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 [204, 194, 62]), 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., [124]). A system in a quantum state
decays to state
,
emitting a quantum of frequency
. The quantum falls a height
in an external gravitational
field and is shifted to frequency
, while the system in state
falls with acceleration
. At the bottom, state
is rebuilt out of state
, the quantum of frequency
,
and the kinetic energy
that state
has gained during its fall. The energy left
over must be exactly enough,
, to raise state
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
to
according to
__________________________________________________________________________________________________
Box 1. The formalism
_____________________________________________________________________________________________________
where and subscript “0” refers to a chosen point in space. If EEP is satisfied,
.
The first successful attempt to prove Schiff’s conjecture more formally was made by Lightman and
Lee [166]. They developed a framework called the formalism that encompasses all metric theories
of gravity and many non-metric theories (see 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
. It characterizes the motion of the charged particles in the external
potential by two arbitrary functions
and
, and characterizes the response of electromagnetic
fields to the external potential (gravitationally modified Maxwell equations) by two functions
and
. The forms of
,
,
, and
vary from theory to theory, but every metric theory satisfies
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 Equation (9) 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 ,
,
, and
reflect different aspects of EEP. For
instance, position or
-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
(see 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
, and the limiting speed of material test particles
, given by
The rate of fall of a composite spherical test body of electromagnetically interacting particles then has the form
where 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
(see Equation (6
)), the results are (TEGP 2.6 (c) [281
]):
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
(see Figure 3
); the null-redshift experiment comparing
hydrogen-maser and SCSO clocks sets a limit on
. Alvarez and
Mann [7, 6, 8, 9, 10] 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.
The 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
,
,
, and
, 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) [281
])
The electrodynamical equations which follow from Equation (17) yield the behavior of rods and clocks,
just as in the full
formalism. For example, the length of a rod which moves with velocity
relative to 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
does not appear in this expression, because only electrostatic interactions are involved, and
appears only in the magnetic sector of the action (17
). The energy and momentum of an
electromagnetically bound body moving with velocity
relative to the rest frame are given by
The electrodynamics given by Equation (17) 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
Equations (18
, 20
); 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 analyzed 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 [279
]).
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,
Kostelecký and collaborators developed a useful and elegant framework for discussing violations of Lorentz
symmetry in the context of the standard model of particle physics [63, 64, 155]. Called the Standard
Model Extension (SME), it takes the standard field theory of particle physics, and
modifies the terms in the action by inserting a variety of tensorial quantities in the quark, lepton, Higgs,
and gauge boson sectors that could explicitly violate LLI. SME extends the earlier classical
and
frameworks, and the
framework of Ni [194] to quantum field theory and particle physics.
The modified terms split naturally into those that are odd under CPT (i.e. that violate CPT)
and terms that are even under CPT. The result is a rich and complex framework, with many
parameters to be analyzed and tested by experiment. Such details are beyond the scope of this
review; for a review of SME and other frameworks, the reader is referred to the Living Review by
Mattingly [182].
Here we confine our attention to the electromagnetic sector, in order to link the SME with the
framework discussed above. In the SME, the Lagrangian for a scalar particle
with charge
interacting with electrodynamics takes the form
The tensor can be decomposed into “electric”, “magnetic”, and “odd-parity” components, by
defining
In the rest frame of the universe, these tensors have some form that is established by the global nature of the solutions of the overarching theory being used. In a frame that is moving relative to the universe, the tensors will have components that depend on the velocity of the frame, and on the orientation of the frame relative to that velocity.
In the case where the theory is rotationally symmetric in the preferred frame, the tensors and
can be expressed in the form
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