## 2.1 The Einstein equivalence principle

The principle of equivalence has historically played an important role in the development of gravitation theory. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraph of the Principia to it. In 1907, Einstein used the principle as a basic element of general relativity. We now regard the principle of equivalence as the foundation, not of Newtonian gravity or of GR, but of the broader idea that spacetime is curved.

One elementary equivalence principle is the kind Newton had in mind when he stated that the property of a body called ``mass'' is proportional to the ``weight'', and is known as the weak equivalence principle (WEP). An alternative statement of WEP is that the trajectory of a freely falling body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition. In the simplest case of dropping two different bodies in a gravitational field, WEP states that the bodies fall with the same acceleration (this is often termed the Universality of Free Fall, or UFF).

A more powerful and far-reaching equivalence principle is known as the Einstein equivalence principle (EEP). It states that:

1. WEP is valid.
2. The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed.
3. The outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed.

The second piece of EEP is called local Lorentz invariance (LLI), and the third piece is called local position invariance (LPI).

For example, a measurement of the electric force between two charged bodies is a local non-gravitational experiment; a measurement of the gravitational force between two bodies (Cavendish experiment) is not.

The Einstein equivalence principle is the heart and soul of gravitational theory, for it is possible to argue convincingly that if EEP is valid, then gravitation must be a ``curved spacetime'' phenomenon, in other words, the effects of gravity must be equivalent to the effects of living in a curved spacetime. As a consequence of this argument, the only theories of gravity that can embody EEP are those that satisfy the postulates of ``metric theories of gravity'', which are:

1. Spacetime is endowed with a symmetric metric.
2. The trajectories of freely falling bodies are geodesics of that metric.
3. In local freely falling reference frames, the non-gravitational laws of physics are those written in the language of special relativity.

The argument that leads to this conclusion simply notes that, if EEP is valid, then in local freely falling frames, the laws governing experiments must be independent of the velocity of the frame (local Lorentz invariance), with constant values for the various atomic constants (in order to be independent of location). The only laws we know of that fulfill this are those that are compatible with special relativity, such as Maxwell's equations of electromagnetism. Furthermore, in local freely falling frames, test bodies appear to be unaccelerated, in other words they move on straight lines; but such ``locally straight'' lines simply correspond to ``geodesics'' in a curved spacetime (TEGP 2.3 [147]).

General relativity is a metric theory of gravity, but then so are many others, including the Brans-Dicke theory. The nonsymmetric gravitation theory (NGT) of Moffat is not a metric theory. Neither, in this narrow sense, is superstring theory (see Sec.  2.3), which, while based fundamentally on a spacetime metric, introduces additional fields (dilatons, moduli) that can couple to material stress-energy in a way that can lead to violations, say, of WEP. So the notion of curved spacetime is a very general and fundamental one, and therefore it is important to test the various aspects of the Einstein Equivalence Principle thoroughly.

A direct test of WEP is the comparison of the acceleration of two laboratory-sized bodies of different composition in an external gravitational field. If the principle were violated, then the accelerations of different bodies would differ. The simplest way to quantify such possible violations of WEP in a form suitable for comparison with experiment is to suppose that for a body with inertial mass , the passive gravitational mass is no longer equal to , so that in a gravitational field g, the acceleration is given by . 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 differently than it does to , a violation of WEP would result. One could then write

where is the internal energy of the body generated by interaction A, and is a dimensionless parameter that measures the strength of the violation of WEP induced by that interaction, and c is the speed of light. A measurement or limit on the fractional difference in acceleration between two bodies then yields a quantity called the ``Eötvös ratio'' given by

where we drop the subscript ``I'' from the inertial masses. Thus, experimental limits on place limits on the WEP-violation parameters .

Many high-precision Eötvös-type experiments have been performed, from the pendulum experiments of Newton, Bessel and Potter, to the classic torsion-balance measurements of Eötvös [58], Dicke [55], Braginsky [31] and their collaborators. In the modern torsion-balance experiments, two objects of different composition are connected by a rod or placed on a tray and suspended in a horizontal orientation by a fine wire. If the gravitational acceleration of the bodies differs, there will be a torque induced on the suspension wire, related to the angle between the wire and the direction of the gravitational acceleration g . If the entire apparatus is rotated about some direction with angular velocity , the torque will be modulated with period . In the experiments of Eötvös and his collaborators, the wire and g were not quite parallel because of the centripetal acceleration on the apparatus due to the Earth's rotation; the apparatus was rotated about the direction of the wire. In the Dicke and Braginsky experiments, g was that of the Sun, and the rotation of the Earth provided the modulation of the torque at a period of 24 hr (TEGP 2.4 (a) [147]). Beginning in the late 1980s, numerous experiments were carried out primarily to search for a ``fifth force'' (see Sec.  2.3), but their null results also constituted tests of WEP. In the ``free-fall Galileo experiment'' performed at the University of Colorado, the relative free-fall acceleration of two bodies made of uranium and copper was measured using a laser interferometric technique. The ``Eöt-Wash'' experiments carried out at the University of Washington used a sophisticated torsion balance tray to compare the accelerations of various materials toward local topography on Earth, movable laboratory masses, the Sun and the galaxy [121, 10], and have recently reached levels of . The resulting upper limits on are summarized in Figure  1 (TEGP 14.1 [147]; for a bibliography of experiments, see [61]).

Figure 1: Selected tests of the weak equivalence principle, showing bounds on , which measures fractional difference in acceleration of different materials or bodies. The free-fall and Eöt-Wash experiments were originally performed to search for a fifth force. The blue band shows current bounds on for gravitating bodies from lunar laser ranging (LURE).

The second ingredient of EEP, local Lorentz invariance, has been tested to high-precision in the ``mass anisotropy'' experiments: The classic versions are the Hughes-Drever experiments, performed in the period 1959-60 independently by Hughes and collaborators at Yale University, and by Drever at Glasgow University (TEGP 2.4 (b) [147]). Dramatically improved versions were carried out during the late 1980s using laser-cooled trapped atom techniques (TEGP 14.1 [147]). A simple and useful way of interpreting these experiments is to suppose that the electromagnetic interactions suffer a slight violation of Lorentz invariance, through a change in the speed of electromagnetic radiation c relative to the limiting speed of material test particles (, chosen to be unity via a choice of units), in other words, (see Sec.  2.2.3). Such a violation necessarily selects a preferred universal rest frame, presumably that of the cosmic background radiation, through which we are moving at about 300 km/s. Such a Lorentz-non-invariant electromagnetic interaction would cause shifts in the energy levels of atoms and nuclei that depend on the orientation of the quantization axis of the state relative to our universal velocity vector, and on the quantum numbers of the state. The presence or absence of such energy shifts can be examined by measuring the energy of one such state relative to another state that is either unaffected or is affected differently by the supposed violation. One way is to look for a shifting of the energy levels of states that are ordinarily equally spaced, such as the four J =3/2 ground states of the Li nucleus in a magnetic field (Drever experiment); another is to compare the levels of a complex nucleus with the atomic hyperfine levels of a hydrogen maser clock. These experiments have all yielded extremely accurate results, quoted as limits on the parameter in Figure  2 . Also included for comparison is the corresponding limit obtained from Michelson-Morley type experiments (for a review, see [75]).

Figure 2: Selected tests of local Lorentz invariance showing the bounds on the parameter , which measures the degree of violation of Lorentz invariance in electromagnetism. The Michelson-Morley, Joos, and Brillet-Hall experiments test the isotropy of round-trip speed of light, the latter experiment using laser technology. The centrifuge, two-photon absorption (TPA) and JPL experiments test the isotropy of light speed using one-way propagation. The remaining four experiments test isotropy of nuclear energy levels. The limits assume a speed of Earth of 300 km/s relative to the mean rest frame of the universe.

Recent advances in atomic spectroscopy and atomic timekeeping have made it possible to test LLI by checking the isotropy of the speed of light using one-way propagation (as opposed to round-trip propagation, as in the Michelson-Morley experiment). In one experiment, for example, the relative phases of two hydrogen maser clocks at two stations of NASA's Deep Space Tracking Network were compared over five rotations of the Earth by propagating a light signal one-way along an ultrastable fiberoptic link connecting them (see Sec.  2.2.3). Although the bounds from these experiments are not as tight as those from mass-anisotropy experiments, they probe directly the fundamental postulates of special relativity, and thereby of LLI (TEGP 14.1 [147], [144]).

The principle of local position invariance, the third part of EEP, can be tested by the gravitational redshift experiment, the first experimental test of gravitation proposed by Einstein. Despite the fact that Einstein regarded this as a crucial test of GR, we now realize that it does not distinguish between GR and any other metric theory of gravity, but is only a test of EEP. A typical gravitational redshift experiment measures the frequency or wavelength shift between two identical frequency standards (clocks) placed at rest at different heights in a static gravitational field. If the frequency of a given type of atomic clock is the same when measured in a local, momentarily comoving freely falling frame (Lorentz frame), independent of the location or velocity of that frame, then the comparison of frequencies of two clocks at rest at different locations boils down to a comparison of the velocities of two local Lorentz frames, one at rest with respect to one clock at the moment of emission of its signal, the other at rest with respect to the other clock at the moment of reception of the signal. The frequency shift is then a consequence of the first-order Doppler shift between the frames. The structure of the clock plays no role whatsoever. The result is a shift

where is the difference in the Newtonian gravitational potential between the receiver and the emitter. If LPI is not valid, then it turns out that the shift can be written

where the parameter may depend upon the nature of the clock whose shift is being measured (see TEGP 2.4 (c) [147] for details).

The first successful, high-precision redshift measurement was the series of Pound-Rebka-Snider experiments of 1960-1965 that measured the frequency shift of gamma-ray photons from Fe as they ascended or descended the Jefferson Physical Laboratory tower at Harvard University. The high accuracy achieved - one percent - was obtained by making use of the Mössbauer 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 and satellites. Figure  3 summarizes the important redshift experiments that have been performed since 1960 (TEGP 2.4 (c) [147]).

Figure 3: Selected tests of local position invariance via gravitational redshift experiments, showing bounds on , which measures degree of deviation of redshift from the formula .

The most precise standard redshift test to date was the Vessot-Levine rocket experiment that took place in June 1976 [131]. 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 masers' frequency stability by monitoring the frequency shift as a function of altitude. A sophisticated 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 the velocity (to evaluate the potential difference , and the special relativistic time dilation). Analysis of the data yielded a limit .

A ``null'' redshift experiment performed in 1978 tested whether the relative rates of two different clocks depended upon position. Two hydrogen maser clocks and an ensemble of three superconducting-cavity stabilized oscillator (SCSO) clocks were compared over a 10-day period. During the period of the experiment, the solar potential changed sinusoidally with a 24-hour period by because of the Earth's rotation, and changed linearly at per day because the Earth is 90 degrees from perihelion in April. However, analysis of the data revealed no variations of either type within experimental errors, leading to a limit on the LPI violation parameter  [130]. This bound has been improved using more stable frequency standards [68, 109]. The varying gravitational redshift of Earth-bound clocks relative to the highly stable Millisecond Pulsar PSR 1937+21, caused by the Earth's motion in the solar gravitational field around the Earth-Moon center of mass (amplitude 4000 km), has been measured to about 10 percent, and the redshift of stable oscillator clocks on the Voyager spacecraft caused by Saturn's gravitational field yielded a one percent test. The solar gravitational redshift has been tested to about two percent using infrared oxygen triplet lines at the limb of the Sun, and to one percent using oscillator clocks on the Galileo spacecraft (TEGP 2.4 (c) [147] and 14.1 (a) [147]).

Modern advances in navigation using Earth-orbiting atomic clocks and accurate time-transfer must routinely take gravitational redshift and time-dilation effects into account. For example, the Global Positioning System (GPS) provides absolute accuracies of around 15 m (even better in its military mode) anywhere on Earth, which corresponds to 50 nanoseconds in time accuracy at all times. Yet the difference in rate between satellite and ground clocks as a result of special and general relativistic effects is a whopping 39  microseconds per day ( from the gravitational redshift, and from time dilation). If these effects were not accurately accounted for, GPS would fail to function at its stated accuracy. This represents a welcome practical application of GR! (For the role of GR in GPS, see [8]; for a popular essay, see [140].)

Local position invariance also refers to position in time. If LPI is satisfied, the fundamental constants of non-gravitational physics should be constants in time. Table  1 shows current bounds on cosmological variations in selected dimensionless constants. For discussion and references to early work, see TEGP 2.4 (c) [147].

 Constant k Limit on per Hubble time  yr Method Fine structure constant H-maser vs. Hg ion clock [109] Rb fountain vs. Cs clock [115] Oklo Natural Reactor [41] 21-cm vs. molecular absorption at Z =0.7 [57] Weak interaction constant 1 Re, K decay rates 0.1 Oklo Natural Reactor [41] 0.06 Big Bang nucleosynthesis [91, 112] e-p mass ratio 1 Mass shift in quasar spectra at Proton g-factor () 21-cm vs. molecular absorption at Z =0.7 [57]

Table 1: Bounds on cosmological variation of fundamental constants of non-gravitational physics. For references to earlier work, see TEGP 2.4 (c) [147].

 The Confrontation between General Relativity and Experiment Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de