegy, 55,58,77,86,104 Energy density of matter, 100 Energy density of radiation, 29,100 Energy, 1,15,35 Energy, kineteic, 31,35,113,123, 128, 139 Energy, potential energy, 24,28,38, 42,54,58 Eratosthenes, 4 Equivqlence principle, 138 F Fitzgerald, 122 Fixed stars, 11,13 Force constant, 67 Force, 1,9 Fujii, 107

167

THE DEADBEAT UNIVERSE G

Galileo, 7,136 General Relatively, 12, 60 Germer, 83 Gilbert, William, 9 Giordano Bruno, 5 God, 5 Gold, 29,101 Gravitational constant, 10,28,32, 105,107,113,130,145 Gravitational potential, 56 Gravitational red-shift, 93 Gravitational tension, 13,31,49, 56,64,87,90,122,135,140,156 H

Hafele, 158 Harmonic motion, 21 Heisenberg, 15,26 heliocentric, 2,8 Heraclides, 2 Hipparchus, 4 Hooke, Robert, 9 Hoskin, M., 6 Hoyle, Sir F., 29,101 Hubble’s law, 43,51,105,134 Hubble, Edvin, 14,42 Huchra, 70 I Inertia, 8,11,30,31 Infinity, 5,13,16,17 Inhalation, 112

Jupiter, 10,56

J K

Källen, 86 Karachentsev, 46 Kazuaki Kuroda, 107 Kembhavi, 101 Kepler, 7,9,18,19,65,87 Kiang, 33

L

Landsberg, 73 Lanzerotti, 71 Large number ratios, 103 Lebach, 63 Light year, 42,145 Long, 107 Lorentz, 12,122 Lorentz-Fitzgerald conraction, 122 Lovell, 3 M Mach, Ernst, 11,13 Mach’s principle, 11 Mass density, 27,50,54,89,100, 105,113,117,120,134,145 Maxwell, 89,119 Michelson, 12,121 Michelson-Morley experiment, 12,121 Microwave radiation, 16 Missing mass problem, 50,53,115 Mohr, 86 Momentum, 15,44,83 Morley, 12,121 Munitz, 4 Mössbauer effect, 157 N Narlikar, 51,101 Newton, Isaac, 9,31,35,39,59, 84,85,110,121 Nicholas (Cardinal) of Cusa, 5,13,17 O Olbers’ paradox, 99 Olbers, Heinrich, Wilhelm, 99 Ozernoy, 46 P Pair production, 29,101 Pais, 155

INDEX Penzias, 16,51 Photon, 87 Plank’s constant, 15,28,44, 46,80,145 Planck, Max, 15,74 Plank’s wavelength, 44 Poincaré, 12 Posidonius, 4 Pound, 157 Q Quantum, 80 Quantized, 70 R

168

T Temperature, 16,27,72 Thales, 2 Thomson, Sir, G.P., 83 Time dilation, 30,32,94,158 Time universal constant s, 32,145 Time, 30 Twin Paradox, 32 U

Uehling, 86 Uncertainty Principle, 16,26

Radian, 22,66 Radiation, 79 Radio polarization, 67 Recession velocities, 16,38,43,49, 96 Red shift, 14,16,26,28,38,42, 59,86,93 Ritz, 122 Rutherford, 149

V Vacuum polarization, 86 van den Bergh, 46 velocity 1,22,26,35 Velocity of light, 12,89,145 Velocity of ligth in gravitational field, 63 Virgo cluster, 97 Virial theorem, 53,113

S Saslaw, 33 Satelite, 158 Segal, 46 Shapiro, 158 Simple harmonic motion, 21 Slipher, 42 Smoot, 73 Snell’s law, 61, 159 Snider, 157 Soldner von, Johann Georg, 59 Soneira, 46 Space station, 159 Special Relativity, 12,39 Spin, 152 Stacey, 107 Star aberration, 10,64 Stefan's law, 72, 87 Stephan Boltzmann's constant, 52,72,87,118,136,145 Stewart, John, 104,118

W Wå hlin, 18,27,43,156 Wavelength of particles, 82 Weight, 8,30 Weyl, Herman, 103, Whittaker, 119 Will, 59 Wilson, 16, 51 X-ray, 46

X

Z Zero point energy, 46,70 Zwicky, F, 48, 115

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