If Schiff's conjecture is correct, then Eötvös experiments may be seen as the direct empirical foundation for EEP, hence for the interpretation of gravity as a curved-spacetime phenomenon. Of course, a rigorous proof of such a conjecture is impossible (indeed, some special counter-examples are known), yet a number of powerful ``plausibility'' arguments can be formulated.
The most general and elegant of these arguments is based upon the assumption of energy conservation. This assumption allows one to perform 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 approach was pioneered by Dicke, Nordtvedt and Haugan (see, e.g. [74]). A system in a quantum state A decays to state B, emitting a quantum of frequency . The quantum falls a height H in an external gravitational field and is shifted to frequency , while the system in state B falls with acceleration . At the bottom, state A is rebuilt out of state B, the quantum of frequency , and the kinetic energy that state B has gained during its fall. The energy left over must be exactly enough, , to raise state A to its original location. (Here an assumption of local Lorentz invariance permits the inertial masses and to be identified with the total energies of the bodies.) If and depend on that portion of the internal energy of the states that was involved in the quantum transition from A to B according to
(violation of WEP), then by conservation of energy, there must be a corresponding violation of LPI in the frequency shift of the form (to lowest order in )
Haugan generalized this approach to include violations of LLI [74], (TEGP 2.5 [147]).
where and subscript ``0'' refers to a chosen point in space. If EEP is satisfied, .
for all U . This consequence follows from the action of electrodynamics with a ``minimal'' or metric coupling:
where the variables are defined in Box 1, and where . By identifying and in a SSS field, and , one obtains Eq. (7). Conversely, every theory within this class that satisfies Eq. (7) can have its electrodynamic equations cast into ``metric'' form. In a given non-metric theory, the functions T, H, and will depend in general on the full gravitational environment, including the potential of the Earth, Sun and Galaxy, as well as on cosmological boundary conditions. Which of these factors has the most influence on a given experiment will depend on the nature of the experiment.
Lightman and Lee then calculated explicitly the rate of fall of a ``test'' body made up of interacting charged particles, and found that the rate was independent of the internal electromagnetic structure of the body (WEP) if and only if Eq. (7) was satisfied. In other words, WEP EEP and Schiff's conjecture was verified, at least within the restrictions built into the formalism.
Certain combinations of the functions T, H, and reflect different aspects of EEP. For instance, position or U -dependence of either of the combinations and signals violations of LPI, the first combination playing the role of the locally measured electric charge or fine structure constant. The ``non-metric parameters'' and (Box 1) are measures of such violations of EEP. Similarly, if the parameter is non-zero anywhere, then violations of LLI will occur. This parameter is related to the difference between the speed of light c, and the limiting speed of material test particles , given by
In many applications, by suitable definition of units, can be set equal to unity. If EEP is valid, everywhere.
The rate of fall of a composite spherical test body of electromagnetically interacting particles then has the form
where and are the electrostatic and magnetostatic binding energies of the body, given by
where , , and the angle brackets denote an expectation value of the enclosed operator for the system's internal state. Eötvös experiments place limits on the WEP-violating terms in Eq. (11), and ultimately place limits on the non-metric parameters and . (We set because of very tight constraints on it from tests of LLI.) These limits are sufficiently tight to rule out a number of non-metric theories of gravity thought previously to be viable (TEGP 2.6 (f) [147]).
The formalism also yields a gravitationally modified Dirac equation that can be used to determine the gravitational redshift experienced by a variety of atomic clocks. For the redshift parameter (Eq. (4)), the results are (TEGP 2.6 (c) [147]):
The redshift is the standard one , independently of the nature of the clock if and only if . Thus the Vessot-Levine rocket redshift experiment sets a limit on the parameter combination (Figure 3); the null-redshift experiment comparing hydrogen-maser and SCSO clocks sets a limit on . Alvarez and Mann [4, 3, 5, 6, 7] extended the formalism to permit analysis of such effects as the Lamb shift, anomalous magnetic moments and non-baryonic effects, and placed interesting bounds on EEP violations.
where . This amounts to using units in which the limiting speed of massive test particles is unity, and the speed of light is c . If , LLI is violated; furthermore, the form of the action above must be assumed to be valid only in some preferred universal rest frame. The natural candidate for such a frame is the rest frame of the microwave background.
The electrodynamical equations which follow from Eq. (15) yield the behavior of rods and clocks, just as in the full formalism. For example, the length of a rod moving through the rest frame in a direction parallel to its length will be observed by a rest observer to be contracted relative to an identical rod perpendicular to the motion by a factor . Notice that c does not appear in this expression. The energy and momentum of an electromagnetically bound body which moves with velocity relative to the rest frame are given by
where , is the sum of the particle rest masses, is the electrostatic binding energy of the system (Eq. (12) with ), and
where
Note that corresponds to the parameter plotted in Figure 2 .
The electrodynamics given by Eq. (15) can also be quantized, so that we may treat the interaction of photons with atoms via perturbation theory. The energy of a photon is times its frequency , while its momentum is . Using this approach, one finds that the difference in round trip travel times of light along the two arms of the interferometer in the Michelson-Morley experiment is given by . The experimental null result then leads to the bound on shown on Figure 2 . Similarly the anisotropy in energy levels is clearly illustrated by the tensorial terms in Eqs. (16) and (18); by evaluating for each nucleus in the various Hughes-Drever-type experiments and comparing with the experimental limits on energy differences, one obtains the extremely tight bounds also shown on Figure 2 .
The behavior of moving atomic clocks can also be analysed in detail, and bounds on can be placed using results from tests of time dilation and of the propagation of light. In some cases, it is advantageous to combine the framework with a ``kinematical'' viewpoint that treats a general class of boost transformations between moving frames. Such kinematical approaches have been discussed by Robertson, Mansouri and Sexl, and Will (see [144]).
For example, in the ``JPL'' experiment, in which the phases of two hydrogen masers connected by a fiberoptic link were compared as a function of the Earth's orientation, the predicted phase difference as a function of direction is, to first order in , the velocity of the Earth through the cosmic background,
where , is the maser frequency, L =21 km is the baseline, and where and are unit vectors along the direction of propagation of the light at a given time and at the initial time of the experiment, respectively. The observed limit on a diurnal variation in the relative phase resulted in the bound . Tighter bounds were obtained from a ``two-photon absorption'' (TPA) experiment, and a 1960s series of ``Mössbauer-rotor'' experiments, which tested the isotropy of time dilation between a gamma ray emitter on the rim of a rotating disk and an absorber placed at the center [144].
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 |