2.3 EEPparticle physics, and 2 Tests of the Foundations 2.1 The Einstein equivalence principle

2.2 Theoretical Frameworks for Analyzing EEP 

2.2.1 Schiff's conjecture 

Because the three parts of the Einstein equivalence principle discussed above are so very different in their empirical consequences, it is tempting to regard them as independent theoretical principles. On the other hand, any complete and self-consistent gravitation theory must possess sufficient 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 connections between the three sub-principles. For instance, the same mathematical 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 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 show up as a violation of local position invariance. Around 1960, 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.

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. [74Jump To The Next Citation Point In The Article]). A system in a quantum state A decays to state B, emitting a quantum of frequency tex2html_wrap_inline3929 . The quantum falls a height H in an external gravitational field and is shifted to frequency tex2html_wrap_inline3933, while the system in state B falls with acceleration tex2html_wrap_inline3937 . At the bottom, state A is rebuilt out of state B, the quantum of frequency tex2html_wrap_inline3933, and the kinetic energy tex2html_wrap_inline3945 that state B has gained during its fall. The energy left over must be exactly enough, tex2html_wrap_inline3949, to raise state A to its original location. (Here an assumption of local Lorentz invariance permits the inertial masses tex2html_wrap_inline3953 and tex2html_wrap_inline3955 to be identified with the total energies of the bodies.) If tex2html_wrap_inline3957 and tex2html_wrap_inline3937 depend on that portion of the internal energy of the states that was involved in the quantum transition from A to B according to

  equation221

(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 tex2html_wrap_inline3965)

  equation224

Haugan generalized this approach to include violations of LLI [74], (TEGP 2.5 [147Jump To The Next Citation Point In The Article]).


Box  tex2html_wrap_inline3967 . The tex2html_wrap_inline3969 formalism
  1. Coordinate system and conventions: tex2html_wrap_inline3971 time coordinate associated with the static nature of the static spherically symmetric (SSS) gravitational field; tex2html_wrap_inline3973 isotropic quasi-Cartesian spatial coordinates; spatial vector and gradient operations as in Cartesian space.
  2. Matter and field variables:
  3. Gravitational potential: tex2html_wrap_inline3995
  4. Arbitrary functions: T (U), H (U), tex2html_wrap_inline4001, tex2html_wrap_inline4003 ; EEP is satisfied if tex2html_wrap_inline4005 for all U .
  5. Action:

    eqnarray249

  6. Non-Metric parameters:

    eqnarray258

    where tex2html_wrap_inline4009 and subscript ``0'' refers to a chosen point in space. If EEP is satisfied, tex2html_wrap_inline4011 .

 

2.2.2 The tex2html_wrap_inline4013 formalism 

The first successful attempt to prove Schiff's conjecture more formally was made by Lightman and Lee [86]. They developed a framework called the tex2html_wrap_inline4015 formalism that encompasses all metric theories of gravity and many non-metric theories (Box  1). It restricts attention to the behavior of charged particles (electromagnetic interactions only) in an external static spherically symmetric (SSS) gravitational field, described by a potential U . It characterizes the motion of the charged particles in the external potential by two arbitrary functions T (U) and H (U), and characterizes the response of electromagnetic fields to the external potential (gravitationally modified Maxwell equations) by two functions tex2html_wrap_inline4001 and tex2html_wrap_inline4003 . The forms of T, H, tex2html_wrap_inline4031 and tex2html_wrap_inline4033 vary from theory to theory, but every metric theory satisfies

  equation271

for all U . This consequence follows from the action of electrodynamics with a ``minimal'' or metric coupling:

  eqnarray277

where the variables are defined in Box  1, and where tex2html_wrap_inline4037 . By identifying tex2html_wrap_inline4039 and tex2html_wrap_inline4041 in a SSS field, tex2html_wrap_inline4043 and tex2html_wrap_inline4045, one obtains Eq. (7Popup Equation). Conversely, every theory within this class that satisfies Eq. (7Popup Equation) can have its electrodynamic equations cast into ``metric'' form. In a given non-metric theory, the functions T, H, tex2html_wrap_inline4031 and tex2html_wrap_inline4033 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. (7Popup Equation) was satisfied. In other words, WEP tex2html_wrap_inline4055 EEP and Schiff's conjecture was verified, at least within the restrictions built into the formalism.

Certain combinations of the functions T, H, tex2html_wrap_inline4031 and tex2html_wrap_inline4033 reflect different aspects of EEP. For instance, position or U -dependence of either of the combinations tex2html_wrap_inline4067 and tex2html_wrap_inline4069 signals violations of LPI, the first combination playing the role of the locally measured electric charge or fine structure constant. The ``non-metric parameters'' tex2html_wrap_inline4071 and tex2html_wrap_inline4073 (Box  1) are measures of such violations of EEP. Similarly, if the parameter tex2html_wrap_inline4075 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 tex2html_wrap_inline3853, given by

  equation306

In many applications, by suitable definition of units, tex2html_wrap_inline3853 can be set equal to unity. If EEP is valid, tex2html_wrap_inline4083 everywhere.

The rate of fall of a composite spherical test body of electromagnetically interacting particles then has the form

  equation311

  eqnarray318

where tex2html_wrap_inline4085 and tex2html_wrap_inline4087 are the electrostatic and magnetostatic binding energies of the body, given by

  equation332

  equation344

where tex2html_wrap_inline4089, tex2html_wrap_inline4091, 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. (11Popup Equation), and ultimately place limits on the non-metric parameters tex2html_wrap_inline4093 and tex2html_wrap_inline4095 . (We set tex2html_wrap_inline4097 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) [147Jump To The Next Citation Point In The Article]).

The tex2html_wrap_inline4015 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 tex2html_wrap_inline3757 (Eq. (4Popup Equation)), the results are (TEGP 2.6 (c) [147Jump To The Next Citation Point In The Article]):

  equation377

The redshift is the standard one tex2html_wrap_inline4103, independently of the nature of the clock if and only if tex2html_wrap_inline4105 . Thus the Vessot-Levine rocket redshift experiment sets a limit on the parameter combination tex2html_wrap_inline4107 (Figure  3); the null-redshift experiment comparing hydrogen-maser and SCSO clocks sets a limit on tex2html_wrap_inline4109 . Alvarez and Mann [4, 3, 5, 6, 7] extended the tex2html_wrap_inline4015 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.

2.2.3 The tex2html_wrap_inline4113 formalism 

The tex2html_wrap_inline4015 formalism can also be applied to tests of local Lorentz invariance, but in this context it can be simplified. Since most such tests do not concern themselves with the spatial variation of the functions T, H, tex2html_wrap_inline4031, and tex2html_wrap_inline4033, but rather with observations made in moving frames, we can treat them as spatial constants. Then by rescaling the time and space coordinates, the charges and the electromagnetic fields, we can put the action in Box  1 into the form (TEGP 2.6 (a) [147Jump To The Next Citation Point In The Article])

  equation396

where tex2html_wrap_inline4125 . This amounts to using units in which the limiting speed tex2html_wrap_inline3853 of massive test particles is unity, and the speed of light is c . If tex2html_wrap_inline3855, 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. (15Popup Equation) yield the behavior of rods and clocks, just as in the full tex2html_wrap_inline4015 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 tex2html_wrap_inline4135 . Notice that c does not appear in this expression. The energy and momentum of an electromagnetically bound body which moves with velocity tex2html_wrap_inline4139 relative to the rest frame are given by

  equation406

where tex2html_wrap_inline4141, tex2html_wrap_inline4143 is the sum of the particle rest masses, tex2html_wrap_inline4085 is the electrostatic binding energy of the system (Eq. (12Popup Equation) with tex2html_wrap_inline4147), and

  equation430

where

  equation442

Note that tex2html_wrap_inline4149 corresponds to the parameter tex2html_wrap_inline3755 plotted in Figure  2 .

The electrodynamics given by Eq. (15Popup Equation) can also be quantized, so that we may treat the interaction of photons with atoms via perturbation theory. The energy of a photon is tex2html_wrap_inline4153 times its frequency tex2html_wrap_inline3833, while its momentum is tex2html_wrap_inline4157 . 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 tex2html_wrap_inline4159 . The experimental null result then leads to the bound on tex2html_wrap_inline4149 shown on Figure  2 . Similarly the anisotropy in energy levels is clearly illustrated by the tensorial terms in Eqs. (16Popup Equation) and (18Popup Equation); by evaluating tex2html_wrap_inline4163 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 tex2html_wrap_inline4149 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 tex2html_wrap_inline4167 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 [144Jump To The Next Citation Point In The Article]).

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 tex2html_wrap_inline4139, the velocity of the Earth through the cosmic background,

  equation468

where tex2html_wrap_inline4171, tex2html_wrap_inline3929 is the maser frequency, L =21 km is the baseline, and where tex2html_wrap_inline4177 and tex2html_wrap_inline4179 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 tex2html_wrap_inline4181 . 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].



2.3 EEPparticle physics, and 2 Tests of the Foundations 2.1 The Einstein equivalence principle

image The Confrontation between General Relativity and Experiment
Clifford M. Will
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