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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 [204194Jump To The Next Citation Point62]), 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., [124Jump To The Next Citation Point]). A system in a quantum state A decays to state B, emitting a quantum of frequency ν. The quantum falls a height H in an external gravitational field and is shifted to frequency ′ ν, while the system in state B falls with acceleration gB. At the bottom, state A is rebuilt out of state B, the quantum of frequency ν ′, and the kinetic energy m B gB H that state B has gained during its fall. The energy left over must be exactly enough, mA gA H, to raise state A to its original location. (Here an assumption of local Lorentz invariance permits the inertial masses mA and mB to be identified with the total energies of the bodies.) If gA and gB depend on that portion of the internal energy of the states that was involved in the quantum transition from A to B according to

( αE ) ( αE ) gA = g 1 + ---A2- , gB = g 1 + ---B2- , EA − EB ≡ h ν (7 ) mAc mBc
(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 2 hν∕mc)
ν′ − ν gH ΔU Z = ---′-- = (1 + α) -2--= (1 + α)--2-. (8 ) ν c c
Haugan generalized this approach to include violations of LLI [124] (TEGP 2.5 [281Jump To The Next Citation Point]).


Box 1. The T H εμ formalism


Coordinate system and conventions:
0 x = t: time coordinate associated with the static nature of the static spherically symmetric (SSS) gravitational field; x = (x,y, z): isotropic quasi-Cartesian spatial coordinates; spatial vector and gradient operations as in Cartesian space.

Matter and field variables:

Gravitational potential:
U (x ).

Arbitrary functions:
T(U ), H (U ), ε(U ), μ(U ); EEP is satisfied if ε = μ = (H∕T )1∕2 for all U.

∑ ∫ ∑ ∫ ∫ I = − m0a (T − Hv2a)1∕2dt + ea A μ(x νa)vμa dt + (8π)−1 (εE2 − μ −1B2) d4x. a a

Non-metric parameters:
2-∂-- 1∕2 2-∂-- 1∕2 −1 Γ 0 = − c0∂U ln[ε(T ∕H ) ]0, Λ0 = − c0∂U ln[μ (T∕H ) ]0, Υ0 = 1 − (T H εμ)0,

where c0 = (T0∕H0 )1∕2 and subscript “0” refers to a chosen point in space. If EEP is satisfied, Γ 0 ≡ Λ0 ≡ Υ0 ≡ 0.


2.2.2 The T H εμ formalism

The first successful attempt to prove Schiff’s conjecture more formally was made by Lightman and Lee [166]. They developed a framework called the TH εμ 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 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 ε(U ) and μ (U ). The forms of T, H, ε, and μ vary from theory to theory, but every metric theory satisfies

( )1∕2 ε = μ = H-- , (9 ) T
for all U. This consequence follows from the action of electrodynamics with a “minimal” or metric coupling:
∫ ∫ ∫ ∑ μ ν 1∕2 ∑ ν μ -1-- √ --- μα νβ 4 I = − m0a (− gμνvava) dt + ea Aμ(xa)va dt − 16π − gg g F μνFαβ d x, (10 ) a a
where the variables are defined in Box 1, and where Fμν ≡ A ν,μ − A μ,ν. By identifying g00 = T and g = H δ ij ij in a SSS field, F = E i0 i and F = ε B ij ijk k, one obtains Equation (9View Equation). Conversely, every theory within this class that satisfies Equation (9View Equation) can have its electrodynamic equations cast into “metric” form. In a given non-metric theory, the functions T, H, ε, and μ 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 Equation (9View Equation) 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, ε, and μ reflect different aspects of EEP. For instance, position or U-dependence of either of the combinations ε(T ∕H )1∕2 and μ(T ∕H )1∕2 signals violations of LPI, the first combination playing the role of the locally measured electric charge or fine structure constant. The “non-metric parameters” Γ 0 and Λ0 (see Box 1) are measures of such violations of EEP. Similarly, if the parameter −1 Υ0 ≡ 1 − (T H εμ)0 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 c 0, given by

( )1 ∕2 −1∕2 T0- c = (ε0μ0) , c0 = H0 . (11 )
In many applications, by suitable definition of units, c0 can be set equal to unity. If EEP is valid, Γ 0 ≡ Λ0 ≡ Υ0 = 0 everywhere.

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

mP a = ----∇U, (12 ) m ES [ ] MS [ ] mP--= 1 + -EB-- 2 Γ − 8Υ + E-B-- 2Λ − 4Υ + ..., (13 ) m M c20 0 3 0 M c20 0 3 0
where ES EB and MS E B are the electrostatic and magnetostatic binding energies of the body, given by
⟨ ∑ ⟩ EES = − 1T 1∕2H − 1ε− 1 eaeb , (14 ) B 4 0 0 0 ab rab ⟨ ⟩ MS 1- 1∕2 − 1 ∑ eaeb EB = − 8T 0 H 0 μ0 r [va ⋅ vb + (va ⋅ nab)(vb ⋅ nab)] , (15 ) ab ab
where rab = |xa − xb|, nab = (xa − xb)∕rab, 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 Equation (13View Equation), and ultimately place limits on the non-metric parameters |Γ 0| < 2 × 10 −10 and |Λ0| < 3 × 10−6. (We set Υ0 = 0 because of very tight constraints on it from tests of LLI; see Figure 2View Image, where δ = − Υ.) 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) [281Jump To The Next Citation Point]).

The T H εμ 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 α (see Equation (6View Equation)), the results are (TEGP 2.6 (c) [281Jump To The Next Citation Point]):

( − 3 Γ + Λ hydrogen hyperfine transition, H -Maser clock, ||{ 0 0 α = − 1(3Γ + Λ ) electromagnetic mode in cavity, SCSO clock, (16 ) || 2 0 0 ( − 2 Γ phonon mode in solid, principal transition in hydrogen. 0

The redshift is the standard one (α = 0 ), independently of the nature of the clock if and only if Γ 0 ≡ Λ0 ≡ 0. Thus the Vessot–Levine rocket redshift experiment sets a limit on the parameter combination |3 Γ 0 − Λ0| (see Figure 3View Image); the null-redshift experiment comparing hydrogen-maser and SCSO clocks sets a limit on |αH − αSCSO | = 32|Γ 0 − Λ0|. Alvarez and Mann [768910] extended the T H εμ 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 2 c formalism

The TH εμ 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, ε, and μ, 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) [281Jump To The Next Citation Point])

∑ ∫ ∑ ∫ ∫ I = − m0a (1 − v2a)1∕2dt + ea A μ(xνa)vμa dt + (8 π)−1 (E2 − c2B2 )d4x, (17 ) a a
where 2 −1 c ≡ H0∕ (T0 ε0μ0 ) = (1 − Υ0 ). This amounts to using units in which the limiting speed c0 of massive test particles is unity, and the speed of light is c. If c ⁄= 1, 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 Equation (17View Equation) yield the behavior of rods and clocks, just as in the full TH εμ formalism. For example, the length of a rod which moves with velocity V 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 1 − V 2∕2 + 𝒪 (V 4). Notice that c does not appear in this expression, because only electrostatic interactions are involved, and c appears only in the magnetic sector of the action (17View Equation). The energy and momentum of an electromagnetically bound body moving with velocity V relative to the rest frame are given by

1 1 E = MR + -MRV 2 + --δM iIjViV j + 𝒪 (M V 4), 2 2 (18 ) P i = M V i + δM ijV j + 𝒪 (M V 3), R I
where MR = M0 − EESB, M0 is the sum of the particle rest masses, EESB is the electrostatic binding energy of the system (see Equation (14View Equation) with T1∕2H ε− 1= 1 0 00), and
( ) [ ] ij 1 4 ES ij ES ij δM I = − 2 -2 − 1 -E B δ + &tidle;EB , (19 ) c 3
⟨ ⟩ ESij 1 ∑ eaeb ( j 1 ) E&tidle;B = − -- ---- niabn ab −--δij . (20 ) 4 ab rab 3
Note that (c− 2 − 1) corresponds to the parameter δ plotted in Figure 2View Image.

The electrodynamics given by Equation (17View 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 ℏ times its frequency ω, while its momentum is ℏ ω∕c. 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 2 − 2 L0 (v ∕c)(c − 1). The experimental null result then leads to the bound on −2 (c − 1) shown on Figure 2View Image. Similarly the anisotropy in energy levels is clearly illustrated by the tensorial terms in Equations (18View Equation, 20View Equation); by evaluating &tidle;EES ij B 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 2View Image.

The behavior of moving atomic clocks can also be analyzed in detail, and bounds on (c−2 − 1 ) 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 2 c 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 [279Jump To The Next Citation Point]).

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

Δ-φ- 4- 2 &tidle;φ ≈ − 3(1 − c )(V ⋅ n − V ⋅ n0), (21 )
where &tidle;φ = 2πνL, ν is the maser frequency, L = 21 km is the baseline, and where n and n0 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 −2 −4 |c − 1| < 3 × 10. 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 [279].

2.2.4 The Standard Model Extension (SME)

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 [6364155]. Called the Standard Model Extension (SME), it takes the standard SU (3) × SU (2) × U (1) 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 T H εμ and c2 frameworks, and the χ − g 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 c2 framework discussed above. In the SME, the Lagrangian for a scalar particle φ with charge e interacting with electrodynamics takes the form

1 [ ] ℒ = [ημν + (kφ)μν](Dμ φ)†Dνφ − m2φ †φ − -- ημαηνβ + (kF)μναβ F μνFαβ, (22 ) 4
where D μφ = ∂μφ + ieA μφ, where (kφ)μν is a real symmetric trace-free tensor, and where (kF )μναβ is a tensor with the symmetries of the Riemann tensor, and with vanishing double trace. It has 19 independent components. There could also be a CPT-odd term in ℒ of the form (k )με A νF αβ A μναβ, but because of a variety of pre-existing theoretical and experimental constraints, it is generally set to zero.

The tensor (kF)μανβ can be decomposed into “electric”, “magnetic”, and “odd-parity” components, by defining

(κDE )jk = − 2(kF )0j0k, jk 1-jpq krs pqrs (23 ) (κHB ) = 2ε ε (kF) , kj jk jpq 0kpq (κDB ) = − (kHE ) = ε (kF) .
In many applications it is useful to use the further decomposition
&tidle;κtr = 1(κDE )jj, 3 1 jk (&tidle;κe+)jk = -(κDE + κHB ) , 2 jk 1- jk 1-jk ii (24 ) (&tidle;κe− ) = 2(κDE − κHB ) − 3δ (κDE ) , (&tidle;κo+)jk = 1(κDB + κHE )jk, 2 1 jk (&tidle;κo− )jk =-(κDB − κHE ) . 2
The first expression is a single number, the next three are symmetric trace-free matrices, and the final is an antisymmetric matrix, accounting thereby for the 19 components of the original tensor (kF )μα νβ.

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 (k )μν φ and μναβ (kF ) can be expressed in the form

( ) (k φ)μν = &tidle;κφ uμ uν + 1ημν , (25 ) 4 (k )μναβ = &tidle;κ (4u[μ ην][αu β] − ημ[αηβ]ν), (26 ) F tr
where [ ] around indices denote antisymmetrization, and where uμ is the four-velocity of an observer at rest in the preferred frame. With this assumption, all the tensorial quantities in Equation (24View Equation) vanish in the preferred frame, and, after suitable rescalings of coordinates and fields, the action (22View Equation) can be put into the form of the 2 c framework, with
( 3 )1 ∕2 ( )1 ∕2 c = 1-−--4&tidle;κφ 1 −-&tidle;κtr- . (27 ) 1 + 14 &tidle;κφ 1 + &tidle;κtr

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