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
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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'.4dc20 92-29555 CIP
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Contents
Preface to Revised Edition page xiii Preface to First Edition xv 1 Introduction 1 2 The Einstein Equivalence Principle and the Foundations of Gravitation Theory 13 2.1 The Dicke Framework 16 2.2 Basic Criteria for the Viability of a Gravitation Theory 18 2.3 The Einstein Equivalence Principle 22 2.4 Experimental Tests of the Einstein Equivalence Principle 24 2.5 Schiff 's Conjecture 38 2.6 The THsu Formalism 45 3 Gravitation as a Geometric Phenomenon 67 3.1 Universal Coupling 67 3.2 Nongravitational Physics in Curved Spacetime 68 3.3 Long-Range Gravitational Fields and the Strong Equivalence Principle 79 4 The Parametrized Post-Newtonian Formalism 86 4.1 The Post-Newtonian Limit 87 4.2 The Standard Post-Newtonian Gauge 96 4.3 Lorentz Transformations and the PPN Metric 99 4.4 Conservation Laws in the PPN Formalism 105 5 Post-Newtonian Limits of Alternative Metric Theories of Gravity 116 5.1 Method of Calculation 116 5.2 General Relativity 121 5.3 Scalar-Tensor Theories 123 5.4 Vector-Tensor Theories 126 5.5 Bimetric Theories with Prior Geometry 130 5.6 Stratified Theories 135 5.7 Nonviable Theories 138 ix Contents x
6 Equations of Motion in the PPN Formalism 142 6.1 Equations of Motion for Photons 143 6.2 Equations of Motion for Massive Bodies 144 6.3 The Locally Measured Gravitational Constant 153 6.4 N-Body Lagrangians, Energy Conservation, and the Strong Equivalence Principle 158 6.5 Equations of Motion for Spinning Bodies 163 7 The Classical Tests 166 7.1 The Deflection of Light 167 7.2 The Time-Delay of Light 173 7.3 The Perihelion Shift of Mercury 176 8 Tests of the Strong Equivalence Principle 184 8.1 The Nordtvedt Effect and the Lunar Eotvos Experiment 185 8.2 Preferred-Frame and Preferred-Location Effects: Geophysical Tests 190 8.3 Preferred-Frame and Preferred-Location Effects: Orbital Tests 200 8.4 Constancy of the Newtonian Gravitational Constant 202 8.5 Experimental Limits on the PPN Parameters 204 9 Other Tests of Post-Newtonian Gravity 207 9.1 The Gyroscope Experiment 208 9.2 Laboratory Tests of Post-Newtonian Gravity 213 9.3 Tests of Post-Newtonian Conservation Laws 215 10 Gravitational Radiation as a Tool for Testing Relativistic Gravity 221 10.1 Speed of Gravitational Waves 223 10.2 Polarization of Gravitational Waves 227 10.3 Multipole Generation of Gravitational Waves and Gravitational Radiation Damping 238 11 Structure and Motion of Compact Objects in Alternative Theories of Gravity 255 11.1 Structure of Neutron Stars 257 11.2 Structure and Existence of Black Holes 264 11.3 The Motion of Compact Objects: A Modified EIH Formalism 266 12 The Binary Pulsar 283 12.1 Arrival-Time Analysis for the Binary Pulsar 287 12.2 The Binary Pulsar According to General Relativity 303 12.3 The Binary Pulsar in Other Theories of Gravity 306 13 Cosmological Tests 310 13.1 Cosmological Models in Alternative Theories of Gravity 312 13.2 Cosmological Tests of Alternative Theories 316 Contents xi
14 An Update 320 14.1 The Einstein Equivalence Principle 320 14.2 The PPN Framework and Alternative Metric Theories of Gravity 331 14.3 Tests of Post-Newtonian Gravity 332 14.4 Experimental Gravitation: Is there a Future? 338 14.5 The Rise and Fall of the Fifth Force 341 14.6 Stellar-System Tests of Gravitational Theory 343 14.7 Conclusions 352 References 353 References to Chapter 14 371 Index 375
Preface to the Revised Edition
Since the publication of the first edition 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 con- frontation 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 high- precision 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 under- stand 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 Post- Newtonian (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 gravita- tional 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 per- forming 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 special- ized 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 Na- tional Science Foundation (PHY 76-21454, PHY 79-20123), the Alfred P. Sloan Foundation (BR 1700), and by a grant from the Mellon Foun- dation.
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 re- analysis 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 re- ceived 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 ac- cepted 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 "scalar- tensor" 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 2 plate of the star field around the location of the radio source 3C48. Al- though 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 per- mit the luminosity to vary systematically over timescales as short as days or hours. Many theorists turned to general relativity and to the strong relativistic gravitational fieldsi t predicts, to provide the mechanism under- lying 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 iden- tification 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 rela- tivity 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" inten- sified 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 Introduction 3 bridging of the gaps between mathematics and physics and mathematics and astrophysics by such workers as Bondi, Dicke, Sciama, Pirani, Pen- rose, 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 pro- gram 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 preci- sion. The next two decades would see further uses of quantum technology in such high-precision tools as atomic clocks, laser ranging, supercon- ducting 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 develop- ment 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 dis- covery in 1964 of the relativistic time-delay effect (Shapiro, 1964), it pro- vided 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 dis- cussions 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 measure- ments of these effects to accuracy within 10%, 1%, or fractions of a per- cent 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 4
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 sim- plicity. His primary goal was to produce a gravitation theory that incor- porated 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 inverse- square 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 obser- vations. 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) 5025'.'6 Venus 211".% Earth 9070 Mars 275 Jupiter 15376 Saturn 773 Others 072
Sum 555770 Observed Advance 559977 Discrepancy 4277 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 the fieldo f 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 con- vinced of the "correctness" of the theory because of its elegance and inter- nal 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 un- known 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 equiv- alence, by Einstein (1911). Beyond that, "the subject [was] still a live one" (Bertotti et al., 1962). The third classical test was actually the first proposed 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 spec- tral 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 gravita- tional red shift the results were inconclusive, and it would be 1962 before a reliable solar red-shift measurement would be made. Similarly incon- clusive 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 complica- tions in their spectra due to scattered light from their companions, reli- able, precise measurements were not possible [see Bertotti et al. (1962) for a review]. Furthermore, by the late 1950s, it was being suggested that the gravita- tional 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*
Result in Approximate Minimum distance from units of Results from number center of Sun Einstein different Eclipse of stars in solar radii prediction analyses
1919 7 2 1.13 + 0.07 1.0 to 1.3 1919 5 2 0.92 + 0.17 1922 92 2.1 0.98 ± 0.06 1.3 to 0.9 1922 145 2.1 1.04 + 0.09 1.2 1922 14 2 0.7 to 1.3 1922 18 2 0.8 to 1.2 1929 17 1.5 1.28 ±0.06 0.9 to 1.2 1936 25 2 1.55 + 0.15 1.55 ± 0.2 1936 8 4 0.7 to 1.2 1947 51 3.3 1.15 ±0.15 1.0 to 1.4 1952 10 2.1 0.97 + 0.06 0.82 ± 0.09 1973" 39 2 0.95 + 0.11 a See Bertotti et al. (1962) for details. b Texas Mauritanian Eclipse Team (1976), Jones (1976). 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 accelera- tion (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 con- fronted 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 ex- pansion 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. Never- theless, 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, Birk- hoff, 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 preci- sion (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 challenging field, with many new experimental and theoretical possibilities. These included new ver- sions 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 oc- curred in the two decades following 1959. In many ways, the years 1960- 1980 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 Penrose paper on spinors experiments Gyroscope precession (Schiff) Pound-Rebka gravitational red-shift experiment Brans-Dicke theory 1962 Discovery of nonsolar x-ray sources Bondi mass-loss formula Discovery of quasar red shifts Kerr metric discovery Princeton Eotvos experiment 1964 Time-delay of light (Shapiro) Pound-Snider red-shift experiment Discovery of 3K microwave Singularity theorems in background general relativity 1966 Reported detection of solar Element production in the oblateness big bang Discovery of pulsars 1968 Planetary radar measurement of time Nordtvedt effect and early delay PPN framework Launch of Mariners 6 and 7 Acquisition of lunar laser echo First radio deflection measurements 1970 Preferred-frame effects CygXl: a black hole candidate Refined PPN framework Mariners 6 and 7 time-delay Area increase of black holes in measurements general relativity 1972 Moscow Eotvos experiment 1974 Quantum evaporation of black holes Discovery of binary pulsar Dipole gravitational radiation in alternative theories 1976 Rocket gravitational red-shift experiment Lunar test of Nordtvedt effect Time delay results from Mariner 9 and Viking 1978 Measurement of orbit period decrease in binary pulsar SS433 1980 Discovery of gravitational lens 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 sim- ilarities 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 div- idend of the Dicke framework is the understanding that gravitational experiments can be divided into two classes. The first consists of experi- ments that test the foundations of gravitation theory, one of these founda- tions being the principle of equivalence. These experiments (Eotvos experiment, Hughes-Drever experiment, gravitational red-shift experi- ment, 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 theo- ries of gravity. Here another theoretical framework was developed that takes up where the Dicke framework leaves off. Known as the "Param- etrized Post-Newtonian" or PPN formalism, it was pioneered by Ken- neth 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), preferred- frame 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 at- tempts 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 Brans- Dicke 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 pre- dictions, 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 Tay- lor, 1975). Here was a system that combined large post-Newtonian grav- itational 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 radio- telescope monitoring of the extremely stable pulsar clock. It was also a system where relativistic gravity and astrophysics became even more in- tertwined than in the case, say, of quasars. In the binary pulsar, relativistic gravitational effects provided a means for accurate measurement of astro- physical 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 electromag- netic 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 ac- count 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 develop- ment 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 "iner- tial mass" mb and "passive gravitational mass" mP, to refer to these quan- tities, 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 gravi- tational field with the same acceleration regardless of their mass or in- ternal 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 gravita- tional 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 HUS AIR of a double density, in a double space, is quadruple in quan- tity; in a triple space, sextuple in quantity. The same thing is to be Tunderstood 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 pen- dulums, 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. tl 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 under- standing of the significance of these principles of equivalence for gravi- tation 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 mani- festation 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 gravi- tation was spelled out in Appendix 4 of Dicke's Les Houches lectures Einstein Equivalence Principle and Gravitation Theory 17
(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-VT<0, V
Riem(>r) = 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). 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 inter- preting 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 tensor field that is approxi- mated 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 vector fieldcouplin g 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 Frame- work 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 Is not relativistic Milne's kinematical relativity Was devised originally to handle certain (Milne, 1948) cosmological problems. Is incomplete: makes no gravitational red-shift prediction Kustaanheimo's various vector Contain a vector gravitational field in flat theories (Kustaanheimo and spacetime. Are incomplete: do not mesh with Nuotio, 1967; Whitrow and the other nongravitational laws of physics Morduch, 1965) (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. Poincare's theory (as Action-at-a-distance theory in flat spacetime. generalized by Whitrow and Is incomplete or inconsistent in the same Morduch, 1965) manner as Kustaanheimo's theories Whitrow-Morduch (1965) Contains a vector gravitational field in flat vector theory spacetime. Is incomplete or inconsistent in the same manner as Kustaanheimo's theories. Birkhoff's (1943) theory Contains a tensor gravitational field used to construct a metric. Violates the Newtonian limit by demanding that p = pc2, i.e. ''sound = "light- Yilmaz's (1971,1973) theory 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 special- izations 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 com- pleteness, 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 geo- metrical 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 re- cent 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 four- momentum conservation and of the relativistic laws of kinematics, elec- tron and muon "g-2" experiments, and tests of esoteric predictions of Lorentz-in variant quantum fieldtheorie s [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 Min- kowski 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 mea- sured 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 experi- mental 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 astro- nomical 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 22
2.3 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 curved- spacetime 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, Brans- Dicke 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 space- time 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 nongravitational fields,suc h as boson fields,fermio n fields,elec - tromagnetic 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 co- efficients, that is, the scalar fields 4>(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 scalar fields {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 (re- definition 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 these fields are one and the same: a unique, symmetric second- rank 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 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 com- parison 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 A A ^ ( + S r, E /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 2 2 v K + a\ t \mC m2c j ' Thus, experimental limits on r\ place limits on the WEP-violation param- eters 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 heuris- tically. Two objects of different composition are connected by a rod of length r, and suspended in a horizontal orientation by a finewir e ("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 Figure 2.2. Schematic arrangement of a torsion-balance Eotvos experi- ment; g is the external gravitational acceleration, and to is the angular velocity vector about which the apparatus is rotated. The unit vectors ew and er are parallel to the wire and rod, respectively. In the Eotvos experi- ments, g was the acceleration toward the Earth, and to was parallel to ew; in the Princeton and Moscow experiments g was that of the Sun, and co was parallel to the Earth's rotation axis. \\W\\\\\ including magnetic levitation (Worden and Everitt, 1974), flotation on liquids (Keiser and Faller, 1979), and free fall in orbit. Experiments to test WEP for individual atoms and elementary particles have been inconclusive or inaccurate, with the exception of neutrons (Fairbank et al, 1974 and Koester, 1976). Table 2.2 discusses various experiments and quotes the limits they set on Y) for different pairs of materials. Future improved tests of WEP must reduce noise due to thermal, seismic, and gravity-gradient effects, and may have to be performed in space using cryogenic techniques. Antici- pated limits on r\ in such experiments range between 10""15 and 10"18 (Worden, 1978). To determine the limits placed on individual parameters r\K by, say, the best of the torsion-balance experiments, we must estimate the co- Table 2.2. Tests of the weak equivalence principle Experiment Reference Method Substances tested Limit on \tj\ Newton Newton (1686) Pendula Various 10"3 Bessel Bessel (1832) Pendula Various 2 x 10"5 Eotvos Eotvos, Pekar, and Fekete (1922) Torsion balance Various 5 x 10"9 Potter Potter (1923) Pendula Various 2 x 10"5 Renner Renner (1935) Torsion balance Various 2 x lO"9 Princeton Roll, Krotkov, and Dicke (1964) Torsion balance Aluminum and gold io-» Moscow Braginsky and Panov (1972) Torsion balance Aluminum and platinum lO"12 Munich Koester (1976) Free fall Neutrons 3 x 10"4 Stanford Worden (1978) Magnetic suspension Niobium, Earth io-4 Boulder Keiser and Faller (1979) Flotation on water Copper, tungsten 4x 10" u Orbital- Worden (1978) Free fall in orbit Various 10~15 - lO"18" " Experiments yet to be performed. Theory and Experiment in Gravitational Physics 28 efficients EA/m for the different interactions and for different materials. For laboratory-sized bodies, the dominant contribution to £A comes from the atomic nucleus. We begin with the strong interactions. The semiempirical mass formula (see, for example, Leighton, 1959) gives Es = -15.74 + 17.SA213 + 23.604 - 2Z)2A~l + 132/1"lS MeV (2.5) where Z and A are the atomic number and mass number, respectively, of the nucleus, and where S = 1 if {Z,A} = {odd, even}, 5 1 if {Z,A} = {even, even} and 8 = 0 if A is odd. Then, Es/mc2 = -1.7 x 1(T2 + 1.9 x 10"2,4"1/3 + 2.5 x 10~2(l - 2Z/A)2 + 1.41 x 10"M-2<5 (2.6) For platinum (Z = 78, A = 195), and aluminum (13,27) the difference in Es/mc2 is approximately 2 x 10"3, so from the limit \n\ < 10"12, we obtain the limit \ns\ < 5 x 10"10. In the case of electromagnetic interactions, we can distinguish among a number of different internal energy contributions, each potentially having its own n* parameter. For the electrostatic nuclear energy, the semiempirical mass formula yields the estimate £ES = 0.71Z(Z - X)A ~1/3 MeV (2.7) Thus, EES/mc2 = 7.6 x 10"4Z(Z - \)A"4/3 (2.8) with the difference for platinum and aluminum being 2.5 x 10"3. The ES 10 resulting limit on n is |T/ES| < 4 X 10" . Another form of electromag- netic energy is magnetostatic, resulting from the nuclear magnetic fields generated by the proton currents. To estimate the nuclear magnetostatic energy requires a detailed shell model computation. For example, the net proton current in any closed angular momentum shell vanishes, hence there is no energy associated with the magnetostatic interaction between such a closed shell and any particle outside the shell. For aluminum and platinum, Haugan and Will (1977) have shown MS 2 7 M 2 7 (£ /mc )A1 = 4.1 x 10~ , (£ 7mc )pt = 2.4 x 10" (2.9) thus \r\us\ < 6 x 10 "6. A third form of electromagnetic energy that has been studied is hyperfine, the energy of interaction between the spins of the nucleons and the magnetic fields generated by the proton and neutron magnetic moments. Computations by Haugan (1978) have yielded the Einstein Equivalence Principle and Gravitation Theory 29 estimate HF 2 2 2 2 £ = (2n/V)ntig pZ + g (A - Z) ] (2.10) where V is the nuclear volume, (iN is the nuclear magneton, and gp = 2.79 and ga = 1.91 are gyromagnetic ratios for the proton and neutron, respectively. Then, Em/mc2 = 2.1 x 10"5[>2Z2 + g2(A - Zf^A'2 (2.11) with the difference between aluminum and platinum being 4 x 10~6; thus|>7HF|<2 x 10" 7. For some time, it was believed that the contribution of the weak interactions to nuclear energy was of the order of a part in 1012, and that the Eotvos experiment was not yet sufficiently accurate to test WEP for weak interactions (see for example, Chiu and Hoffman, 1964; Dicke, 1964a). However, these estimates took into account only the parity non- conserving parts of the weak interactions, which make no contribu- tion to the energy of a nucleus in its ground state, to first order in the weak-interaction coupling constant Gw. On the other hand, the parity- conserving parts of the weak interactions do contribute at first order in Gw and yield a value E/me2 ~ 10"8 (Haugan and Will, 1976). Specifically, in the Weinberg-Salam model for weak and electromagnetic interactions, the result is E^/mc2 = 2.2 x 10"8(iVZM2)[l + g(N,Z)], 2 2 g(N,Z) = 0.295[i(iV - Z) /ATZ + 4 sin 0W 2 2 + (Z/N) sin 0W(2 sin 0W - 1)] (2.12) where N = (A Z) is the neutron number, and where 0W ~ 20° is the "Weinberg" angle. For aluminum and platinum, the difference is 2 x 10~10, yielding |>?w| < 10~2. Gravitational interactions are specifically excluded from WEP and EEP. In Chapter 3, we shall extend these two principles to incorporate local gravitational effects, thereby defining the Gravitational Weak Equiv- alence Principle (GWEP) and the Strong Equivalence Principle (SEP). These two principles will be useful in classifying alternative metric theories of gravity. In any case, for laboratory Eotvos experiments, gravitational interactions are totally irrelevant, since for an atomic nucleus a 2 2 39 E /mc ~ Gmp/Raucleusc ~ 10" To test for gravitational effects in GWEP, it will be necessary to employ planetary objects and planetary Eotvos experiments (Section 8.1). Theory and Experiment in Gravitational Physics 30 (b) Tests of Local Lorentz Invariance Any experiment that purports to test special relativity (Section 2.2) also tests some aspect of Local Lorentz Invariance, since every Earth- bound laboratory resides in a gravitational field (although it is only par- tially in free fall). However, very few of these experiments have been used to make quantitative tests of LLI in the same way that Eotvos experiments have been used to test WEP. For example, although elementary-particle experimental results are consistent with the validity of Lorentz invariance in the description of high-energy phenomena, they are not "clean" tests because in many cases it is unlikely that a violation of Lorentz invariance could be distinguished from effects due to the complicated strong and weak interactions. For instance, the observed violation of conservation of four momentum in beta decay was found to be due not to a violation of LLI, but to the emission of a hitherto unknown particle, the neutrino. However, there is one experiment that can be interpreted as a "clean" test of Local Lorentz invariance, and an ultrahigh precision one at that. This is the Hughes-Drever experiment, performed in 1959-60 indepen- dently by Hughes and collaborators at Yale University and by Drever at Glasgow University (Hughes et al., 1960; Drever, 1961). In the Glasgow version, the experiment examined the J = § ground state of the 7Li nu- cleus in an external magnetic field. The state is split into four levels by the magnetic field, with equal spacing in the absence of external perturba- tions, so the transition line is a singlet. Any external perturbation asso- ciated with a preferred direction in space (the velocity of the Earth relative to the mean rest frame of the universe, for example) that has a quadrupole (/ = 2) component will destroy the equality of the energy spacing and split the transition lines. Using NMR techniques, the experiment set a limit of 0.04 Hz (1.7 x 10"16 eV) on the separation in frequency (energy) of the lines. One interpretation of this result is that it sets a limit on a possible anisotropy 3m\j in the inertial mass of the 7Li nucleus: |5mjJc2| ;$ 1.7 x 10"16 eV. If any of the forms of internal energy of the 7Li nucleus suffered a breakdown of Local Lorentz Invariance, one would expect a contribution to 5m{J of the form dmij ~ X 5AEx/c2 (2.13) A where <5A is a dimensionless parameter that measures the strength of anisotropy induced by interaction A. Using formulae from Section 2.4(a), we can then make estimates of EA for 7Li and obtain the following limits Einstein Equivalence Principle and Gravitation Theory 31 on<5\ |<5S| < 1(T23, |<5HF| < 5 x 1(T22, |£ES| < 10-22) |gW| < 5 x 1Q-18 (2.14) Notice that the magnetostatic energy for 7Li is zero, since the proton shell structure is ls1/2lp3/2 and there is no magnetostatic interaction either within the closed s-shell (/ = 0) or between that shell and the valence proton. Because of the remarkably small size of these limits, the Hughes- Drever experiment has been called the most precise null experiment ever performed. If Local Lorentz Invariance is violated, then there must be a preferred rest frame, presumably that of the mean rest frame of the universe, or, equivalently of the cosmic microwave background, in which the local laws of physics take on their special form. Deviations from this form would then depend on the velocity of the laboratory relative to the pre- ferred frame. Since the anisotropy is a quadrupole effect, one would expect it to be proportional to the square of the velocity w of the laboratory. If <>o is a parameter that measures the "bare" strength of LLI violation, then one would expect For the motion of the Earth relative to the universe rest frame, w2 ~ 10 ~6. Limits on the <5Q can then be inferred from Equation (2.14). As a special case of this general argument, the Hughes-Drever experiment has also been interpreted as a test of the existence of additional long-range tensor fields that couple directly to matter (Peebles and Dicke, 1962; Peebles, 1962). Other experiments that can be interpreted as tests of LLI include var- ious ether-drift experiments, such as the Turner-Hill experiment (Dicke, 1964a; Haugan, 1979). (c) Tests of Local Position Invariance The two principal tests of Local Position Invariance are gravita- tional red-shift experiments that test the existence of spatial dependence on the outcomes of local experiments, and measurements of the constancy of the fundamental nongravitational constants that test for temporal dependence. Theory and Experiment in Gravitational Physics 32 Gravitational Red-Shift Experiments A typical gravitational red-shift experiment measures the frequency or wavelength shift Z = Av/v = AA/A between two identical frequency standards (clocks) placed at rest at different heights in a static gravitational field. To illustrate how such an experiment tests LPI, we shall assume that the remaining parts of EEP, namely WEP and Local Lorentz Invariance, are valid. (In Sec- tions 2.5 and 2.6, we shall discuss this question under somewhat different assumptions.) WEP guarantees that there exist local freely falling frames whose acceleration g relative to the static gravitational field is the same as that of test bodies. Local Lorentz Invariance guarantees that in these frames, the proper time measured by an atomic clock is related to the Minkowski metric by c2dx2 oc - r\^dx% dx\ oc c2dt\ - dx\ - dy\ - dz% (2.15) where x% are coordinates attached to the freely falling frame. However, in a local freely falling frame that is momentarily at rest with respect to the atomic clock, we permit its rate to depend on its location (violation of Local Position Invariance), that is, relative to an arbitrarily chosen atomic time standard based on a clock whose fundamental structure is different than the one being analyzed, the proper time between ticks is given by T = T(O) (2.16) where O is a gravitational potential whose gradient is related to the test- body acceleration by g = £V. Now the emitter, receiver, and gravitational field are assumed to be static, therefore in a static coordinate system (ts,xs), the trajectories of successive wave crests of emitted signal are identical except for a time translation Ats from one crest to the next. Thus, the interval of time Afs between ticks (passage of wave crests) of the emitter and of the receiver must be equal (otherwise there would be a build up or depletion of wave crests between the two clocks, in violation of our assumption that the situation is static). The static coordinates are not freely falling coordi- nates, but are accelerated upward (in the +z direction) relative to the 2 freely falling frame, with acceleration g. Thus, for \gts/c\ ~ |#zs/c | « 1 (i.e., for g uniform over the distance between the clocks), a sequence of Lorentz transformations yields (MTW, Section 6.6) 2 ctF = (zs + c /g)sinh{gts/c), 2 zF = (zs + c /g)cosh(gts/c), xF=xs, yF = ys (217) Einstein Equivalence Principle and Gravitation Theory 33 Thus, the time measured by the atomic clocks (relative to the standard clock) is given by c2dx2 = T2( For small separations, Az = zrec zem, we can expand T( T(4>rec) = T0 + ^r^Az (2.20) where T0 = T( 2 1 where a =C C~ TJ)/T0 and where AC/ = g Az = g(zrec zem). If there is no location dependence in the clock rate, then a = 0, and the red shift is the standard prediction, i.e., Z = AU/c2 (2.22) An alternative version of this argument assumes the validity of both LLI and LPI and shows that, if the red shift is given by Equation (2.22), then the acceleration of the local frames in which Lorentz and Position Invariance hold is the same as that of test bodies, i.e., the local frames are freely falling frames (Thorne and Will, 1971). Although there were several attempts following the publication of the general theory of relativity to measure the gravitational red shift of spec- tral lines from white dwarf stars, the results were inconclusive (see Bertotti et al., 1962 for a review). The first successful, high-precision red-shift mea- surement was the series of Pound-Rebka-Snider experiments of 1960-65, which measured the frequency shift of y-ray photons from Fe57 as they ascended or descended the Jefferson Physical Laboratory tower at Har- vard University. The high accuracy achieved (1%) was obtained by making use of the Mossbauer effect to produce a narrow resonance line whose shift could be accurately determined. Other experiments since 1960 measured the shift of spectral lines in the Sun's gravitational field and the change in rate of atomic clocks transported aloft on aircraft, rockets, Table 2.3. Gravitational red-shift experiments Experiment Reference Method Limit on |a| Pound-Rebka-Snider Pound and Rebka (1960) Fall of photons from io-2 Pound and Snider (1965) Mossbauer emitters Brault Brault (1962) Solar spectral lines 5 x 10"2 Jenkins Jenkins (1969) Crystal oscillator clocks 9x 10"2 on GEOS-1 satellite Snider Snider (1972,1974) Solar spectral lines 6 x 10"2 Jet-Lagged Clocks (A) Hafele and Keating (1972a,b) Cesium beam clocks on 10"1 jet aircraft Jet-Lagged Clocks (B) Alley (1979) Rubidium clocks on 2 x 10"2 jet aircraft Vessot-Levine Rocket Vessot and Levine (1979) Hydrogen maser on rocket 2 x 10"4 Red-shift Experiment Vessot et al. (1980) Null Red-shift Experiment Turneaure et al. (1983) Hydrogen maser vs. SCSO io-2 Close Solar Probe" Nordtvedt (1977) Hydrogen maser or SCSO 10-6« on satellite ' Experiments yet to be performed. Einstein Equivalence Principle and Gravitation Theory 35 and satellites. Table 2.3 summarizes the important red-shift experiments that have been performed since 1960. Recently, however, a new era in red-shift experiments has been ushered in with the development of frequency standards of ultrahigh stability - parts in 1015 to 1016 over averaging times of 10 to 100 s and longer. Ex- amples are hydrogen-maser clocks (Vessot, 1974), superconducting-cavity stabilized oscillator (SCSO) clocks (Stein, 1974; Stein and Turneaure, 1975), and cryogenically cooled monocrystals of dielectric materials such as silicon and sapphire (McGuigan et al., 1978). The first such experi- ment was the Vessot-Levine Rocket Red-shift Experiment that took place in June, 1976. A hydrogen-maser clock was flown on a rocket to an altitude of about 10,000 km and its frequency compared to a similar clock on the ground. The experiment took advantage of the high frequency stability of hydrogen-maser clocks (parts in 1015 over 100 s averaging times) by monitoring the frequency shift as a function of altitude. A so- phisticated data acquisition scheme accurately eliminated all effects of the first-order Doppler shift due to the rocket's motion, while tracking data were used to determine the payload's location and velocity (to eval- uate the potential difference AU, and the second-order Doppler shift). Analysis of the data yielded a limit (Vessot and Levine, 1979; Vessot et al., 1980) |a| < 2 x 10"4 (2.23) Coincidentally, the Scout rocket that carried the maser aloft stood 22.6 m in its gantry, almost exactly the height of the Harvard Tower. In an inter- planetary version of this experiment, a stable clock (H-maser or SCSO clock) would be flown on a spacecraft in a very eccentric solar orbit (closest approach ~4 solar radii); such an experiment could test a to a part in 106 (Nordtvedt, 1977) and could conceivably look for "second- order" red-shift effects of O(AC/)2 (Jaffe and Vessot, 1976). Advances in stable clocks have also made possible a new type of red- shift experiment that is a direct test of Local Position Invariance (LPI): a "null" gravitational red-shift experiment that compares two different types of clocks, side by side, in the same laboratory. If LPI is violated, then not only can the proper ticking rate of an atomic clock vary with position, but the variation must depend on the structure and composition of the clock, otherwise all clocks would vary with position in a universal way and there would be no operational way to detect the effect (since one clock must be selected as a standard and ratios taken relative to that clock). Theory and Experiment in Gravitational Physics 36 Thus, we must write for a given clock type A, A A 2 T = t\U) = T (1 - a AU/c ) (2.24) Then a comparison of two different clocks at the same location would measure TA/TB = (TA/tB)o [ 1 _ (aA _ aB) A [//C2] £.25) A B s tne where (T /T )0 i constant ratio between the two clock times observed at a chosen initial location. A null red-shift experiment of this type was performed in April, 1978 at Stanford University. The rates of two hydrogen maser clocks and of an ensemble of three SCSO clocks were compared over a 10 day period (Turneaure et al., 1983). During this time, the solar potential U changed sinusoidally with a 24 hour period by 3 x 10~13 because of the Earth's rotation, and changed linearly at 3 x 10""12 per day because the Earth is 90° from perihelion in April. However, analysis of the data set an upper limit on both effects, leading to a limit on the LPI violation parameter |aH_ascso|< 10-2 (2.26) The art of atomic timekeeping has advanced to such a state that it may soon be necessary to take red-shift and Doppler-shift corrections into account in making comparisons between timekeeping installations at different altitudes and latitudes. Constancy of the constants The other key test of Local Position Invariance is the constancy of the nongravitational constants over cos- mological timescales (we delay discussion of the gravitational "constant" until Section 8.5). We shall not review here the various theories and proposals, originating with Dirac, that permit variable fundamental con- stants [for detailed review and references, see Dyson (1972)], rather we shall cite the most recent observational evidence (Table 2.4). The observa- tions range from comparisons of spectral lines in distant galaxies and quasars, to measurements of isotopic abundances of elements in the solar system, to laboratory comparisons of atomic clocks. Recently, Shlyakhter (1976a,b) has made significant improvements in the limits on variations in the electromagnetic, weak, and strong coupling constants by studying isotopic abundances in the "Oklo Natural Reactor," a sustained U235 fission reactor that evidently occurred in Gabon, Africa nearly two billion years ago (Maurette, 1976). Measurements of ore samples yielded an abnormally low value for the ratio of two isotopes of samarium (Sm149/ Table 2.4. Limits on cosmological variation of nongravitational constants Limit on kjk per Hubble time Limit from 2 x 1010 yr Oklo reactor 1 1 Constant k (H0 = 55kms" Mpc" ) Method Reference (Shlyakhter, 1976a,b) Fine structure Constant: a = e2/hc 4 x 10"4 Re187 ft decay rate over Dyson (1972) geological time 8 x 10"2 Mgll fine structure and Wolfe, Brown, and Roberts 10"7 21 cm line in radio (1976) source at Z = 0.5 8 x 10"2 SCSO clock vs. cesium Turneaure and Stein (1976) beam clock Weak Interaction Constant: 3 187 40 2 P = g(mlc/h 2 Re , K decay rates Dyson (1972) 2 x 10~ Electron-Proton Mass Ratio: mjmp 1 Mass shift in quasar Pagel (1977) spectral lines (Z ~ 2) Proton Gyromagnetic Factor: 1 g me/mp 10" Mgll, 21 cm line Wolfe, Brown, and Roberts (1976) Strong Interactions: gl 8 x 10"2 Nuclear stability Davies (1972) 8 x 10"9 Theory and Experiment in Gravitational Physics 38 Sm147). Neither of these isotopes is a fission product, but Sm149 can be depleted by a dose of neutrons. Estimates of the neutron fluence (inte- grated dose) during the reactor's "on" phase, combined with the measured abundance anomaly yielded a value for the neutron capture cross section for Sm149 two billion years ago which agrees with the modern value. However, the capture cross section is extremely sensitive to the energy of a low-lying level (E ~ 0.1 eV) of Sm149, so that a variation of only 20 x 10 ~3 eV in this energy over 109 years would change the capture cross section from its present value by more than the observed amount. By estimating the contributions of strong, electromagnetic, and weak interactions to this energy, Shlyakhter obtained the limits on the rate of variation of the corresponding coupling constants shown in Table 2.4, column 5 (see also Dyson, 1978). 2.5 Schiff 's Conjecture Because the three parts of the Einstein Equivalence Principle dis- cussed above are so very different in their empirical consequences, it is tempting to regard them as independent theoretical principles. However, any complete and self-consistent gravitation theory must possess suffi- cient mathematical machinery to make predictions for the outcomes of experiments that test each principle, and because there are limits to the number of ways that gravitation can be meshed with the special relativistic laws of physics, one might not be surprised if there were theoretical con- nections between the three subprinciples. For instance, the same mathe- matical formalism that produces equations describing the free fall of a hydrogen atom must also produce equations that determine the energy levels of hydrogen in a gravitational field, and thereby determine the ticking rate of a hydrogen maser clock. Hence a violation of EEP in the fundamental machinery of a theory that manifests itself as a violation of WEP might also be expected to emerge as a violation of Local Position Invariance. Around 1960, Leonard I. Schiff conjectured that this kind of connection was a necessary feature of any self-consistent theory of gravity. More precisely, Schiff's conjecture states that any complete, self-consistent theory of gravity that embodies WEP necessarily embodies EEP. In other words, the validity of WEP alone guarantees the validity of Local Lorentz and Position Invariance, and thereby of EEP. This form of Schiff's con- jecture is an embellished classical version of his original 1960 quantum mechanical conjecture (Schiff, 1960a). His interest in this conjecture was rekindled in November, 1970 by a vigorous argument with Kip S. Thorne at a conference on experimental gravitation held at the California Insti- Einstein Equivalence Principle and Gravitation Theory 39 tute of Technology. Unfortunately, his untimely death in January, 1971 cut short his renewed effort. If Schiff's conjecture is correct, then the Eotvos experiments may be seen as the direct empirical foundation for EEP, and for the interpretation of gravity as a curved-spacetime phenomenon. Some authors, notably Schiff, have gone further to argue that if the conjecture is correct, then gravitational red-shift experiments are weak tests of gravitation theory, compared to the more accurate Eotvos experiment. For these reasons, much effort has gone into "proving" Schiff's conjecture. Of course, a rigorous proof of such a conjecture is impossible, yet a number of powerful "plausibility" arguments using a variety of assumptions can be formulated. The most general and elegant of these arguments is based upon the assumption of energy conservation. This assumption allows one to per- form very simple cyclic gedanken experiments in which the energy at the end of the cycle must equal that at the beginning of the cycle. This ap- proach was pioneered by Dicke (1964a), and subsequently generalized by Nordtvedt (1975) and Haugan (1979). Specifically, we restrict attention to theories of gravity in which there is a conservation law of energy for nongravitating "test" systems that reside in given static and external gravitational fields. To guarantee the existence of such a law, it is sufficient for the theory to be based on an invariant action principle [cf. Dicke's constraint (2)], but it is not necessary. We consider an idealized composite body made up of structureless test particles that interact by some nongravitational force to form a bound system. For a system that moves sufficiently slowly in a weak, static gravitational field,th e laws governing its motion can be put into a quasi- Newtonian form (we assume the theory has a Newtonian limit); in par- ticular, the conserved energy function Ec associated with the conservation law can be assumed to have the general form 2 2 2 2 Ec = MKc - MRU(X) + |MRF + O(MRU , MRV\MRUV ) (2.27) where X and V are quasi-Newtonian coordinates of the center of mass of the body, MR is the "rest" energy of the body, U is the external gravita- tional potential, and c is a fundamental speed used to convert units of mass into units of energy. If EEP is violated, we must allow for the possibility that the speed of light and the limiting speed of material particles may differ in the presence of gravity; to maintain this possibility we do not set c = 1 automatically in Equation (2.27) (see also footnote, p. 24). Note that V is the velocity relative to some preferred frame. In Theory and Experiment in Gravitational Physics 40 problems involving external, static gravitational potentials, the preferred frame is generally the rest frame of the external potential, while in problems involving cosmological gravitational effects where localized potentials can be ignored, the preferred frame is that of the universe rest frame. [In problems involving both kinds of effects, the simple form of Equa- tion (2.27) no longer holds.] The possible occurrence of EEP violations 2 arises when we write the rest energy MRc in the form 2 2 MRc = Moc - £B(X, V) (2.28) where Mo is the sum of the rest masses of the structureless constituent particles, and £B is the binding energy of the body. It is the position and velocity dependence of £B, a dependence that in general is a function of the structure of the system, which signals the breakdown of EEP. Roughly speaking, an observer in a freely falling frame can monitor the binding energy of the system, thereby detecting the effects of his location and velocity in local nongravitational experiments. Haugan (1979) has made this more precise by showing that in fact it is the possible functional difference in £B(X, V) between the system under study and a "standard" system arbitrarily chosen as the basis for the units of measurement that leads to measurable effects. Because the location and velocity dependence in £B is a result of the external gravitational environment, it is useful to expand it in powers of U and V2. To an order consistent with the quasi-Newtonian approximation in Equation (2.27), we write iJ 1 2 £B(X,V) = El + 8myU (X) - tfntfV'V + O(E%U ,...) (2.29) where UiJ is the external gravitational potential tensor [cf. Equa- tion (4.28)]; it is of the same order as U and satisfies U" = U. The quan- tities <5w# and bm\' are called the anomalous passive gravitational and inertial mass tensors, respectively. They are expected to be of order t\E%, where r\ is a dimensionless parameter that characterizes the strength of EEP-violating effects; they depend upon the detailed internal structure of the composite body. Summation over repeated spatial indices i, j is assumed. The conserved energy can thus be written, to quasi-Newtonian order, 2 2 Ec = (M0c - £g) - [(Mo - E°c- )S» 2 ^'J + O(M0U ,...) (2.30) We first give examples of violations of Local Position and Lorentz Invariance generated by £B(X, V). Consider two different systems at rest in the gravitational potential. Each system makes a transition from one Einstein Equivalence Principle and Gravitation Theory 41 quantum energy level to another, and emits a quantum of frequency v = AEc/h. The ratio of the two frequencies is given from Equation (2.30) by 2 L (AEE)x (AEg)2Jc In the case where <5mj/ oc diJ, the quantity in square brackets can be identified as the coefficient a. 1 a2 in Equation (2.25). Thus the anomalous passive gravitational mass tensor dmtf produces preferred-location effects in a null gravitational red-shift experiment. Consider the same two systems far from gravitating matter, but moving relative to the universe rest frame with velocity V. Then the ratio of the two frequencies is given by jAQSmiV A(<5m{V2l V'VJ 1 2 (A£g)21 \ (A£°)x (A£°)2 J c Thus, the anomalous inertial mass tensor produces preferred-frame effects in an experiment such as the Hughes-Drever experiment, where the two systems in question are two excited states of 7Li nuclei of different azi- muthal quantum numbers in an external magnetic field. In this case, because the Zeeman splitting is the same for all levels in the absence of a preferred-frame effect, (A£B)X = (A£B)2, however because of the possible anisotropy in dm?, one would expect A(dm\J) to differ for transitions between different pairs of levels [for further details, see Section 2.6]. Thus dmy is responsible for violations of Local Position Invariance and <5m{J is responsible for violations of Local Lorentz Invariance. In order to verify Schiff s conjecture, it remains only to show that 5mj,J and dm\3 also produce violations of WEP. To do this we make use of a cyclic gedanken experiment first used by Dicke (1964a). We begin with a set of n free particles of mass m0 at rest at X = h. From Equation (2.27), 2 2 the conserved energy is simply nm0c [l t/(h)/c ]. We then form a composite body and release the binding energy £B(h,0), in the form of free particles of rest mass m0, stored in a massless reservoir. The conserved 2 2 energy of the composite body is [nmoc £B(h,0)][l - [7(h)/c ] and 2 that of the reservoir is £B(h,0)[l t/(h)/c ]. The composite body falls freely to X = 0 with an acceleration assumed to be A = g 4- <5A while the stored test particles fall with acceleration g = Vt/ (by definition). At X = 0 we bring both systems to rest, and place the energies thus gained, 2 J i i 2 - [nm0 - £B(0, V)/c ]A h - dni g h , and - EB(h, 0)g h/c into the reservoir (we have assumed g, h, and V are parallel). Dropping terms of order (g h)2, we see that the reservoir now contains conserved Theory and Experiment in Gravitational Physics 42 energy 2 2 2 £B(h,O)[l - C/(0)/c ] - £jjg h/c - (nmo - E°/c )A h - &nitfh> 2 From this we extract enough energy £B(0,0)[l t/(0)/c ] to disassemble the composite system into its n constituents, and enough energy nmog h to give the particles sufficient kinetic energy to return to their initial state of rest at X = h. The cycle is now closed, and if energy is to be con- served, the reservoir must be empty. To quasi-Newtonian order, this implies 2 £B(h, 0) - £B(0,0) - (nm0 - E$/c )SA h - <5m{W = 0 (2.33) Since lJ £B(h, 0) - £B(0,0) = dmy\U h (2.34) we obtain k A' = g> + (5mi /MR)U{f - (<5mp/MRV (2.35) 2 where MR s nm0 £B/c . The first term is the universal gravitational acceleration that would be expected in a theory satisfying WEP. The remaining terms depend upon the body's structure through the anomalous mass tensors in £B(X, V). Hence a violation of Local Lorentz or Position Invariance implies a violation of WEP. Equivalently, WEP {dm^ = dm{k = 0) implies Local Lorentz and Position Invariance. Equivalently, WEP implies EEP. The gravitational red-shift experiment can also be studied within this framework, using a cyclic gedanken experiment suggested by Nordtvedt (1975). The cycle begins as before with a set of n free particles of mass m0 at rest at X = h. We form a composite body and release the binding 2 energy £B(h,0)[l L/(h)/c ] in the form of a massless quantum which propagates to X = 0. Its energy there, compared to the energy £B(0,0)[l l/(0)/c2] of a quantum emitted from an identical system at X = 0, is 2 assumed to be given by (1 - Z)£B(0,0)[l - t/(0)/c ]. This energy is stored in a reservoir. Our goal is to evaluate the red shift Z. The body is then allowed to fall freely to X = 0, where it is brought to rest, with the kinetic energy of motion, 2 -{nm0 - £B(0,V)/c ]A h - <5mJW added to the reservoir. If we substitute for A from Equation (2.35), we see that the reservoir now contains energy (to quasi-Newtonian order) 2 2 (1 - Z)£B(0,0)[l - t/(0)/c ] - [rnno - £B(0,0)/c ]g h - Einstein Equivalence Principle and Gravitation Theory 43 2 We extract from the reservoir enough energy £B(0,0)[l - t/(O)/c ] to disassemble the system, and enough energy nmog h to return the n free particles to the starting point. Again, conservation of energy requires that the reservoir be empty and therefore that Z must satisfy (to first order in g h) - ZE% + Elg h/c2 - dmyVUiJ h = 0 (2.36) or Z = [At/ - (8r4Jc2/E$) AUir\/c2 (2.37) where AC/sgh and AUiJ = Vt/° h. By a similar analysis one can show that the second-order Doppler shift between an emitter moving at velocity V and a receiver at rest, relative to a preferred universe rest frame, is given by (Haugan, 1979) 2 2 ZD = - i V /c + ttSnf/E&VV' (2.38) Thus, the simple assumption of energy conservation has allowed us to "prove" Schiff's conjecture, as well as elucidate the empirical consequences of possible violations of the three aspects of the Einstein Equivalence Principle. Thorne, Lee, and Lightman (1973) have proposed a more qualitative "proof" of Schiff's conjecture for that class of gravitation theories that are based on an invariant action principle, so-called Lagrangian-based theories of gravity. They begin by defining the concept of "universal coupling": a generally covariant Lagrangian-based theory is universally coupled if it can be put into a mathematical form (representation) in which the action for matter and nongravitational fields /NG contains precisely one gravitational field: a symmetric, second-rank tensor # with signature + 2 that reduces to J/ when gravity is turned off; and when ^ is replaced by if, 7NG becomes the action of special relativity. Clearly, among all Lagrangian-based theories, one is universally coupled if and only if it is a metric theory (for details see Thorne, Lee, and Lightman, 1973). Let us illustrate this point with a simple example. Consider a Lagrangian-based theory of gravity that possesses a globally flat back- ground metric i\ and a symmetric, second-rank tensor gravitational field h. The nongravitational action for charged point particles of rest mass m0 and charge e, and for electromagnetic fields has the form /NG = h + /in, + hm (2-39) Theory and Experiment in Gravitational Physics 44 where 2 Io= -m0 jdr, dx = -(rj^ + h^)dx"dx\ ^^ d4x (2.40) where F^ = AVfll A^, and where IMNIWI1 ' (2.41) We work in a coordinate system in which if = diag( 1,1,1,1). To see whether this theory is universally coupled, the obvious step is to assume that the single gravitational field i/^v is given by "/V = V + V (2.42) This would make Jo and Jinl appear universally coupled. However, in the electromagnetic Lagrangian, we obtain, for example, tf* _ W = yj,** _ }f% + O(h3) (2.43) where ||^""|| = ||^||"1. Thus, there is no way to combine */ and h^ into a single gravitational field in ING, hence the theory is not universally coupled. To see that the theory is also not a metric theory, we transform to a frame in which at an event 2P, Note that in this frame, h^ ^ 0 in general, thus the action can be put into the form /NG = ^SRT + A/ (2.44) where 1 1 2 JSRT = -m0 Jdx + e JAndx* - (len)' j^F^F^-fj) ! d*x, AI = + O(h3F2) (2.45) where fcflC = h*9]? ¥= 0. So in a local Lorentz frame, the laws of physics are not those of special relativity, so the theory is not a metric theory. Notice that in this particular case, for weak gravitational fields (|fy,v| « 1), the theory is metric to first order in h^, while the deviations from metric form occur at second order in h^. In the next section, we shall present a Einstein Equivalence Principle and Gravitation Theory 45 mathematical framework for examining a class of theories with non- universal coupling and for making quantitative computations of its empirical consequences. Consider now all Lagrangian-based theories of gravity, and assume that WEP is valid. WEP forces 7NG to involve one and only one gravita- tional field (which must be a second-rank tensor ty which reduces to r\ far from gravitating matter). If /NG were to involve some other gravita- tional fields 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 non- metric (nonuniversal) and metric coupling. This theory should be simple, yet capable of making quantitative predictions for the outcomes of ex- periments. One such "model" theory is the THe/x formalism, devised by Lightman and Lee (1973a). It restricts attention to the motions and elec- tromagnetic interactions of charged structureless test particles in an ex- ternal, 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 /NG 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 ex- ternal 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 en- compass 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 /NG 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 /NG 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 form- alism as presented in equations (2.46). Later, we shall discuss the physical restrictions built into it, and shall apply it to the interpretation of experi- ment. 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 transforma- tion, correct to first order in the quantities g0? and gj, x, assumed small, is 2 2 x = Hy\x + |tf 0-»Togof + ±Ho ' H ^2xg0 x - gox )] (2.47) where the subscript (0) and superscript (') on the functions T, H, E, and fi denote s To = T(x* = x = 0), r0 = ^r/a The action Io in the new coordinates then has the form 2 il2 a 2 'o = -I>oa fd - v a) dt{\ + O[(x ) ]} (2.50) /2 where va = dxjdf. Note that our choice of the multiplicative factors Tj 12 and HQ resulted in unit coefficients in /0, making it look exactly like that of special relativity. Similarly the actions /int and Iem can be rewritten in the new coordinates, with the result /in, = 2X UfV? dt{\ + O[(XS)2]}, (2.51) 1 l 2 1 Im = (Snr eoT o' Ho l 1/2 - To 'HOEE VO '^(l - JT'OTO HE Aogo x) 1/2 1 + H0TE r0fg0 -(E x S)(l - To 'HoEE VO )}^ + [corrections of order (x*)2] (2.52) where ii A% = (dx*/dx )Aa E = *At - A o, 6 = V x A (2.53) Theory and Experiment in Gravitational Physics 48 and Ao = (2ro/T'o)(^o * + iT'oTo ' - iH'oHo *) (2.54) Let us now examine the consequences for EEP of physics governed by > j ING. Focus on the form of JNG at the event 0 (x = t = 0), since local test experiments are assumed to take place in vanishingly small regions sur- rounding 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 2 /NG= -X>oaf(l -v a) 1/2 2 X 2 + (8TT)- hoT^Ho J[£ - (To 'Hoeo Vo )B ] 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 1 1 l To /fo£oVo = l or eofio=To Ho (2.56) Since we have not specified the event 0>, this condition must hold through- out 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., l 2 To 'Hoeo Vo = (clight/c0) (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. Equa- tion (2.46)] 1 2 1/2 c0 = (To/Ho) ' , clight = (EoAio)- (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 lfi move relative to this frame. Thus, the quantity 1 ToHo oMo 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 pre- ferred-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) 1 Even if the theory is locally Lorentz invariant (TQ H0EQ 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 vf ri*'"'ri FllvFltf. The nonuniversality of this coupling violates EEP and leads to position-dependent effects, for example, in gravitational red- shift 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 1/2 1/4 /4 e*a = ettso To H S , Af = A^T^H^ (2.60) then the action, (2.55), takes the form 2 2 4 + (8TT)- * j(E* - B* ) d x (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 I2 XI2 E0TI HQ is independent of 9. In the latter case, the units of charge can be effectively chosen so that everywhere in spacetime, l 2 112 soT ol Ho = 1 (2.62) Note, however, that if LPI alone is satisfied, one can renormalize the I2 1/2 charge and vector potential to make either £0TQ HQ = 1 or l2 1/2 fioTl Ho = 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 112 e0 = n0 = (Ho/T0) , 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 com- posite 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 Fo = Ao = 0, and U i.e., l l2 1/2 2 m eoT o Ho = const, noTl' Ho = const, EoHo^HoTo1 (2.64) Again, the units of charge can be normalized so that 1 2 e0 = Ho = (H0/T0) ' , 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 X (d/dt)(HW~ \) + {W- V(T - Hvl) = aL(xa), aL(x0) = (ea/mOa){VAo(xa) + V[va A(xfl)] - dA{xa)/dt}, V (EE) = 4np, Vx(^»B) = 4TTJ + d{eE)/dt (2.67) 2 112 3 3 where W = {T- Hv a) ,p = Y.aea<5 (x - xfl), J = £aeaya5 (x - xa), and aL is the Lorentz acceleration of particle a. These equations are used to Einstein Equivalence Principle and Gravitation Theory 51 calculate the acceleration from rest of a bound test body consisting of charged point particles. A number of approximations are necessary to make the computation tractable. First, the functions T, H, e, and fi, considered to be functions of 1 2 v ~ e /mor « 1 where r is a typical interparticle distance. By analogy with the post- Newtonian expansion to be described in Chapter 4, we call this a post- Coulombian 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 accelera- 2 2 tion), O(v ) (Coulomb interparticle acceleration), O(gf0t; ) (post-Cou- lombian gravitational acceleration), O(v4) (post-Coulombian interparticle acceleration), O(g0v*) (post-post-Coulombian gravitational acceleration), 2 and so on. To O(g0v ), we obtain o + itf'oHo'goi;2 1 l 2 + (T'oTo - H'0H0- )g0 vavfl + Ty H» XW (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 2 1 1 V 40 - ej^o.oo = 4ns~ p-e~ \e- (\A0 - A,o), V2A - e/xA.oo = ~^nJ + (sp)-xV(sp)V A - ^V/z x (V x A) (2.71) These equations can be solved iteratively by writing (0) (1) Ao = A^ + A%\ A = A + A (2.72) Theory and Experiment in Gravitational Physics 52 ( ( y where A v/A ° ~ O(g0), and solving for each term to an appropriate order in v2. The result is Ao= -4> (0) A = A + 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 (2-75) A = m 2J wOaXa> m = ZJ m0a a 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) a ab where xab = xa xb, rab = |xa(,|, and the double sum over a and b excludes the case a = b. The final result is 2 l 2 { 1 1 a d X /dt = g - ^[TE '% roje /m) J 1/2 1 1 + 0 '[ro eo (l-ToHo eo/Xo)] x (JEjf/m + <5yEE» (2.78) where Vo (2-79) Einstein Equivalence Principle and Gravitation Theory 53 where Fo is given by Equation (2.54), and where Ef? = The first term gl is the universal acceleration of a neutral test body (gov- erned by Io alone); the other two terms depend on the body's electro- magnetic self-energy and self-energy tensor. These terms vanish for all bodies (i.e., WEP is satisfied) if and only if (2.81) at any event 0>, which is equivalent to Equation (2.63). Hence WEP => EEP and Schiff's conjecture is verified, at least within the confines of the THEH formalism. It is useful to define the gravitational potential U whose gradient yields the test-body acceleration g; modulo a constant x U(x)= ~^T'0Ho g0-x (2.82) If the functions T, H, e, and fi are now considered as functions of U instead of 2 To= -c 0( ]|x=0 (2.83) (c) Energy conservation and anomalous mass tensors Because the THefi formalism is based on an action principle, it possesses conservation laws, in particular a conservation law for energy, and so is amenable to analysis using the conserved-energy framework described in Section 2.5. The main products of that framework are the anomalous inertial mass tensor Sm[J and passive gravitational mass tensor Sm'j obtained from the conserved energy. These two quantities then yield expressions for violations of WEP, Local Lorentz Invariance, and Local Position Invariance. As a concrete example (Haugan, 1979), we consider a classical bound system of two charged particles. As in the above "proof" of Schiff's con- jecture we work to post-Coulombian order and to first order in g0 x. We first formulate the equations of motion in terms of a truncated action ?NG = h + An, (2-84) Theory and Experiment in Gravitational Physics 54 where 7im is rewritten entirely in terms of particle coordinates by substitut- ing the post-Coulombian solutions for A^, Equation (2.73), into Iint. Variation of 7NG with respect to particle coordinates then yields the com- plete particle equations of motion. We identify a Lagrangian L using the definition 7NG = $Ldt (2.85) We next make a change of variables in L from xu x2, vl5 and v2, to the center of mass and relative variables X = (m^ + m2x2)/m, x = xt x2, V = dX/dt, v = dx/dt (2.86) where m = mx + m2, n = mlm2/m. A Hamiltonian H is constructed from L using the standard technique Pj s dL/dV{ pi = BL/dv3, H s PJVj + pV - L (2.87) The result is 2 2 l 2 H = Tj' m(l + iTiTo »g0 X) + n' HE P /2m /2 2 2 eie2/r)go X - Tj Ho [(p P) 2 2 2 o \e,e2lr)\P + (n P) ]/2m } + hT0li0Ho \m2 - m1)(e1e2/r)(p P + ii pfi + O(p4) + O(P4) (2.88) where n = x/r. The post-Coulombian terms O(p4) and O(P)4 neglected in Equation (2.88) do not couple the internal motion and the center-of-mass motion and thus do not lead to violations of EEP. We now average H over several timescales for the internal motions of the bound two-body system, assumed short compared to the timescale for the center-of-mass motion. The average is simplified using virial theorems obtained from Hamilton's equations for the internal variables derived from H. The relevant expres- sions are + post-Coulombian terms), (2.89) H nj(n p)]> (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 The conserved energy function Ec used in Section 2.5 is then defined to be l2 Ec = Tl Ho\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 1/2 M = m + Substitution of these formulae into Equation (2.35) for the center-of-mass acceleration of the system yields precisely Equation (2.78). One advantage of the Hamiltonian approach is that it can also be applied to quantum systems (Will, 1974c). This is especially useful in discussing gravitational red-shift experiments since it is transitions be- tween quantized energy levels that produce the photons whose red shifts are measured. For the idealized gravitational red shift experiments dis- cussed in Section 2.5, only the anomalous passive mass tensor 5m^ is needed. The simplest quantum system of interest is that of a charged Theory and Experiment in Gravitational Physics 56 particle (electron) moving in a given external electromagnetic potential of a charged particle (proton) at rest in the SSS field, i.e., a hydrogen atom. For such a system the truncated Lagrangian [Equation (2.85)] has the form 2 112 L = - me(T0 - Hov ) - eA^if (2.97) where m0 = me and e + \e\ for the electron. We shall ignore the spatial variation of T, H, e, and \i across the atom, hence we evaluate each at x = 0. The Hamiltonian obtained from L is given by /2 2 2 1/2 H = n [me + Ho > + eA| ] + eA0 (2.98) where Pj = dL/dvK Introducing the Dirac matrices where / is the two-dimensional unit matrix and , EES = ( £ Q") (2-8°)