Theory and Experiment in Gravitational Physics

This is a revised edition of a classic and highly regarded book, first published in 1981, giving a comprehensive survey of the intensive research and testing of general relativity that has been conducted over the last three decades. As a foundation for this survey, the book first introduces the important principles of gravitation theory, developing the mathematical formalism that is necessary to carry out specific computations so that theoretical predictions can be compared with experimental findings. A completely up-to-date survey of experimental results is included, not only discussing Einstein's "classical" tests, such as the deflection of light and the perihelion shift of Mercury, but also new solar system tests, never envisioned by Einstein, that make use of the high precision space and laboratory technologies of today. The book goes on to explore new arenas for testing gravitation theory in black holes, neutron stars, gravitational waves and cosmology. Included is a systematic account of the remarkable "binary pulsar" PSR 1913+16, which has yielded precise confirmation of the existence of gravitational waves. The volume is designed to be both a working tool for the researcher in gravitation theory and experiment, as well as an introduction to the subject for the scientist interested in the empirical underpinnings of one of the greatest theories of the twentieth century. Comments on the previous edition: "consolidates much of the literature on experimental gravity and should be invaluable to researchers in gravitation" Science "a c»ncise and meaty book . . . and a most useful reference work . . . researchers and serious students of gravitation should be pleased with it" Nature

Theory and Experiment in Gravitational Physics Revised Edition

THEORY AND EXPERIMENT IN GRAVITATIONAL PHYSICS CLIFFORD M.WILL McDonnell Center for the Space Sciences, Department of Physics Washington University, St Louis

Revised Edition

[CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE u n i v e r s i t y p r e s s

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521439732 © Cambridge University Press 1981, 1993 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1981 First paperback edition 1985 Revised edition 1993 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication Data

Will, Clifford M. Theory and experiment in gravitational physics / Clifford M. Will. Rev. ed. p. cm. Includes bibliographical references and index. ISBN 0 521 43973 6 1. Gravitation. I. Title. QC178.W47 1993 53i'.4—dc20 92-29555 CIP ISBN 978-0-521-43973-2 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

To Leslie

Contents

1 2 2.1 2.2 2.3 2.4 2.5 2.6 3 3.1 3.2 3.3 4 4.1 4.2 4.3 4.4 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Preface to Revised Edition Preface to First Edition Introduction The Einstein Equivalence Principle and the Foundations of Gravitation Theory The Dicke Framework Basic Criteria for the Viability of a Gravitation Theory The Einstein Equivalence Principle Experimental Tests of the Einstein Equivalence Principle Schiff 's Conjecture The THsu Formalism Gravitation as a Geometric Phenomenon Universal Coupling Nongravitational Physics in Curved Spacetime Long-Range Gravitational Fields and the Strong Equivalence Principle The Parametrized Post-Newtonian Formalism The Post-Newtonian Limit The Standard Post-Newtonian Gauge Lorentz Transformations and the PPN Metric Conservation Laws in the PPN Formalism Post-Newtonian Limits of Alternative Metric Theories of Gravity Method of Calculation General Relativity Scalar-Tensor Theories Vector-Tensor Theories Bimetric Theories with Prior Geometry Stratified Theories Nonviable Theories

page xiii xv 1 13 16 18 22 24 38 45 67 67 68 79 86 87 96 99 105 116 116 121 123 126 130 135 138

ix

Contents 6 6.1 6.2 6.3 6.4 6.5 7 7.1 7.2 7.3 8 8.1 8.2 8.3 8.4 8.5 9 9.1 9.2 9.3 10 10.1 10.2 10.3 11 11.1 11.2 11.3 12 12.1 12.2 12.3 13 13.1 13.2

Equations of Motion in the PPN Formalism Equations of Motion for Photons Equations of Motion for Massive Bodies The Locally Measured Gravitational Constant N-Body Lagrangians, Energy Conservation, and the Strong Equivalence Principle Equations of Motion for Spinning Bodies The Classical Tests The Deflection of Light The Time-Delay of Light The Perihelion Shift of Mercury Tests of the Strong Equivalence Principle The Nordtvedt Effect and the Lunar Eotvos Experiment Preferred-Frame and Preferred-Location Effects: Geophysical Tests Preferred-Frame and Preferred-Location Effects: Orbital Tests Constancy of the Newtonian Gravitational Constant Experimental Limits on the PPN Parameters Other Tests of Post-Newtonian Gravity The Gyroscope Experiment Laboratory Tests of Post-Newtonian Gravity Tests of Post-Newtonian Conservation Laws Gravitational Radiation as a Tool for Testing Relativistic Gravity Speed of Gravitational Waves Polarization of Gravitational Waves Multipole Generation of Gravitational Waves and Gravitational Radiation Damping Structure and Motion of Compact Objects in Alternative Theories of Gravity Structure of Neutron Stars Structure and Existence of Black Holes The Motion of Compact Objects: A Modified EIH Formalism The Binary Pulsar Arrival-Time Analysis for the Binary Pulsar The Binary Pulsar According to General Relativity The Binary Pulsar in Other Theories of Gravity Cosmological Tests Cosmological Models in Alternative Theories of Gravity Cosmological Tests of Alternative Theories

x 142 143 144 153 158 163 166 167 173 176 184 185 190 200 202 204 207 208 213 215 221 223 227 238 255 257 264 266 283 287 303 306 310 312 316

Contents 14 An Update 14.1 The Einstein Equivalence Principle 14.2 The PPN Framework and Alternative Metric Theories of Gravity 14.3 Tests of Post-Newtonian Gravity 14.4 Experimental Gravitation: Is there a Future? 14.5 The Rise and Fall of the Fifth Force 14.6 Stellar-System Tests of Gravitational Theory 14.7 Conclusions References References to Chapter 14 Index

xi 320 320 331 332 338 341 343 352 353 371 375

Preface to the Revised Edition

Since the publication of thefirstedition of this book in 1981, experimental gravitation has continued to be an active and challenging field. However, in some sense, the field has entered what might be termed an Era of Opportunism. Many of the remaining interesting predictions of general relativity are extremely small effects and difficult to check, in some cases requiring further technological development to bring them into detectable range. The sense of a systematic assault on the predictions of general relativity that characterized the "decades for testing relativity" has been supplanted to some extent by an opportunistic approach in which novel and unexpected (and sometimes inexpensive) tests of gravity have arisen from new theoretical ideas or experimental techniques, often from unlikely sources. Examples include the use of laser-cooled atom and ion traps to perform ultra-precise tests of special relativity, and the startling proposal of a "fifth" force, which led to a host of new tests of gravity at short ranges. Several major ongoing efforts continued nonetheless, including the Stanford Gyroscope experiment, analysis of data from the Binary Pulsar, and the program to develop sensitive detectors for gravitational radiation observatories. For this edition I have added chapter 14, which presents a brief update of the past decade of testing relativity. This work was supported in part by the National Science Foundation (PHY 89-22140). Clifford M. Will 1992

xm

Preface to First Edition

For over half a century, the general theory of relativity has stood as a monument to the genius of Albert Einstein. It has altered forever our view of the nature of space and time, and has forced us to grapple with the question of the birth and fate of the universe. Yet, despite its subsequently great influence on scientific thought, general relativity was supported initially by very meager observational evidence. It has only been in the last two decades that a technological revolution has brought about a confrontation between general relativity and experiment at unprecedented levels of accuracy. It is not unusual to attain precise measurements within a fraction of a percent (and better) of the minuscule effects predicted by general relativity for the solar system. To keep pace with these technological advances, gravitation theorists have developed a variety of mathematical tools to analyze the new highprecision results, and to develop new suggestions for future experiments made possible by further technological advances. The same tools are used to compare and contrast general relativity with its many competing theories of gravitation, to classify gravitational theories, and to understand the physical and observable consequences of such theories. The first such mathematical tool to be thoroughly developed was a "theory of metric theories of gravity" known as the Parametrized PostNewtonian (PPN) formalism, which was suited ideally to analyzing solar system tests of gravitational theories. In a series of lectures delivered in 1972 at the International School of Physics "Enrico Fermi" (Will, 1974, referred to as TTEG), I gave a detailed exposition of the PPN formalism. However, since 1972, significant progress has been made, on both the experimental and theoretical sides. The PPN formalism has been refined, and new formalisms have been developed to deal with other aspects of xv

Preface to First Edition

xvi

gravity, such as nonmetric theories of gravity, gravitational radiation, and the motion of condensed objects. A irecent review article (Will, 1979)1 summarizes the principal results of these new developments, but gives none of the physical or mathematical details. Since 1972, there has been a need for a complete treatment of techniques for analyzing gravitation theory and experiment. To fill this need I have designed this study. It analyzes in detail gravitational theories, the theoretical formalisms developed to study them, and the contact between these theories and experiments. I have made no attempt to analyze every theory of gravity or calculate every possible effect; instead I have tried to present systematically the methods for performing such calculations together with relevant examples. I hope such a presentation will make this book useful as a working tool for researchers both in general relativity and in experimental gravitation. It is written at a level suitable for use as either a reference text in a standard graduate-level course on general relativity or, possibly, as a main text in a more specialized course. Not the least of my motivations for writing such a book is the fact that it was my "centennial project" for 1979 - the 100th anniversary of Einstein's birth. It is a pleasure to thank Bob Wagoner, Martin Walker, Mark Haugan, and Francis Everitt for helpful discussions and critical readings of portions of the manuscript. Ultimate responsibility for errors or omissions rests, of course, with the author. For his constant support and encouragement, I am grateful to Kip Thorne. Victoria LaBrie performed her usual feats of speedy and accurate typing of the manuscript. Thanks also go to Rose Aleman for help with the typing. Preparation of this book took place while the author was in the Physics Department at Stanford University, and was supported in part by the National Aeronautics and Space Administration (NSG 7204), the National Science Foundation (PHY 76-21454, PHY 79-20123), the Alfred P. Sloan Foundation (BR 1700), and by a grant from the Mellon Foundation. 1

See also Will (1984).

Introduction

On September 14,1959,12 days after passing through her point of closest approach to the Earth, the planet Venus was bombarded by pulses of radio waves sent from Earth. Anxious scientists at Lincoln Laboratories in Massachusetts waited to detect the echo of the reflected waves. To their initial disappointment, neither the data from this day, nor from any of the days during that month-long observation, showed any detectable echo near inferior conjunction of Venus. However, a later, improved reanalysis of the data showed a bona fide echo in the data from one day: September 14. Thus occurred the first recorded radar echo from a planet. On March 9, 1960, the editorial office of Physical Review Letters received a paper by R. V. Pound and G. A. Rebka, Jr., entitled "Apparent Weight of Photons." The paper reported the first successful laboratory measurement of the gravitational red shift of light. The paper was accepted and published in the April 1 issue. In June, 1960, there appeared in volume 10 of the Annals of Physics a paper on "A Spinor Approach to General Relativity" by Roger Penrose. It outlined a streamlined calculus for general relativity based upon "spinors" rather than upon tensors. Later that summer, Carl H. Brans, a young Princeton graduate student working with Robert H. Dicke, began putting the finishing touches on his Ph.D. thesis, entitled "Mach's Principle and a Varying Gravitational Constant." Part of that thesis was devoted to the development of a "scalartensor" alternative to the general theory of relativity. Although its authors never referred to it this way, it came to be known as the Brans-Dicke theory. On September 26,1960, just over a year after the recorded Venus radar echo, astronomers Thomas Matthews and Allan Sandage and co-workers at Mount Palomar used the 200-in. telescope to make a photographic

Theory and Experiment in Gravitational Physics plate of the star field around the location of the radio source 3C48. Although they expected to find a cluster of galaxies, what they saw at the precise location of the radio source was an object that had a decidedly stellar appearance, an unusual spectrum, and a luminosity that varied on a timescale as short as 15 min. The name quasistellar radio source or "quasar" was soon applied to this object and to others like it. These disparate and seemingly unrelated events of the academic year 1959-60, in fields ranging from experimental physics to abstract theory to astronomy, signaled a new era for general relativity. This era was to be one in which general relativity not only would become an important theoretical tool of the astrophysicist, but would have its validity challenged as never before. Yet it was also to be a time in which experimental tools would become available to test the theory in unheard-of ways and to unheard-of levels of precision. The optical identification of 3C48 (Matthews and Sandage, 1963) and the subsequent discovery of the large red shifts in its spectral lines and in those of 3C273 (Schmidt, 1963; Greenstein and Matthews, 1963),presented theorists with the problem of understanding the enormous outpourings of energy (1047 erg s"1) from a region of space compact enough to permit the luminosity to vary systematically over timescales as short as days or hours. Many theorists turned to general relativity and to the strong relativistic gravitationalfieldsit predicts, to provide the mechanism underlying such violent events. This was the first use of the theory's strong-field aspect (outside of cosmology), in an attempt to interpret and understand observations. The subsequent discovery of pulsars and the possible identification of black holes showed that it would not be the last. However, the use of relativistic gravitation in astrophysical model building forced theorists and experimentalists to address the question: Is general relativity the correct relativistic theory of gravitation? It would be difficult to place much confidence in models for such phenomena as quasars and pulsars if there were serious doubt about one of the basic underlying physical theories. Thus, the growth of "relativistic astrophysics" intensified the need to strengthen the empirical evidence for or against general relativity. The publication of Penrose's spinor approach to general relativity (Penrose, 1960) was one of the products of a new school of relativity theorists that came to the fore in the late 1950s. These relativists applied the elegant, abstract techniques of pure mathematics to physical problems in general relativity, and demonstrated that these techniques could also aid in the work of their more astrophysically oriented colleagues. The

2

Introduction

3

bridging of the gaps between mathematics and physics and mathematics and astrophysics by such workers as Bondi, Dicke, Sciama, Pirani, Penrose, Sachs, Ehlers, Misner, and others changed the way that research (and teaching) in relativity was carried out, and helped make it an active and exciting field of physics. Yet again the question had to be addressed: Is general relativity the correct basis for this research? The other three events of 1959-60 contributed to the rebirth of a program to answer that question, a program of experimental gravitation that had been semidormant for 40 years. The Pound-Rebka (1960) experiment, besides verifying the principle of equivalence and the gravitational red shift, demonstrated the powerful use of quantum technology in gravitational experiments of high precision. The next two decades would see further uses of quantum technology in such high-precision tools as atomic clocks, laser ranging, superconducting gravimeters, and gravitational-wave detectors, to name only a few. Recording radar echos from Venus (Smith, 1963) opened up the solar system as a laboratory for testing relativistic gravity. The rapid development during the early 1960s of the interplanetary space program made radar ranging to both planets and artificial satellites a vital new tool for probing relativistic gravitational effects. Coupled with the theoretical discovery in 1964 of the relativistic time-delay effect (Shapiro, 1964), it provided new and accurate tests of general relativity. For the next decade and a half, until the summer of 1974, the solar system would be the sole arena for high-precision tests of general relativity. Finally, the development of the Brans-Dicke (1961) theory provided a viable alternative to general relativity. Its very existence and agreement with experimental results demonstrated that general relativity was not a unique theory of gravity. Many even preferred it over general relativity on aesthetic and" theoretical grounds. At the very least, it showed that discussions of experimental tests of relativistic gravitational effects should be carried on using a broader theoretical framework than that provided by general relativity alone. It also heightened the need for high-precision experiments because it showed that the mere detection of a small general relativistic effect was not enough. What was now required was measurements of these effects to accuracy within 10%, 1%, or fractions of a percent and better, to distinguish between competing theories of gravitation. To appreciate more fully the regenerative effect that these events had on gravitational theory and its experimental tests, it is useful to review briefly the history of experimental gravitation in the 45 years following the publication of the general theory of relativity.

Theory and Experiment in Gravitational Physics In deriving general relativity, Einstein was not particularly motivated by a desire to account for unexplained experimental or observational results. Instead, he was driven by theoretical criteria of elegance and simplicity. His primary goal was to produce a gravitation theory that incorporated the principle of equivalence and special relativity in a natural way. In the end, however, he had to confront the theory with experiment. This confrontation was based on what came to be known as the "three classical tests." One of these tests was an immediate success - the ability of the theory to account for the anomalous perihelion shift of Mercury. This had been an unsolved problem in celestial mechanics for over half a century, since the discovery by Leverrier in 1845 that, after the perturbing effects of the planets on Mercury's orbit had been accounted for, and after the effect of the precession of the equinoxes on the astronomical coordinate system had been subtracted, there remained in the data an unexplained advance in the perihelion of Mercury. The modern value for this discrepancy is 43 arc seconds per century (Table 1.1). A number of ad hoc proposals were made in an attempt to account for this excess, including, among others, the existence of a new planet, Vulcan, near the Sun; a ring of planetoids; a solar quadrupole moment; and a deviation from the inversesquare law of gravitation (for a review, see Chazy, 1928). Although these proposals could account for the perihelion advance of Mercury, they either involved objects that were detectable by direct optical observation, or predicted perturbations on the other planets (for example, regressions of nodes, changes in orbital inclinations) that were inconsistent with observations. Thus, they were doomed to failure. General relativity accounted Table 1.1. Perihelion advance of Mercury Cause of advance

Rate (arc s/century)

General precession (epoch 1900) Venus Earth Mars Jupiter Saturn Others

5025'.'6 211".% 9070 275 15376 773 072

Sum Observed Advance Discrepancy

555770 559977 4277

4

Introduction

5

for the anomalous shift in a natural way without disturbing the agreement with other planetary observations. This result would go unchallenged until 1967. The next classical test, the deflection of light by the Sun, was not only a success, it was a sensation. Shortly after the end of World War I, two expeditions set out from England: one for Sobral, in Brazil; and one for the island of Principe off the coast of Africa. Their goal was to measure the deflection of light as predicted by general relativity -1.75 arc seconds for a ray that grazes the Sun. The observations had to be made in the path of totality of a solar eclipse, during which the Moon would block the light from the Sun and reveal thefieldof stars behind it. Photographic plates taken of the star field during the eclipse were compared with plates of the same field taken when the Sun was not present, and the angular displacement of each star was determined. The results were 1.13 + 0.07 times the Einstein prediction for the Sobral expedition, and 0.92 ±0.17 for the Principe expedition (Dyson et al., 1920). The announcement of these results confirming the theory caught the attention of a war-weary public and helped make Einstein a celebrity. But Einstein was so convinced of the "correctness" of the theory because of its elegance and internal consistency that he is said to have remarked that he would have felt sorry for the Almighty if the results had disagreed with the theory (see Bernstein, 1973). Nevertheless, the experiments were plagued by possible systematic errors, and subsequent independent analyses of the Sobral plates yielded values ranging from 1.0 to 1.3 times the general relativity value. Later eclipse expeditions made very little improvement (Table 1.2). The main sources of error in such optical deflection experiments are unknown scale changes between eclipse and comparison photographic plates, and the precarious conditions, primarily associated with bad weather and exotic locales, under which such expeditions are carried out. By 1960, the best that could be said about the deflection of light was that it was definitely more than 0'.'83, or half the Einstein value. This was the amount predicted from a simple Newtonian argument, by Soldner in 1801 (Lenard, 1921),1 or from an extension of the principle of equivalence, by Einstein (1911). Beyond that, "the subject [was] still a live one" (Bertotti et al., 1962). The third classical test was actually thefirstproposed by Einstein (1907): the gravitational red shift of light. But by contrast with the other two 1

In 1921, the physicist Philipp Lenard, an avowed Nazi, reprinted Soldner's paper in the Annalen der Physik in an effort to discredit Einstein's "Jewish" science by showing the precedence of Soldner's "Aryan" work.

Theory and Experiment in Gravitational Physics

6

tests, there was no reliable confirmation of it until the 1960 Pound-Rebka experiment. One possible test was a measurement of the red shift of spectral lines from the Sun. However, 30 years of such measurements revealed that the observed shifts in solar spectral lines are affected strongly by Doppler shifts due to radial mass motions in the solar photosphere. For example, the frequency shift was observed to vary between the center of the Sun and the limb, and to depend on the line strength. For the gravitational red shift the results were inconclusive, and it would be 1962 before a reliable solar red-shift measurement would be made. Similarly inconclusive were attempts to measure the gravitational red shift of spectral lines from white dwarfs, primarily from Sirius B and 40 Eridani B, both members of binary systems. Because of uncertainties in the determination of the masses and radii of these stars, and because of possible complications in their spectra due to scattered light from their companions, reliable, precise measurements were not possible [see Bertotti et al. (1962) for a review]. Furthermore, by the late 1950s, it was being suggested that the gravitational red shift was not a true test of general relativity after all. According to Leonard I. Schiff and Robert H. Dicke, the gravitational red shift was a consequence purely of the principle of equivalence, and did not test the field equations of gravitational theory. Schiff took the argument one step Table 1.2. Optical measurements of light deflection by the Suri*

Eclipse

Approximate number of stars

Minimum distance from center of Sun in solar radii

Result in units of Einstein prediction

1919 1919 1922 1922 1922 1922 1929 1936 1936 1947 1952 1973"

7 5 92 145 14 18 17 25 8 51 10 39

2 2 2.1 2.1 2 2 1.5 2 4 3.3 2.1 2

1.13 + 0.07 0.92 + 0.17 0.98 ± 0.06 1.04 + 0.09 0.7 to 1.3 0.8 to 1.2 1.28 ±0.06 1.55 + 0.15 0.7 to 1.2 1.15 ±0.15 0.97 + 0.06 0.95 + 0.11

a b

See Bertotti et al. (1962) for details. Texas Mauritanian Eclipse Team (1976), Jones (1976).

Results from different analyses 1.0 to 1.3 1.3 to 0.9 1.2 0.9 to 1.2 1.55 ± 0.2 1.0 to 1.4 0.82 ± 0.09

Introduction

7

further and suggested that the gravitational red-shift experiment was superseded in importance by the more accurate Eotvos experiment, which verified that bodies of different composition fall with the same acceleration (Schiff, 1960a; Dicke, 1960). Other potential tests of general relativity were proposed, such as the Lense-Thirring effect, an orbital perturbation due to the rotation of a body, and the de Sitter effect, a secular motion of the perigee and node of the lunar orbit (Lense and Thirring, 1918; de Sitter, 1916), but the prospects for ever detecting them were dim. Cosmology was the other area where general relativity could be confronted with observation. Initially the theory met with success in its ability to account for the observed expansion of the universe, yet by the 1940s there was considerable doubt about its applicability. According to pure general relativity, the expansion of the universe originated in a dense primordial explosion called the "big bang." The age of the universe since the big bang could be determined by extrapolating the expansion of the universe backward in time using the field equations of general relativity. However, the observed values of the present expansion rate were so high that the inferred age of the universe was shorter than that of the Earth. One result of this doubt was the rise in popularity during the 1950s of the steady-state cosmology of Herman Bondi, Thomas Gold, and Fred Hoyle. This model avoided the big bang altogether, and allowed for the expansion of the universe by the continuous creation of matter. By this means, the universe would present the same appearance to all observers for all time. But by the late 1950s, revisions in the cosmic distance scale had reduced the expansion rate by a factor of five, and had thereby increased the age of the universe in the big bang model to a more acceptable level. Nevertheless, cosmological observations were still in no position to distinguish among different theories of gravitation or of cosmology [for a detailed technical and historical review, see Weinberg (1972), Chapter 14]. Meanwhile, a small "cottage industry" had sprung up, devoted to the construction of alternative theories of gravitation. Some of these theories were produced by such luminaries as Poincare, Whitehead, Milne, Birkhoff, and Belinfante. Many of these authors expressed an uneasiness with the notions of general covariance and curved spacetime, which were built into general relativity, and responded by producing "special relativistic" theories of gravitation. These theories considered spacetime to be "special relativistic" at least at a background level, and treated gravitation as a Lorentz-invariant field on that background. As of 1960, it was possible

Theory and Experiment in Gravitational Physics

8

to enumerate at least 25 such alternative theories, as found in the primary research literature between 1905 and 1960 [for a partial list, see Whitrow and Morduch (1965)]. Thus, by 1960, it could be argued that the validity of general relativity rested on the following empirical foundation: one test of moderate precision (the perihelion shift, approximately 1%), one test of low precision (the deflection of light, approximately 50%), one inconclusive test that was not a real test anyway (the gravitational red shift), and cosmological observations that could not distinguish between general relativity and the steady-state theory. Furthermore, a variety of alternative theories laid claim to viability. In addition, the attitude toward the theory seemed to be that, whereas it was undoubtedly of importance as a fundamental theory of nature, its observational contacts were limited to the classical tests and cosmology. This view was present for example in the standard textbooks on general relativity of this period, such as those by Mcller (1952), Synge (1960), and Landau and Lifshitz (1962). As a consequence, general relativity was cut off from the mainstream of physics. It was during this period that one young, beginning graduate student was advised not to enter this field, because general relativity "had so little connection with the rest of physics and astronomy" (his name: Kip S. Thorne). However, the events of 1959-60 changed all that. The pace of research in general relativity and relativistic astrophysics began to quicken and, associated with this renewed effort, the systematic high-precision testing of gravitational theory became an active and challengingfield,with many new experimental and theoretical possibilities. These included new versions of old tests, such as the gravitational red shift and deflection of light, with accuracies that were unthinkable before 1960. They also included brand new tests of gravitational theory, such as the gyroscope precession, the time delay of light, and the "Nordtvedt effect" in lunar motion, that were discovered theoretically after 1959. Table 1.3 presents a chronology of some of the significant theoretical and experimental events that occurred in the two decades following 1959. In many ways, the years 19601980 were the decades for testing relativity. Because many of the experiments involved the resources of programs for interplanetary space exploration and observational astronomy, their cost in terms of money and manpower was high and their dependence upon increasingly constrained government funding agencies was strong. Thus, it became crucial to have as good a theoretical framework as possible for comparing the relative merits of various experiments, and for pro-

Introduction Table 1.3. A chronology: 1960-80 Time

Experimental or observational events

Theoretical events

1960

Hughes-Drever mass-anisotropy experiments Pound-Rebka gravitational red-shift experiment Discovery of nonsolar x-ray sources Discovery of quasar red shifts Princeton Eotvos experiment

Penrose paper on spinors Gyroscope precession (Schiff)

1962

1964

Brans-Dicke theory Bondi mass-loss formula Kerr metric discovery Time-delay of light (Shapiro)

Pound-Snider red-shift experiment Discovery of 3K microwave background

Singularity theorems in general relativity

1966

1968

Reported detection of solar oblateness Discovery of pulsars Planetary radar measurement of time delay Launch of Mariners 6 and 7 Acquisition of lunar laser echo First radio deflection measurements

Element production in the big bang Nordtvedt effect and early PPN framework

1970 CygXl: a black hole candidate Mariners 6 and 7 time-delay measurements 1972 1974

Moscow Eotvos experiment Discovery of binary pulsar

1976

1978

1980

Rocket gravitational red-shift experiment Lunar test of Nordtvedt effect Time delay results from Mariner 9 and Viking Measurement of orbit period decrease in binary pulsar SS433 Discovery of gravitational lens

Preferred-frame effects Refined PPN framework Area increase of black holes in general relativity Quantum evaporation of black holes Dipole gravitational radiation in alternative theories

Theory and Experiment in Gravitational Physics

10

posing new ones that might have been overlooked. Another reason that such a theoretical framework was necessary was to make some sense of the large (and still growing) number of alternative theories of gravitation. Such a framework could be used to classify theories, elucidate their similarities and differences, and compare their predictions with the results of experiments in a systematic way. It would have to be powerful enough to be used to design and assess experimental tests in detail, yet general enough not to be biased in favor of general relativity. A leading exponent of this viewpoint was Robert Dicke (1964a). It led him and others to perform several high-precision null experiments which greatly strengthened our faith in the foundations of gravitation theory. Within this viewpoint one asks general questions about the nature of gravity and devises experiments to test them. The most important dividend of the Dicke framework is the understanding that gravitational experiments can be divided into two classes. The first consists of experiments that test the foundations of gravitation theory, one of these foundations being the principle of equivalence. These experiments (Eotvos experiment, Hughes-Drever experiment, gravitational red-shift experiment, and others, many performed by Dicke and his students) accurately verify that gravitation is a phenomenon of curved spacetime, that is, it must be described by a "metric theory" of gravity. General relativity and Brans-Dicke theory are examples of metric theories of gravity. The second class of experiments consists of those that test metric theories of gravity. Here another theoretical framework was developed that takes up where the Dicke framework leaves off. Known as the "Parametrized Post-Newtonian" or PPN formalism, it was pioneered by Kenneth Nordtvedt, Jr. (1968b), and later extended and improved by Will (1971a), Will and Nordtvedt (1972), and Will (1973). The PPN framework takes the slow motion, weak field, or post-Newtonian limit of metric theories of gravity, and characterizes that limit by a set of 10 real-valued parameters. Each metric theory of gravity has particular values for the PPN parameters. The PPN framework was ideally suited to the analysis of solar system gravitational experiments, whose task then became one of measuring the values of the PPN parameters and thereby delineating which theory of gravity is correct. A second powerful use of the PPN framework was in the discovery and analysis of new tests of gravitation theory, examples being the Nordtvedt effect (Nordtvedt 1968a), preferredframe effects (Will, 1971b) and preferred-location effects (Will, 1971b, 1973). The Nordtvedt effect, for instance, is a violation of the equality of acceleration of massive bodies, such as the Earth and Moon, in an

Introduction

11

external field; the effect is absent in general relativity but present in many alternative theories, including the Brans-Dicke theory. The third use of the PPN formalism was in the analysis and classification of alternative metric theories of gravitation. After 1960, the invention of alternative gravitation theories did not abate, but changed character. The crude attempts to derive Lorentz-invariant field theories described previously were mostly abandoned in favor of metric theories of gravity, whose development and motivation were often patterned after that of the BransDicke theory. A "theory of gravitation theories" was developed around the PPN formalism to aid in their systematic study. The PPN formalism thus became the standard theoretical tool for analyzing solar system experiments, looking for new tests, and studying alternative metric theories of gravity. One of the central conclusions of the two decades of testing relativistic gravity in the solar system is that general relativity passes every experimental test with flying colors. But by the middle 1970s it became apparent that the solar system could no longer be the sole testing ground for gravitation theories. One reason was that many alternative theories of gravity agreed with general relativity in their post-Newtonian limits, and thereby also agreed with all solar system experiments. But they did not necessarily agree in other predictions, such as cosmology, gravitational radiation, neutron stars, or black holes. The second reason was the possibility that experimental tools, such as gravitational radiation detectors, would ultimately be available to perform such extra-solar system tests. This suspicion was confirmed in the summer of 1974 with the discovery by Joseph Taylor and Russell Hulse of the binary pulsar (Hulse and Taylor, 1975). Here was a system that combined large post-Newtonian gravitational effects, highly relativistic gravitational fields associated with the pulsar, and the possibility of the emission of gravitational radiation by the binary system, with ultrahigh precision data obtained by radiotelescope monitoring of the extremely stable pulsar clock. It was also a system where relativistic gravity and astrophysics became even more intertwined than in the case, say, of quasars. In the binary pulsar, relativistic gravitational effects provided a means for accurate measurement of astrophysical parameters, such as the mass of a neutron star. The role of the binary pulsar as a new arena for testing relativistic gravity was cemented in the winter of 1978 with the announcement (Taylor et al., 1979) that the rate of change of the orbital period of the system had been measured. The result agreed with the prediction of general relativity for the rate of orbital energy loss due to the emission of gravitational radiation. But it

Theory and Experiment in Gravitational Physics

12

disagreed violently with the predictions of most alternative theories, even those with post-Newtonian limits identical to general relativity. As a young student of 17 at the Poly technical Institute of Zurich, Einstein studied closely the work of Helmholtz, Maxwell, and Hertz, and ultimately used his deep understanding of electromagnetic theory as a foundation for special and general relativity. He appears to have been especially impressed by Hertz's confirmation that light and electromagnetic waves are one and the same (Schilpp, 1949). The electromagnetic waves that Hertz studied were in the radio part of the spectrum, at 30 MHz. It is amusing to note that, 60 years later, the decades for testing relativistic gravity began with radio waves, the 440 MHz waves reflected from Venus, and ended with radio waves, the signals from the binary pulsar, observed at 430 MHz. During these two decades, that closed on the centenary of Einstein's birth, the empirical foundations of general relativity were strengthened as never before. But this does not end the story. The confrontation between general relativity and experiment will proceed, using new tools, in new arenas. Whether or not general relativity will continue to survive is a matter of speculation for some, pious hope for another group, and supreme confidence for others. Regardless of one's theoretical prejudices, it can certainly be agreed that gravitation, the oldest known, and in many ways most fundamental interaction, deserves an empirical foundation second to none. Throughout this book, we shall adopt the units and conventions of Misner, Thorne, and Wheeler, 1973 (hereafter referred to as MTW). Although we have attempted to produce a reasonably self-contained account of gravitation theory and gravitational experiments, the reader's path will be greatly smoothed by a familiarity with at least the equivalent of "track 1" of MTW. A portion of the present book (Chapters 4-9) is patterned after the author's 1972 Varenna lectures "The Theoretical Tools of Experimental Gravitation" (Will, 1974a, hereafter referred to as TTEG), with suitable modification and updating. An overview of this book without the mathematical details is provided by the author's "The Confrontation between Gravitation Theory and Experiment" (Will, 1979). Other useful reviews of this subject are of three types: (i) semipopular: Nordtvedt (1972), Will (1972, 1974b); (ii) technical: Richard (1975), Brill (1973), Rudenko (1978); (iii) "early": Dicke (1964a,b), Bertotti et al. (1962). The reader is referred to these works for background or for different points of view.

The Einstein Equivalence Principle and the Foundations of Gravitation Theory

The Principle of Equivalence has played an important role in the development of gravitation theory. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraphs of the Principia to a detailed discussion of it (Figure 2.1). He also reported there the results of pendulum experiments he performed to verify the principle. To Newton, the Principle of Equivalence demanded that the "mass" of any body, namely that property of a body (inertia) that regulates its response to an applied force, be equal to its "weight," that property that regulates its response to gravitation. Bondi (1957) coined the terms "inertial mass" mb and "passive gravitational mass" mP, to refer to these quantities, so that Newton's second law and the law of gravitation take the forms F = m,a,

F = mPg

where g is the gravitational field. The Principle of Equivalence can then be stated succinctly: for any body mP = m1 An alternative statement of this principle is that all bodies fall in a gravitational field with the same acceleration regardless of their mass or internal structure. Newton's equivalence principle is now generally referred to as the "Weak Equivalence Principle" (WEP). It was Einstein who added the key element to WEP that revealed the path to general relativity. If all bodies fall with the same acceleration in an external gravitational field, then to an observer in a freely falling elevator in the same gravitational field, the bodies should be unaccelerated (except for possible tidal effects due to inhomogeneities in the gravitational field, which can be made as small as one pleases by working in a sufficiently small elevator). Thus insofar as their mechanical motions are

Figure 2.1. Title page and first page of Newton's Principia.

PHILOSOPHISE NATURALIS

PRINCIPIA MATHEMATICA Autore JS. UEfFTON, Trin. CM. Cantab. Soc. Mathefeos Profeflbre Lucafuoto, & Sodetatis Regalis Sodali.

IMPRIMATUR S. P E P Y S, Reg. Soc. P R R S E S. Jutii 5. 1686.

L 0 N D I N /, Juflii Societatis Regia ac Typis Jofepbi Streater. Proftat apud plures Bibliopolas. Anno MDCLXXXVIl.

14

Figure 2.1 (continued)

MATHEMATICAL PRINCIPLES OF

NATURAL PHILOSOPHY1 D eft

nitions

DEFINITION I The quantity of matter is the measure of the same, arising from its density and bulk, conjointly.2

T

HUS AIR of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction, and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter. DEFINITION

IIs

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple. t l Appendix, Note 10.] [ 2 Appendix, Note 11.]

[ 3 Appendix, Note 12.]

CO

15

Theory and Experiment in Gravitational Physics

16

concerned, the bodies will behave as if gravity were absent. Einstein went one step further. He proposed that not only should mechanical laws behave in such an elevator as if gravity were absent but so should all the laws of physics, including, for example, the laws of electrodynamics. This new principle led Einstein to general relativity. It is now called the "Einstein Equivalence Principle" (EEP). Yet, it is only relatively recently that we have gained a deeper understanding of the significance of these principles of equivalence for gravitation and experiment. Largely through the work of Robert H. Dicke, we have come to view principles of equivalence, along with experiments such as the Eotvos experiment, the gravitational red-shift experiment, and so on, as probes more of the foundations of gravitation theory, than of general relativity itself. This viewpoint is part of what has come to be known as the Dicke Framework described in Section 2.1, allowing one to discuss at a very fundamental level the nature of space-time and gravity. Within it one asks questions such as: Do all bodies respond to gravity with the same acceleration? Does energy conservation imply anything about gravitational effects? What types of fields, if any, are associated with gravitation-scalar fields, vector fields, tensor fields... ? As one product of this viewpoint, we present in Section 2.2 a set of fundamental criteria that any potentially viable theory should satisfy, and as another, we show in Section 2.3 that the Einstein Equivalence Principle is the foundation for all gravitation theories that describe gravity as a manifestation of curved spacetime, the so-called metric theories of gravity. In Section 2.4 we describe the empirical support for EEP from a variety of experiments. Einstein's generalization of the Weak Equivalence Principle may not have been a generalization at all, according to a conjecture based on the work of Leonard Schiff. In Section 2.5, we discuss Schiif 's conjecture, which states that any complete and self-consistent theory of gravity that satisfies WEP necessarily satisfies EEP. Schiff's conjecture and the Dicke Framework have spawned a number of concrete theoretical formalisms, one of which is known as the THsu formalism, presented in Section 2.6, for comparing and contrasting metric theories of gravity with nonmetric theories, analyzing experiments that test EEP and WEP, and proving Schiff's conjecture. 2.1

The Dicke Framework

The Dicke Framework for analyzing experimental tests of gravitation was spelled out in Appendix 4 of Dicke's Les Houches lectures

Einstein Equivalence Principle and Gravitation Theory (1964a). It makes two main assumptions about the type of mathematical formalism to be used in discussing gravity: (i) Spacetime is a four-dimensional differentiable manifold, with each point in the manifold corresponding to a physical event. The manifold need not a priori have either a metric or an affine connection. The hope is that experiment will force us to conclude that it has both. (ii) The equations of gravity and the mathematical entities in them are to be expressed in a form that is independent of the particular coordinates used, i.e., in covariant form. Notice that even if there is some physically preferred coordinate system in spacetime, the theory can still be put into covariant form. For example, if a theory has a preferred cosmic time coordinate, one can introduce a scalar field T{0>) whose numerical values are equal to the values of the preferred time t: T(0>) = t{0>),

0> a point in spacetime

If spacetime is endowed with a metric, one might also demand that VT be a timelike vector field and be consistently oriented toward the future (or the past) throughout spacetime by imposing the covariant constraints VT-VTr) = 0

and by defining covariant derivatives and contractions with respect to i\. In most cases, this covariance is achieved at the price of the introduction into the theory of "absolute" or "prior geometric" elements (T, i/), that are not determined by the dynamical equations of the theory. Some authors regard the introduction of absolute elements as a failure of general covariance (Einstein would be one example), however we shall adopt the weaker assumption of coordinate invariance alone. (For further discussion of prior geometry, see Section 3.3.) Having laid down this mathematical viewpoint [statements (i) and (ii) above] Dicke then imposes two constraints on all acceptable theories of gravity. They are: (1) Gravity must be associated with one or more fields of tensorial character (scalars, vectors, and tensors of various ranks).

17

Theory and Experiment in Gravitational Physics

18

(2) The dynamical equations that govern gravity must be derivable from an invariant action principle. These constraints strongly confine acceptable theories. For this reason we should accept them only if they are fundamental to our subsequent arguments. For most applications of the Dicke Framework only the first constraint is often needed. It is a fact, however, that the most successful gravitation theories are those that satisfy both constraints. The Dicke Framework is particularly useful for designing and interpreting experiments that ask what types of fields are associated with gravity. For example, there is strong evidence from elementary particle physics for at least one symmetric second-rank tensorfieldthat is approximated by the Minkowski metric i\ when gravitational effects can be ignored. The Hughes-Drever experiment rules out the existence of more than one second-rank tensor field, each coupling directly to matter, and various ether-drift experiments rule out a long-range vectorfieldcoupling directly to matter. No experiment has been able to rule out or reveal the existence of a scalar field, although several experiments have placed limits on specific scalar-tensor theories (Chapters 7 and 8). However, this is not the only powerful use of the Dicke Framework. 2.2

Basic Criteria for the Viability of a Gravitation Theory

The general unbiased viewpoint embodied in the Dicke Framework has allowed theorists to formulate a set of fundamental criteria that any gravitation theory should satisfy if it is to be viable [we do not impose constraints (1) and (2) above]. Two of these criteria are purely theoretical, whereas two are based on experimental evidence. (i) It must be complete, i.e., it must be capable of analyzing from "first principles" the outcome of any experiment of interest. It is not enough for the theory to postulate that bodies made of different material fall with the same acceleration. The theory must incorporate a complete set of electrodynamic and quantum mechanical laws, which can be used to calculate the detailed behavior of bodies in gravitational fields. This demand should not be extended too far, however. In areas such as weak and strong interaction theory, quantum gravity, unified field theories, spacetime singularities, and cosmic initial conditions, even special and general relativity are not regarded as being complete or fully developed. We also do not regard the presence of "absolute elements" and arbitrary parameters in gravitational theories as a sign of incompleteness, even though they are generally not derivable from "first principles," rather we

Einstein Equivalence Principle and Gravitation Theory

19

view them as part of the class of cosmic boundary conditions. Fortunately, so simple a demand as one that the theory contain a set of gravitationally modified Maxwell equations is sufficiently telling that many theories fail this test. Examples are given in Table 2.1. (ii) It must be self-consistent, i.e., its prediction for the outcome of every experiment must be unique, i.e., when one calculates the predictions by two different, though equivalent methods, one always gets the same

Table 2.1. Basically nonviable theories of gravitation - a partial list Theory and references

Comments"

Newtonian gravitation theory Milne's kinematical relativity (Milne, 1948)

Is not relativistic Was devised originally to handle certain cosmological problems. Is incomplete: makes no gravitational red-shift prediction Contain a vector gravitational field in flat spacetime. Are incomplete: do not mesh with the other nongravitational laws of physics (viz. Maxwell's equations) except by imposing them on the flat background spacetime. Are then inconsistent: give different results for light propagation for light viewed as particles and light viewed as waves. Action-at-a-distance theory in flat spacetime. Is incomplete or inconsistent in the same manner as Kustaanheimo's theories Contains a vector gravitational field in flat spacetime. Is incomplete or inconsistent in the same manner as Kustaanheimo's theories. Contains a tensor gravitational field used to construct a metric. Violates the Newtonian limit by demanding that p = pc2, i.e.

Kustaanheimo's various vector theories (Kustaanheimo and Nuotio, 1967; Whitrow and Morduch, 1965)

Poincare's theory (as generalized by Whitrow and Morduch, 1965) Whitrow-Morduch (1965) vector theory Birkhoff's (1943) theory

''sound

Yilmaz's (1971,1973) theory

=

"light-

Contains a tensor gravitational field used to construct a metric. Is mathematically inconsistent: functional dependence of metric on tensor field is not well defined.

° These theories are nonviable in their present form. Future modifications or specializations might make some of them viable. If I have misinterpreted any theory here I apologize to its proponents, and urge them to demonstrate explicitly its completeness, self-consistency, and compatibility with special relativity and Newtonian gravitation theory.

Theory and Experiment in Gravitational Physics

20

results. An example is the bending of light computed either in the geometrical optics limit of Maxwell's equations or in the zero-rest-mass limit of the motion of test particles. Furthermore, the system of mathematical equations it proposes should be well posed and self-consistent. Table 2.1 shows some theories that fail this criterion. (iii) It must be relativistic, i.e., in the limit as gravity is "turned off" compared to other physical interactions, the nongravitational laws of physics must reduce to the laws of special relativity. The evidence for this comes largely from high-energy physics and from a variety of optical ether-drift experiments. Since these experiments are performed at high energies and velocities and over very small regions of space and time, the effects of gravity on their outcome are negligible. Thus we may treat such experiments as if they were being performed far from all gravitating matter. The evidence provided by these experiments is of two types. First are experiments that measure space and time intervals directly, e.g., measurements of the time dilation of systems ranging from atomic clocks to unstable elementary particles, experiments that verify the velocity of light is independent of the velocity of the source for sources ranging from pions at 99.98% of the speed of light to pulsating binary x-ray sources at 10" 3 of the speed of light [for a thorough review and reference list, see Newman et al. (1978)] and Michelson-Morley-type experiments [for recent high-precision results, see Trimmer et al. (1973) and Brillet and Hall (1979); see also Mansouri and Sexl (1977a,b,c) for theoretical discussion]. Second are experiments which reveal the fundamental role played by the Lorentz group in particle physics, including verifications of fourmomentum conservation and of the relativistic laws of kinematics, electron and muon "g-2" experiments, and tests of esoteric predictions of Lorentz-in variant quantumfieldtheories [Lichtenberg (1965), Blokhintsev (1966), Newman et al. (1978), Combley et al. (1979), and Cooper et al. (1979)]. The fundamental theoretical object that enters these laws is the Minkowski metric i\, with a signature of + 2, which has orthonormal tetrads related by Lorentz transformations, and which determines the ticking rates of atomic clocks and the lengths of laboratory rods. If we view q as a field [Dicke statement (ii)], then we conclude that there must exist at least one second-rank tensor field in the Universe, a symmetric tensor ^, which reduces to r\ when gravitational effects can be ignored. Let us examine what particle physics experiments do and do not tell us about the tensor field V- First, they do not guarantee the existence of global Lorentz frames, i.e., coordinate systems extending throughout

Einstein Equivalence Principle and Gravitation Theory

21

spacetime in which (-1,1,1,1) Nor do they demand that at each event 2P, there exist local frames related by Lorentz transformations, in which the laws of elementary-particle physics take on their special form. They only demand that, in the limit as gravity is "turned off," the nongravitational laws of physics reduce to the laws of special relativity. Second, elementary-particle experiments do tell us that the times measured by atomic clocks in the limit as gravity is turned off depend only on velocity, not upon acceleration. The measured squared interval, ds2 = i^dx^dx", is independent of acceleration. Equivalently, but more physically, the time interval measured by a clock moving with velocity vJ relative to a coordinate system in the absence of gravity is ds = (-q^tordx*)112

= dt(l - |v|2)1/2

independent of the clock's acceleration d2xi/dt2. (For a review of experimental tests, see Newman et al., 1978.) We shall henceforth assume the existence of the tensor field $. (iv) It must have the correct Newtonian limit, i.e., in the limit of weak gravitational fields and slow motions, it must reproduce Newton's laws. Massive amounts of empirical data support the validity of Newtonian gravitation theory (NGT), at least as an approximation to the "true" relativistic theory of gravity. Observations of the motions of planets and spacecraft agree with NGT down to the level (parts in 108) at which post-Newtonian effects can be observed. Observations of planetary, solar, and stellar structure support NGT as applied to bulk matter. Laboratory Cavendish experiments provide support for NGT for small separations between gravitating bodies. One feature of NGT that has recently come under experimental scrutiny is the inverse-square force law. Despite one claim to the contrary (Long, 1976), there seems to be no hard evidence for a deviation from this law (other than those produced by post-Newtonian effects) over distances ranging from a few centimeters to several astronomical units (see Mikkelson and Newman, 1977; Spero et al., 1979; Paik, 1979; Yu et al., 1979; Panov and Frontov, 1979; and, Hirakawa et al., 1980). Thus, to at least be viable, a gravitation theory must be complete, self-consistent, relativistic, and compatible with NGT. Table 2.1 shows examples of theories that violate one or more of these criteria.

Theory and Experiment in Gravitational Physics 2.3

22

The Einstein Equivalence Principle The Einstein Equivalence Principle is the foundation of all curved spacetime or "metric" theories of gravity, including general relativity. It is a powerful tool for dividing gravitational theories into two distinct classes: metric theories, those that embody EEP, and nonmetric theories, those that do not embody EEP. For this reason, we shall discuss it in some detail and devote the next section (Section 2.4) to the supporting experimental evidence. We begin by stating the Weak Equivalence Principle in more precise terms than those used before. WEP states that if an uncharged test body is placed at an initial event in spacetime and given an initial velocity there, then its subsequent trajectory will be independent of its internal structure and composition. By "uncharged test body" we mean an electrically neutral body that has negligible self-gravitational energy (as estimated using Newtonian theory) and that is small enough in size so that its coupling to inhomogeneities in external fields can be ignored. In the same spirit, it is also useful to define "local nongravitational test experiment" to be any experiment: (i) performed in a freely falling laboratory that is shielded and is sufficiently small that inhomogeneities in the external fields can be ignored throughout its volume, and (ii) in which self-gravitational effects are negligible. For example, a measurement of the fine structure constant is a local nongravitational test experiment; a Cavendish experiment is not. The Einstein Equivalence Principle then states: (i) WEP is valid, (ii) the outcome of any local nongravitational test experiment is independent of the velocity of the (freely falling) apparatus, and (iii) the outcome of any local nongravitational test experiment is independent of where and when in the universe it is performed. This principle is at the heart of gravitation theory, for it is possible to argue convincingly that if EEP is valid, then gravitation must be a curvedspacetime phenomenon, i.e., must satisfy the postulates of Metric Theories of Gravity. These postulates state: (i) spacetime is endowed with a metric g, (ii) the world lines of test bodies are geodesies of that metric, and (iii) in local freely falling frames, called local Lorentz frames, the nongravitational laws of physics are those of special relativity. General relativity, BransDicke theory, and the Rosen bimetric theory are metric theories of gravity (Chapter 5); the Belinfante-Swihart theory (Section 2.6) is not. The argument proceeds as follows. The validity of WEP endows spacetime with a family of preferred trajectories, the world lines of freely falling test bodies. In a local frame that follows one of these trajectories,

Einstein Equivalence Principle and Gravitation Theory

23

test bodies have unaccelerated motions. Furthermore, the results of local nongravitational test experiments are independent of the velocity of the frame. In two such frames located at the same event, 9, in spacetime but moving relative to each other, all the nongravitational laws of physics must make the same predictions for identical experiments, that is, they must be Lorentz invariant. We call this aspect of EEP Local Lorentz Invariance (LLI). Therefore, there must exist in the universe one or more second-rank tensor fields i/t(1), ij/(2\ . . . , that reduce in a local freely falling frame to fields that are proportional to the Minkowski metric, (j)(1\^)tl, 0 (2) (^)«J,..., where 4>(A\0>) are scalar fields that can vary from event to event. Different members of this set of fields may couple to different nongravitationalfields,such as bosonfields,fermionfields,electromagnetic fields, etc. However, the results of local nongravitational test experiments must also be independent of the spacetime location of the frame. We call this Local Position Invariance (LPI). There are then two possibilities, (i) The local versions of ijf{A) must have constant coefficients, that is, the scalarfields4>(A\^) must be constants. It is therefore possible by a simple universal rescaling of coordinates and coupling constants (such as the unit of electric charge) to set each scalar field equal to unity in every local frame, (ii) The scalarfieldsiA)({A\0>) = cA4>(0>). If this is true, then physically measurable quantities, being dimensionless ratios, will be location independent (essentially, the scalar field will cancel out). One example is a measurement of the fine structure constant; another is a measurement of the length of a rigid rod in centimeters, since such a measurement is a ratio between the length of the rod and that of a standard rod whose length is defined to be one centimeter. Thus, a combination of a rescaling of coupling constants to set the cA's equal to unity (redefinition of units), together with a "conformal" transformation to a new field ij/ = cj>~ V. guarantees that the local version of if/ will be ij. In either case, we conclude that there exist fields that reduce to r\ in every local freely falling frame. Elementary differential geometry then shows that thesefieldsare one and the same: a unique, symmetric secondrank tensor field that we now denote g. This g has the property that it possesses a family of preferred worldlines called geodesies, and that at each event $* there exist local frames, called local Lorentz frames, that follow these geodesies, in which

<W^) = 1** + 0(Y |X« - x\0>)\\

dgjdx* = 0,

at 0>

Theory and Experiment in Gravitational Physics

24

However, geodesies are straight lines in local Lorentz frames, as are the trajectories of test bodies in local freely falling frames, hence the test bodies move on geodesies of g and the Local Lorentz frames coincide with the freely falling frames. We shall discuss the implications of the postulates of metric theories of gravity in more detail in Chapter 3. Because EEP is so crucial to this conclusion about the nature of gravity, we turn now to the supporting experimental evidence. 2.4

Experimental Tests of the Einstein Equivalence Principle (a) Tests of the Weak Equivalence Principle

A direct test of WEP is the Eotvos experiment, the comparison of the acceleration from rest of two laboratory-sized bodies of different composition in an external gravitational field. If WEP were invalid, then the accelerations of different bodies would differ. The simplest way to quantify such possible violations of WEP in a form suitable for comparison with experiment is to suppose that for a body of inertial mass m,, the passive mass mP is no longer equal to mv Now the inertial mass of a typical laboratory body is made up of several types of mass energy: rest energy, electromagnetic energy, weak-interaction energy, and so on. If one of these forms of energy contributes to mP differently than it does to m,, a violation of WEP would result. One could then write ™p = m, + I r]AEA/c2

(2.1)

A

where EA is the internal energy of the body generated by interaction A, and nA is a dimensionless parameter that measures the strength of the violation of WEP induced by that interaction, and c is the speed of light.1 For two bodies, the acceleration is then given by

^

( + S r,AEA/m2Ag

(2.2)

where we have dropped the subscript I on mj and m2. 1 Throughout this chapter we shall avoid units in which c = 1. The reason for this is that if EEP is not valid then the speed of light may depend on the nature of the devices used to measure it. Thus, to be precise we should denote c as the speed of light as measured by some standard experiment. Once we accept the validity of EEP in Chapter 3 and beyond, then c has the same value in every local Lorentz frame, independently of the method used to measure it, and thus can be set equal to unity by appropriate choice of units.

Einstein Equivalence Principle and Gravitation Theory

25

A measurement or limit on the relative difference in acceleration then yields a quantity called the "Eotvos ratio" given by

K + a\

t

\mC2

m2c2j

v

'

Thus, experimental limits on r\ place limits on the WEP-violation parameters rjA. Many high-precision Eotvos-type experiments have been performed, from the pendulum experiments of Newton, Bessel, and Potter to the classic torsion-balance measurements of Eotvos, Dicke, and Braginsky and their collaborators. The latter experiments can be described heuristically. Two objects of different composition are connected by a rod of length r, and suspended in a horizontal orientation by afinewire ("torsion balance"). If the gravitational acceleration of the bodies differs, there will be a torque N induced on the suspension wire, given by N = tjr(g x ew) • er where g is the gravitational acceleration, and ew and er are unit vectors along the wire and rod, respectively (see Figure 2.2). If the entire apparatus is rotated about a direction 7 HF | that couples to electromagnetism in a Lagrangian term of the form ^""F^F^, where s*™11 is the completely antisymmetric Levi-Civita symbol. Ni has argued that such a term, while violating EEP, does not violate WEP, although it does have the observable effect of producing an anomalous torque on systems of electromagnetically bound charged particles. Whether this torque then can lead to observable WEP violations is an open question at present. 2.6

The THs/i Formalism

The discussion of Schiff's conjecture presented in the previous section was very general, and perhaps gives compelling evidence for the validity of the conjecture. However, because of the generality of those arguments, there was little quantitative information. For example, no means was presented to compute explicitly the anomalous mass tensors (5mj/ and 8m\J for various systems. In order to make these ideas more concrete, we need a model theory of the nongravitational laws of physics in the presence of gravity that incorporates the possibility of both nonmetric (nonuniversal) and metric coupling. This theory should be simple, yet capable of making quantitative predictions for the outcomes of experiments. One such "model" theory is the THe/x formalism, devised by Lightman and Lee (1973a). It restricts attention to the motions and electromagnetic interactions of charged structureless test particles in an external, static, and spherically symmetric (SSS) gravitational field. It

Theory and Experiment in Gravitational Physics

46

assumes that the nongravitational laws of physics can be derived from an action / N G given by f NG

=

Jo + hat + Iem> \{T -

E2 - li-^d'x

(2.46)

(we use units here in which x and t both have units of length) where mOa, ea, and x£(t) are the rest mass, charge, and world line of particle a, x° = t, v»a = dxljdt, E = \A0 - A o, B = (V x A), and where scalar products between 3-vectors are taken with respect to the Cartesian metric 8ij. The functions T, H, e, and n are assumed to be functions of a single external gravitational potential 4>, but are otherwise arbitrary. For an SSS field in a given theory, T, H, e, and /x will be particular functions of O. It turns out that, for SSS fields, equations (2.46) are general enough to encompass all metric theories of gravitation and a wide class of nonmetric theories, such as the Belinfante-Swihart (1957) theory and the nonmetric theory discussed in Section 2.5. In many cases, the form of / N G in equation (2.46) is valid only in special coordinate systems ("isotropic" coordinates in the case of metric theories of gravity). An example of a theory that does not fit the THsfi form of / N G is the Naida-Capella nonmetric theory (see Lightman and Lee, 1973a for discussion). Cases such as this must then be analyzed on an individual basis. For an "en" formalism, see Dicke (1962). (a) Einstein Equivalence Principle in the THe/x formalism We begin by exploring in some detail the properties of the formalism as presented in equations (2.46). Later, we shall discuss the physical restrictions built into it, and shall apply it to the interpretation of experiment. In order to examine the Einstein Equivalence Principle in this formalism we must work in a local freely falling frame. But we do not yet know whether WEP is satisfied by the THsn theory (and suspect that it is not, in general), so we do not know to which freely falling trajectories local frames should be attached. We must therefore arbitrarily choose a set of trajectories: the most convenient choice is the set of trajectories of neutral test particles, i.e., particles governed only by the action l0, since

Einstein Equivalence Principle and Gravitation Theory

47

their trajectories are universal and independent of the mass mOa. We make a transformation to a coordinate system x" = (?, x) chosen according to the following criteria: (i) the origins of both coordinate systems coincide, that is, for a selected event 3P, xx{@) - x\0>) = 0, (ii) at 0>, a neutral test body has zero acceleration in the new coordinates, i.e., d2xJ'/dt2^ = 0, and in the neighborhood of 9 the deviations from zero acceleration are quadratic in the quantities Ax* = x* — x%0>), and (iii) the motion of the neutral-test body is derivable from an action Jo. The required transformation, correct to first order in the quantities g0? and gj, • x, assumed small, is x = Hy\x

+ |tf 0-»Togof2 + ±Ho ' H^2xg 0 • x - gox2)]

(2.47)

where the subscript (0) and superscript (') on the functions T, H, E, and fi denote To = T(x* = xs = 0),

r 0 = ^r/a(xj = t = 0), since local test experiments are assumed to take place in vanishingly small regions surrounding 3". Because such experiments are designed to be electrically neutral overall, we can assume that the E and B fields do not extend outside this region. Then at 9

/NG= - X > o a f ( l -v2a) + (8TT)- hoT^Ho

1/2

J [ £ 2 - (To 'Hoeo Vo X)B2] d*x

(2.55)

We first see that, in general, /NG violates Local Lorentz Invariance. A simple Lorentz transformation of particle coordinates and fields in 7NG shows that JNG is a Lorentz invariant if and only if To 1 /f o £oVo 1 = l

or eofio=TolHo

(2.56)

Since we have not specified the event 0>, this condition must hold throughout the SSS spacetime. Notice that the quantity (TQ lHtfo Vo x ) 1/2 plays the role of the speed of light in the local frame, or more precisely, of the ratio of the speed of light clight to the limiting speed c0 of neutral test particles, i.e., To 'Hoeo Vo l = (clight/c0)2

(2.57)

Our units were chosen in such a way that, in the local freely falling frame, c0 = 1; equivalently, in the original THsfi coordinate system [cf. Equation (2.46)] c0 = (To/Ho)1'2,

clight = (EoAio)-1/2

(2.58)

These speeds will be the same only if Equation (2.56) is satisfied. If not, then the rest frame of the SSS field is a preferred frame in which / NG takes its THe/j. form, and one can expect observable effects in experiments that move relative to this frame. Thus, the quantity 1 — ToHo lfioMo plays the role of a preferred-frame parameter: if it is zero everywhere, the formalism is locally Lorentz invariant; if it is nonzero anywhere, there will be preferred-frame effects there. As we shall see, the Hughes-Drever experiment provides the most stringent limits on this preferred-frame parameter.

Einstein Equivalence Principle and Gravitation Theory

49

Next, we observe that / NG is locally position invariant if and only if o 1/2 = [constant, independent of 9\ o 1/2 = [constant, independent of ^>]

(2.59)

Even if the theory is locally Lorentz invariant (TQ 1H0EQ VO * = 1> independent of &) there may still be location-dependent effects if the quantities in Equation (2.59) are not constant. This would correspond, for example, to the situation discussed in Section 2.3, in which different parts of the local physical laws in a freely falling frame couple to different multiples of the Minkowski metric; in this case, free particle motion coupling to 7 itself, electrodynamics coupling to the position-dependent tensor i\* =. eTi/2H'i/2ti in the manner given by the field Lagrangian ri*'"'rivfFllvFltf. The nonuniversality of this coupling violates EEP and leads to position-dependent effects, for example, in gravitational redshift experiments (also see Section 2.4). An alternative way to characterize these effects in the case where Local Lorentz Invariance is satisfied is to renormalize the unit of charge and the vector potential at each event & according to e*a = ettso 1/2To 1/4 H S/4,

Af = A^T^H^

(2.60)

then the action, (2.55), takes the form

+ (8TT)- * j(E*2 - B*2) d4x

(2.61)

This action has the special relativistic form, except that the physically measured charge e* now depends on location via Equation (2.60), unless E0TII2HQXI2 is independent of 9. In the latter case, the units of charge can be effectively chosen so that everywhere in spacetime, soTlol2Ho112

= 1

(2.62)

Note, however, that if LPI alone is satisfied, one can renormalize the charge and vector potential to make either £0TQI2HQ 1/2 = 1 or fioTll2Ho 1/2 = 1, but not both, thus in general LLI need not be satisfied. Combining Equations (2.56), (2.59), and (2.62), we see that a necessary and sufficient condition for both Local Lorentz and Position Invariance to be valid is e0 = n0 = (Ho/T0)112, for all events 9 (2.63)

Theory and Experiment in Gravitational Physics

50

Consider now the terms in / NG , in Equations (2.50)-(2.52) that depend on the first-order displacements x, t from the event 9. These occur only in / em , and presumably produce polarizations of the electromagnetic fields of charged bodies proportional to the external "acceleration" g0 = V4>. One would expect these polarizations to result in accelerations of composite bodies made up of charged test particles relative to the local freely falling frame (i.e., relative to neutral test particles), in other words, to result in violations of WEP. These terms are absent if F o = Ao = 0, and U i.e., eoTlol2Ho

1/2

= const, noTl'2Ho EoHo^HoTo1

m

= const, (2.64)

Again, the units of charge can be normalized so that e0 = Ho = (H0/T0)1'2, for all 9

(2.65)

But this condition also guarantees Local Lorentz and Position Invariance. Thus, within the THe/j. formalism, for SSS fields [Equation (2.65)] => WEP, [Equation (2.65)] => EEP

(2.66)

However, the above discussion suggests that WEP alone may guarantee Equation (2.65) and thereby EEP. We can demonstrate this directly by carrying out an explicit calculation of the acceleration of a composite test body within the THsfi framework. The resulting restricted proof of Schiff's conjecture was first formulated by Lightman and Lee (1973a). (b) Proof of Schiff's conjecture We work in the global THsfi coordinate system in which JNG has the form Equation (2.46). Variation of / NG gives a complete set of particle equations of motion and "gravitationally modified" Maxwell (GMM) equations, given by (d/dt)(HW~ \ ) + {W- XV(T - Hvl) = aL(xa), aL(x0) = (ea/mOa){VAo(xa) + V[va • A(xfl)] - dA{xa)/dt}, V • (EE) = 4np, V x ( ^ » B ) = 4TTJ + d{eE)/dt where W = {T-

(2.67)

Hv2a)112,p = Y.aea\x=0. As long as the body is small compared to the scale over which d> varies, we can assume that g0 • x « 1, and work to first order in g0 • x. Second, we assume that the internal particle velocities and electromagnetic fields are sufficiently small so we can expand the equations of motion and GMM equations in terms of the small quantities v1 ~ e2/mor « 1 where r is a typical interparticle distance. By analogy with the postNewtonian expansion to be described in Chapter 4, we call this a postCoulombian expansion; for the purpose of the present discussion we shall work to first post-Coulombian order. We expect the single particle acceleration to contain terms that are O(g0) (bare gravitational acceleration), O(v2) (Coulomb interparticle acceleration), O(gf0t;2) (post-Coulombian gravitational acceleration), O(v4) (post-Coulombian interparticle acceleration), O(g0v*) (post-post-Coulombian gravitational acceleration), and so on. To O(g0v2), we obtain o + itf'oHo'goi;2 + (T'oTo 1 - H'0H0- l)g0 • vavfl + Ty2H» X W

(2-69)

To write the Lorentz acceleration aL(xa) directly in terms of particle coordinates, we must obtain the vector potential A^ in this form to an appropriate order. In a gauge in which £Mo,o - V • A = 0

(2.70)

the GMM equations take the form V 2 4 0 - ej^o.oo = 4ns~1p-e~1\e2

V A - e/xA.oo = ~^nJ

(\A0 - A,o), x

+ (sp)- V(sp)V • A - ^ V / z x (V x A) (2.71)

These equations can be solved iteratively by writing Ao = A^ + A%\

A = A(0) + A(1)

(2.72)

Theory and Experiment in Gravitational Physics

52

where A(v/A(°y ~ O(g0), and solving for each term to an appropriate order in v2. The result is A o = -4> A = A(0) + O(0O)

(2.73)

where

a

The resulting single-particle acceleration is inserted into a definition of center of mass. It turns out that to post-Coulombian order, it suffices to use the simple center-of-mass definition Y-

-iv

-v

2 J wOaXa>

A = m

a

m

(2-75) m

= ZJ

0a

a

We then compute d2X/dt2, substituting the single-particle equations of motion to the necessary order, and using the fact that, at t = 0, X = 0, dX/dt = 0. The resulting expression is simplified by the use of virial theorems that relate internal structure-dependent quantities to each other via total time derivatives of other internal quantities. As long as we restrict attention to bodies in equilibrium, these time derivatives can be assumed to vanish when averaged over intervals of time long compared to internal timescales. Errors generated by our choice of center-of-mass definitions similarly vanish. To post-Coulombian order, the required virial relation is =o (2.76) where angular brackets denote a time average, and where (2.77) ab

a

where xab = xa — xb, rab = |xa(,|, and the double sum over a and b excludes the case a = b. The final result is d2Xl/dt2 = g{-

^[TE1'%1rojea/m)

+ 0 J '[ro 1/2 eo 1 (l-T o Ho 1 eo/Xo)] x (JEjf/m +

(2.89) (2.90)

Einstein Equivalence Principle and Gravitation Theory

55

Notice that although the post-Coulombian corrections in Equation (2.89) may depend on the center of mass variables P or X, this dependence does not affect the form of if; it is only the explicit dependence on P and X in Equation (2.88) that generates the center-of-mass motion. The resulting average Hamiltonian is then rewritten in terms of V using VJ = d/dPJ. The conserved energy function Ec used in Section 2.5 is then defined to be Ec = Tll2Ho\H}, so that at lowest order, Ec = mc\ = m(ToHo *). The result is Ec = M(T0Ho ') + i x (£«%

)]

|[

^^o}

xT'oHo'go-X

(2.91)

where M = m +