The Einstein Theory of Relativity

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DO

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marke4

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THE EINSTEIN

THEORY

OF RELATIVITY

Boofcs

by L

/?.

ancf H. G. Lieber

NON-EUCLIDEAN GEOMETRY GALOIS AND THE THEORY OF GROUPS THE EDUCATION OF

MITS

C.

T.

THE EINSTEIN THEORY OF RELATIVITY MITS,

WITS

AND LOGIC

INFINITY

Books of drawings by H. G. Lieber

GOODBYE

MR.

MAN, HELLO MR. NEWMAN

(WITH INTRODUCTION BY

L.

R.

LIEBER)

COMEDIE INTERNATIONALE

THE EINSTEIN

THEORY

OF RELATIVITY Text By

LILLIAN

R.

LIEBER

Drawings By

HUGH GRAY

LIEBER

HOLT, RINEHART

New

AND WINSTON

York / Chicago / San Francisco

Copyright, 1936, 1945, by

L.

R.

and

H. G. Lieber

All rights reserved, including the right to repro-

duce In

of

this

book or portions thereof in any form. Holt, Rinehart and Winston

Canada, Canada,

Limited.

First Edition

October 1945 Second Printing, April 1946 Third Printing, November 1946 Fourth Printing, November 1947 Fifth Printing, May 1950 S/xtfi Printing, March 1954 Seventh Printing, November 1 957 Eighth Printing, July 1958 Ninth Printing, July 1959 Tenth Printing, April 1960 Eleventh Printing, April 1961 Twelfth Printing, April 1964 Thirteenth Printing, November 1966 Firsf Printing,

1975

85251-0115 Printed in the United States of America

To

FRANKLIN DELANO ROOSEVELT who saved

the world from those forces

of evil which sought to destroy

Art and Science and the very Dignity of

Man.

PREFACE In

this

book on

the Einstein Theory of Relativity

the attempt is made to introduce just enough

mathematics to

HELP

NOT to HINDER

and

the lay reader/ "lay" can of course apply to various domains of knowledge

perhaps then

we should

say:

the layman in Relativity.

Many

"popular" discussions of

Relativity,

without any mathematics at have been written. But we doubt whether

all,

even the best of these can possibly give to a novice an adequate idea of

what

What in

it is

all

about.

very clear when expressed mathematical language is

sounds "mystical"

in

ordinary language.

On

the other hand,

there are

many

discussions,

including Einstein's own papers, which are accessible to the

experts only. vii

We

believe that

there

is

a class of readers

who

can get very little out of either of these two kinds of discussion readers who

know enough about

mathematics to follow a simple mathematical presentation of a domain new to them, built from the ground up, with sufficient details to bridge the gaps that exist

FOR THEM

in

both

the popular and the expert presentations. This book is an attempt to satisfy the needs of this

kind of reader.

viii

CONTENTS PREFACE Part

I.

II.

III.

I

-THE

SPECIAL THEORY

INTRODUCTION

3

The Michelson-Morley Experiment

8

IV. The

V. The

Remedy

31

Solution of the Difficulty

39

VI. The Result of Applying the VII.

VIII.

20

Re-Examination of the Fundamental Ideas

The

Four-Dimensional tinuum

Some Consequences

44

Remedy

Space-Time

Con-

57

of the Theory of

69

Relativity

IX.

A

Point of Logic and a

83

Summary

87

The Moral Part

A

II

-THE GENERAL THEORY

GUIDE TO PART

91

II

95

X. Introduction

101

XI. The Principle of Equivalence XII. XIII.

A

107

Non-Euclidean World!

113

The Study of Spaces

XIV. What

XV. The

Is

127

a Tensor?

Effect

on Tensors

of a

Change

Coordinate System

XVI.

A Very Helpful Simplification ix

in the 1

41

150

160

XVII. Operations with TenseXVIII.

A Physical

167

Illustration

XIX. Mixed Tensors

XX.

Contraction and Differentiation

XXI. The XXII. XXIII.

Our

Last Detour

200

at Last

the Curvature Tensor?

206

Einstein's

Law

of Gravitation

21 3

of Einstein's

Law

of Gravitation

Is

The Big G's or

219

with Newton's

XXVII.

How

Can the

Einstein

Law

of Gravitation Be

227

Tested?

XXVIII. Surmounting the

XXIX. "The Proof

Difficulties

237

Pudding"

255

the Path of a Planet

266

of the

XXX. More About

XXXI. The Perihelion

272

of Mercury

XXXII.

Deflection of a

Ray

XXXIII.

Deflection of a

Ray of

XXXIV. The

78

191

The Curvature Tensor

XXVI. Comparison

1

187

Little g's

XXIV. Of What Use

XXV.

173

of Light Light, cont.

Third of the "Crucial"

Phenomena

XXXV. Summary

283

289 299 303

The Moral

Would You Like

THE ATOMIC

276

to

Know?

BOMB

310 318

Parti

THE SPECIAL THEORY

INTRODUCTION.

I.

In order to appreciate the fundamental importance

of Relativity, it is

how

necessary to it

Whenever in

know

arose. 11

a "revolution

takes place,

any domain, always preceded by

it is

some maladjustment producing

a tension,

which ultimately causes a break, followed by a greater stability at least for the

time being.

What was

the maladjustment in Physics the latter part of the 19th century, which led to the creation of in

11

the "revolutionary

Let us summarize It

Relativity

it

Theory?

briefly:

has been assumed that

space is filled with ether,* through which radio waves and all

light

waves

are transmitted

any modern child *This ether

is

talks quite glibly

of course

NOT the

chemical ether

which surgeons use! ft is it

not a liquid, solid, or gas,

has never been seen

by anybody,

presence is only conjectured because of the need for some medium to transmit radio and light waves. its

3

11

about "wave-lengths in connection with the radio.

Now,

there

if

is

an ether,

does it surround the earth and travel with it, or does it remain stationary while the earth travels through it?

Various known (acts* indicate that the ether does

NOT travel

with the earth.

THROUGH

the earth is moving If, then, there must be an "ether wind/'

person riding on a bicycle

just as a

through feels an

still air,

air

wind blowing

in his face.

And

so an experiment was performed and Morley (see p. 8) Michelson by in 1887, to detect this ether wind/and much to the surprise of everyone, no ether wind was observed.

unexpected result was explained by Dutch physicist, Lorentz, in 1 895, in a way which will be described

This a

in

Chapter

II.

The search for the ether wind was then resumed by means of other kinds of experiments.! *See the

by A. in

M article

Aberration of Light",

S.

Eddington, the Encyclopedia Britannica, 14th ed.

tSec the article "Relativity" by James Jeans, also in the Enc. Brit. 14th ed.

4

the ether,

But, again and again, to the consternation of the physicists, no ether wind could be detected,' until

it

seemed

that 1

nature was in a "conspiracy' to prevent our finding this effect!

At

this

point

Einstein took

and decided

up the problem, that 11

a natural "conspiracy must be a natural

LAW

And

operating. to answer the question

what is he proposed

as to

his Theory of two papers, 1905 and the other

in

Relativity,

in

published

one

this law,

in

1915.*

He first found it necessary to re-examine the fundamental ideas upon which classical physics was based, and proposed certain vital changes in them. He

then

made

A

VERY LIMITED NUMBER OF MOST REASONABLE ASSUMPTIONS

from which he deduced his theory. So fruitful did his analysis prove to be that

by means

of

it

he succeeded

in:

(1) Clearing up the fundamental ideas. (2) Explaining the Michelson-Morley experiment in a much more rational way than

had previously been done. *Both

now

published

in

one volume

including also the papers

by

Lorentz and Minkowski, to which

see

we

SOME

shall refer later/

INTERESTING READING, page 324.

Doing away with

(3)

other outstanding difficulties in physics. (4) Deriving a

NEW LAW OF GRAVITATION

much more adequate than the Newtonian one (See Part II.: The General Theory) and which led to several important predictions

which could be verified by experiment; and which have been so verified since then.

(5) Explaining

QUITE INCIDENTALLY a famous discrepancy in astronomy which had worried the astronomers for

many

(This also

years is discussed in

The General Theory). Thus, the Theory of Relativity had a profound philosophical bearing of physics, on as well as explaining

ALL

many SPECIFIC that

outstanding difficulties

had seemed to be entirely

UNRELATED, and

of further increasing our of the physical world by suggesting a number of

NEW experiments which NEW discoveries. No

knowledge

have led to

other physical theory

been so powerful though based on so FEW assumptions. has

As we

shall see.

THE MICHELSON-MORLEY EXPERIMENT*

II.

On

page 4 we referred to

the problem that

Michelson and Morley Let us

now

set themselves.

see

what experiment they performed and what was the startling result. In

order to get the idea of the experiment

very clearly it

will

in

mind,

be helpful

first

to consider the following simple problem,

which can be solved

by any boy who

has studied

elementary algebra:

Imagine a river in which there is a current flowing with velocity v, in the direction indicated

by the

Now

which would take longer

for a

man

From

A

to

to

'Published

swim

B and back in

arrow:

to

A

,

the

Philosophical Magazine, vol. 24, (1887).

8

or

from if

A

C and back

to

the distances

AB

to

A

,

AB and AC are

equal,

being parallel to the current,

AC perpendicular to it? Let the man's rate of swimming

and

in still

be represented by c / then, when swimming against the from

A

his rate

8

to

water

current,

,

would be only

v

c

f

whereas,

when swimming with from 8 back to A , his rate

the current,

+

would, of course, be c

v.

Therefore the time required to swim from A to fi

would be a/(c v), where a represents the distance and the time required for the trip

from

would be a/(c

8

+

to

A

v).

Consequently, the time for the round

ti

or

Now

ti

let us

= =

trip

-

a/(c

2

2ac/(c

would be

/)

-

+ v

a/(c

+

v)

2

).

see

how

long the round

from

A

to

AB ;

trip

C and back

to

A

would take. If he headed directly toward C , the current would carry him downstream, and he would land at some point to the left of C in the figure on p. 8. Therefore, in order to arrive at

C

,

9

he should head just far

for

some point

D

enough upstream

to counteract the effect of the current.

In

other words, the water could

be kept

until

he swam

own

from

A

if

to

D

at his

still

rate c

,

and then the current were suddenly allowed to operate, carrying him at the rate v from D to C (without his making any further effort), then the effect would obviously be the same as his going directly from

A

iojC 2

2

with a velocity equal to Vc' v/ as is obvious from the right triangle:

ex

\r

Consequently/ the time required for the journey from

A

to

C

2

would be a/Vc^- v , where a is the distance from Similarly, in

going back from C to easy to see that,

A,

it is

10

A

to

C

method

the same

by

of reasoning, 2

_

the time would again be a/Vc Hence the time for the round trip

from

A

to

C and back

would be fa

In

let us write ft for

Then we

get: ti

and

fa

=

c/

ti

2

.

_ A,

to

= 2a/vV -

order to compare

v

and

Vc

f-

2

v

y\

more

easily,

2 .

2

2a/3

/c

2a/3/c.

Assuming that v is less than c , 2 2 v being obviously less than and c

Vc

the

2

v

2

is

therefore less than c

and consequently ft is greater than (since the denominator is

less than

Therefore that IT

t\

the numerator). is greater than

1

c

2

,

,

^

fa ,

is,

TAKES LONGER TO

SWIM UPSTREAM AND BACK

THAN TO SWIM THE SAME DISTANCE ACROSS-STREAM

AND

BACK.

But what has all this to do with the Michelson-Morley experiment? In that experiment, to B: a ray of light was sent from

A

C-r

^-

HB 11

At 8 there was a mirror which reflected the light back to so that the ray of light makes the round trip from just as the

A,

AioB and

back,

swimmer did

the problem described above. since the entire apparatus shares the motion of the earth, in

Now,

which is moving through space, supposedly through a stationary ether/ thus creating an ether wind in the opposite direction, (namely, the direction indicated above), this experiment seems entirely analogous to the problem of the swimmer.

And,

therefore/ as before/ ti

and

ti

= 2a0Yc = 2a|S/c.

0) (2)

is now the velocity of light, the time required for the light to C and back to to go from

Where

and

c

*2 is

A

A

(being reflected from another mirror at If/ ti

Q.

therefore, and t> are found experimentally/

then

by dividing (1) the value of /? would

And

since

by (2), be easily obtained.

= c/V c

2

-T

2 ,

c being the known velocity of light, the value of v could be calculated.

That

is,

THE ABSOLUTE VELOCITY

OF THE EARTH

would thus become known. Such was the plan of the experiment.

Now

what actually happened?

12

The experimental values of t\ and were found to be the SAME, instead of

ti

ti

being greater than ti was a most disturbing \

this

Obviously

result,

quite out of harmony with the reasoning given above.

The Dutch

physicist, Lorentz, then suggested the following explanation of Michelson's strange result:

Lorentz suggested that matter,

owing

and

this

to

its

WHEN

SHRINKS

electrical structure, IT

IS

MOVING,

contraction occurs

ONLY IN THE DIRECTION OF of shrinkage The he assumes to be in the ratio of 1/ff

MOTION.*

AMOUNT

2

v (where /3 has the value c/Vc Thus a sphere of one inch radius ellipsoid when shortest semi-axis

becomes an with

its

(now only

it is

2

,

as before).

moving,

1//3 inches long)

*The two papers by Lorentz on this subject are included in the volume mentioned in the footnote on page 5. In the first of these papers Lorentz mentions that the explanation proposed here occurred also to Fitzgerald.

Hence

it

is

often referred to as

the "Fitzgerald contraction" or the "Lorentz contraction* or 1

11

the "Lorentz-Fitzgerald contraction.

13

in

the direction of motion,

thus:

a erection

Applying this idea Michelson-Morlcy experiment,

to the

the distance

AB (=

on

a)

becomes a/jS , and ti becomes 2a/3/c

p. 8,

/

2

instead of 2a/3 /c , so that now ft t2

=

just as the

One

,

experiment requires.

might ask

how

it is

that Michelson did not

observe the shrinkage? Why did not his measurements show that

AB was

shorter than

AC

(See the figure on p. 8)? The obvious answer is that the measuring rod itself contracts

when applied to AB, so that one is not aware of the shrinkage. To

this

explanation

of the

Michelson-Morley experiment the natural objection may be raised that an explanation for the express

which

is

purpose

14

invented

of smoothing out a certain and assumes a correction of is

JUST too

And

difficulty,

the right amount, to be satisfying.

artificial

Poincare, the French mathematician/

raised this very natural objection.

Consequently, Lorentz undertook to examine his contraction

hypothesis

other connections, to see whether it is in harmony also with facts other than in

the Michelson-Morley experiment. He then published a second paper

in

1904,

giving the result of this investigation.

To present let us

first

this result in a clear

re-state the

form

argument

as follows:

vt

T

Consider a set of axes, X and Y, supposed to be fixed in the stationary ether, and another set X' and Y' , attached to the earth and moving with it,

15

with velocity v

X

Let

and

7

V"

Now

,

as indicated

move along X, move parallel to

above

V.

suppose an observer on the

earth,

say Michelson, is

trying to

the time

measure

takes a ray of light to B , to travel from it

A

A

and 8 being fixed points on the moving axis X . both

r

At

the

moment

when the

ray of light

starts at

A

f

that

Y and Y

suppose coincide, and A coincides with D / and while the light has been traveling to B the axis

V

moved

has

the distance vt

,

and B has reached the position shown in the figure on p. 1 5, t

being the time

it

takes for this to happen.

DB = x and AB =

Then, we have x' if

=

x

x',

vt.

(3)

only another way of expressing what was said on p. where the time for

This

is

first part of the journey was said to be equal to a/(c

9

the

And, as we saw there, this way of thinking of did

NOT agree

Applying now

the

v).*

phenomenon

with the experimental facts. the contraction hypothesis

we are now designating a by x', = t , or x = ct we have x'/(c v)

*Since

vf.

But the distance the light has traveled is x , and x

=

ct,

consequently x'

=x-

vt

is

16

equivalent to a/(c

v)

=

t.

proposed by Lorentz, x should be divided by /3, so that equation (3) becomes r

x7/3 or

x'

Now

- vt - vt). (x

=

x

=

when Lorentz examined

as stated

on

1

p.

(4)

other fads,

5,

he found that equation (4) was quite in harmony with ail these but that he was now obliged

facts,

to introduce a further correction

expressed by the equation (5)

where /3 , t , v , x , and c have the same meaning as before But what is t?! Surely the time measurements in the two systems are not different:

Whether the

origin

is

at

D

or at

A

should not affect the

TIME-READINGS. In t'

other words, as Lorentz saw it, 11 sort of "artificial time

was a

introduced only for mathematical reasons,

because in

it

helped to give

harmony with the

But obviously

NO As

had

t

results

facts.

for

him

PHYSICAL MEANING.

Jeans, the English physicist, puts

"If the observer

it:

could be persuaded

to measure time in this

artificial way, wrong to begin with and then making them gain or lose permanently, the effect of his supposed artificiality

setting his clocks

17

would

just

counterbalance

the effects of his motion 11

through the ether

!*

Thus, the equations finally proposed

by Lorentz

are:

Note

x'

=

z'

=

(x

-

vt)

that

since the axes attached to the earth (p. are moving along the X-axis,

1

5)

obviously the values of y and z (z

being the third dimension)

are the

same

The equations are

known

/

and z

just

given

as

,

respectively.

as

THE LORENTZ TRANSFORMATION, since they show how to transform a set of values of x , y , z , t into a set x',

y, z,

t'

coordinate system moving with constant velocity in a

v,

along the X-axis, with respect to the

unprimed coordinate system.

And,

as

we

saw,

whereas the Lorentz transformation really expressed the facts correctly, it seemed to have

NO

PHYSICAL MEANING,

*See the

article

on

Relativity in the

Encyclopedia Britannica, 14th edition.

19

and was merely a set of empirical equations.

Let us

now

see what Einstein did.

RE-EXAMINATION OF THE

III.

FUNDAMENTAL

IDEAS.

As Einstein regarded the situation, the negative result of the Michelson-Morley experiment, as well as of other experiments

which seemed to indicate a "conspiracy" on the part of nature against man's efforts to obtain knowledge of the physical world (see p. 5),

these negative results,

according to Einstein, did not merely demand explanations of a certain number of isolated difficulties,

but the situation was so serious that a

complete examination

of fundamental ideas

was necessary. In

he

other words, felt that there was something

fundamentally and radically wrong in

physics,

rather than a

And

mere

superficial difficulty.

so he undertook to re-examine

such fundamental notions as

our ideas of

LENGTH

and

TIME and MASS.

His exceedingly reasonable examination

20

is

most illuminating,

as

we

But

shall

first

why

now

let us

see.

remind the reader

and mass

length, time

are fundamental,

Everyone knows

that

VELOCITY depends upon the distance (or LENGTH) traversed in a given

TIME,

hence the unit of velocity

DEPENDS UPON the units of

LENGTH

and TIME.

Similarly,

since acceleration

is

the change in velocity in a unit of time, hence the unit of acceleration

DEPENDS UPON the units of velocity and time, and therefore ultimately upon the units of

LENGTH

and TIME.

Further,

since force

is

measured

by the product

of

mass and acceleration, the unit of force

DEPENDS UPON the units of mass and acceleration,

and hence ultimately the units of

MASS, LENGTH And so on. In all

upon

and TIME,

other words,

measurements

depend

in

MASS, LENGTH That

is

physics

primarily on

and TIME.

why 21

the system of units ordinarily used is called the "C.G.S."

where C stands

for

system,^ "centimeter"

(the unit of length), stands for "gram" (the unit of mass), and 5 stands for "second" (the unit of time)/

G

these being the fundamental units from which all the others are derived.

Let us

now

return to

Einstein's re-examination of

these fundamental units.

Suppose

that

two observers

wish to compare their measurements of time. If they are near each other

they can, of course/ look and compare them.

at

each other's watches

If they are far apart/ they can still compare each other's readings

BY

MEANS OF

SIGNALS,

say light signals or radio signals/ 1 that is/ any "electromagnetic wave* which can travel through space. Let us/ therefore/ imagine that

one observer/ f , is on the earth/ and the other/ 5 / on the sun/ and imagine that signals are sent as follows:

By his own watch/ 5 sends a message to which reads "twelve o'clock/" f receives this message say/ eight minutes later;* *Since the sun is about from the earth,

93 000 000

miles

light travels about 186 000 miles per second, the time for a light (or radio) signal to travel from the sun to the earth/

and

is

approximately eight minutes.

22

now, it

if

when

his

watch agrees with that of S n f


+

2

y

= (xj 67

+

(yj

(although x does

and y does in

So,

NOT

NOT

equal equal /).

x',

three dimensions,

x2

+

2

y

+

z

2

= (x7

+

(yj

+

(zj

and, similarly, as we have seen on p. 66, the "interval" between two events, in our four-dimensional

space-time world of events, remains the same, no matter which of the two observers,

K or

/C',

measures That

is

it.

to say,

although K and K' do not agree on some things/ as, for

example/ and time measurements, agree on other things:

their length

they (1) (2)

DO

The statement of their LAWS (see The "interval" between events,

p,

51)

Etc.

In other words/ although length and time

are

no longer

INVARIANTS,

the Einstein theory, other quantities, in

like the

ARE

space-time interval between two events,

invariants

in this

theory.

These

invariants are the quantities

68

which have the

SAME

value

for all observers,*

and may therefore be regarded as the realities of the universe.

Thus, from this point of view, the things that we see or measure

NOT

are the realities, since various observers

do not of the

get the same measurements same objects,

but rather certain mathematical relationships

between the measurements 2 2 2 2 r ) z (Like x y

+

+

+

are the realities, since they are the

same

for all observers.*

We in

shall see,

discussing

The General Theory of

how

Relativity,

fruitful

Minkowski's view-point of a four-dimensional Space-Time

World

proved to be.

VIII.

SOME CONSEQUENCES OF THE THEORY OF

We if

RELATIVITY.

have seen that

two observers,

K and K , move f

relatively to each other

*Ail observers moving relatively to each other with

UNIFORM

velocity (see p. 56).

69

with constant velocity, measurements of length and time

their

are different;

and, on page 29,

we promised In this

also to

show

measurements of mass are

that their

chapter

we

different.

shall discuss

mass measurements, as well as other

measurements which

depend upon these fundamental ones.

We

already

know

that

if

an object moves

direction parallel to the relative motion of K and in a

K

,

then the Lorentz transformation gives the relationship

between the length and time measurements of K and K'.

We

also

in a

direction

know

that

PERPENDICULAR motion of K and K

to

f

the relative there

is

NO difference

LENGTH and,

in

the

measurements (See footnote on

p. 50),

in this case,

the relationship between the time measurements may be found as follows:

For this

PERPENDICULAR

direction

Michelson argued that the time would be t2

= 2a/c

(seep. 12).

Now is

this argument supposed to be from the point of view

of an observer

DOES take

who

the motion into account,

70

and hence already contains the "correction" factor /}/ hence, replacing to by t'f the expression t' = 2a/3/c represents the time in

the perpendicular direction

as

K tells

K SHOULD be written. it

K

Whereas

,

in his

own

system,

would, of course, write

=

t

2a/c

for his "true" time,

t.

Therefore t'

=

|8f

gives the relationship sought above, from the point of view of K.

From a

this

we

see that

body moving

with velocity u

PERPENDICULAR appear to K and K to

in this

direction,

will

have

different velocities:

Thus,

Since u

=

where

and

c/

d/t and

=

u'

c/'/f'

represent the distance traversed by the object