**5. Electromagnetism in curved spacetime and the Einstein equivalence principle**

In the earth laboratory, where variation of gravity is small, we can use standard Maxwell equations together with Lorentz force law for ordinary measurements and experiments. However, in precision experiments on earth, in space, in the astrophysical situation or in the cosmological setting, the gravity plays an important role and is non-negligible. In the remaining part of this chapter, we address to the issue of electromagnetism in gravity and more empirical tests of electromagnetism and special relativity. The standard way of including gravitational effects in electromagnetism is to use the comma-goes-to-semicolon rule, i. e., the principle of equivalence (the minimal coupling rule). This is the essence of Einstein Equivalence Principle (EEP) which states that everywhere in the 4-dimensional spacetime, locally, the physics is that of special relativity. This guarantees that the 4 dimensional geometry can be described by a metric *gij* which can be transformed into the Minkowski metric locally. In curved spacetime, *ηij* is replaced by *gij* with partial derivative (comma) replaced by the covariant derivative in the *gij* metric (semi-colon) in the Lagrangian density for a system of charged particles. When this is done the Lagrangian density becomes

$$L\_{l} = -\begin{pmatrix} 1/\{16\text{ri}\} \end{pmatrix} \chi\_{\mathbb{CR}^{\bar{l}\bar{l}\bar{l}}} F\_{\bar{l}\bar{l}} \ F\_{\bar{l}l} - A\_{k} j^{k} \text{(-g)} \begin{pmatrix} \text{-}g \end{pmatrix} \text{(}\text{d}s\text{)} / \text{(}\text{d}t \text{)} \delta(\text{x-x}),\tag{79}$$

where the GR (General Relativity) constitutive tensor *χGRijkl* is given by

$$\chi\_{\mathbb{CP}^{\mathbb{Z}^{\text{j}}}} = (\cdot \mathbb{g})^{1/2} \left[ \left( 1/2 \right) \, \text{g}^{\text{ik}} \, \text{g}^{\text{il}} \text{ - (1/2)} \, \text{g}^{\text{il}} \, \text{g}^{\text{kj}} \right] \, \tag{80}$$

and *g* is the determinant of *gij*. In general relativity or metric theories of gravity where EEP holds, the line element near a world point (event) *P* is given by

$$ds^2 = \mathbf{g}\_{\overline{\mathbb{M}}} \, d\mathbf{x}^i \, d\mathbf{x}^j = \mathbf{g}\_{AB} \, d\mathbf{x}^A \, d\mathbf{x}^B = \left[\eta\_{AB} + \mathbf{O}((\Delta \mathbf{x}^{\mathbb{C}})^2)\right] \, d\mathbf{x}^A \, d\mathbf{x}^B,\tag{81}$$

where {*xi* } is an arbitrary coordinate system, {*xA*} is a locally inertial frame, and *gij* & *gAB* are the metric tensor in their respective frames. According to the definition of locally inertial frame, we have *gAB* = *ηAB* + O((Δ*xC*)2). Therefore, in the locally inertial system near *P*, special relativity holds up to the curvature ambiguity, and the definition of rods and clocks is the same as in the special relativity including local quantum mechanics and electromagnetism.

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 57

with *ηij* the Minkowski metric and |*χ*(1)*ijkl*| << 1 for all *i, j, k,* and *l*. The small special relativity violation (constant part), if any, is put into the *χ*(1)*ijkl*'s. In this field the dispersion

 *K*1 =*χ*(1)1010 - 2*χ*(1)1013 +*χ*(1)1313, (88)

 *K*2 =*χ*(1)2020 - 2*χ*(1)2023+*χ*(1)2323, (89)

 *K* =*χ*(1)1020 - *χ*(1)1023-*χ*(1)1320+*χ*(1)1323. (90) Photons with two different polarizations propagate with different speeds *V*± = *ω*±/*k* and would split in 4-dimensional spacetime. The conditions for no splitting (no retardation) is *ω*<sup>+</sup>

 *K*1 = *K*2, *K* = 0. (91)

*Constraints from no birefringence.* The condition for no birefringence (no splitting, no retardation) for electromagnetic wave propagation in all directions in the weak field limit gives ten independent constraint equations on the constitutive tensor *χijkl*'s. With these ten

where *H* = det (*Hij*) and *Hij* is a metric which generates the light cone for electromagnetic propagation (Ni, 1983a, 1984a,b). Note that (92) has an axion degree of freedom, *φeijkl*, and a 'dilaton' degree of freedom, *ψ.* Lämmerzahl and Hehl (2004) have shown that this nonbirefringence guarantees, without approximation, Riemannian light cone, i.e., Eq. (92) holds without the assumption of weak field also. To fully recover EEP, we need (i) good constraints from no birefringence, (ii) good constraints on no extra physical metric, (iii) good constraints on no *ψ* ('dilaton'), and (iv) good constraints on no *φ* (axion) or no pseudoscalar-

Eq. (92) is verified empirically to high accuracy from pulsar observations and from polarization measurements of extragalactic radio sources. With the null-birefringence observations of pulsar pulses and micropulses before 1980, the relations (92) for testing EEP are empirically verified to 10-14 – 10-16 (Ni, 1983a, 1984a, 1984b). With the present pulsar observations, these limits would be improved; a detailed such analysis is given by Huang (2002). Analyzing the data from polarization measurements of extragalactic radio sources, Haugan and Kauffmann (1995) inferred that the resolution for null-birefringence is 0.02 cycle at 5 GHz. This corresponds to a time resolution of 4 × 10-12 s and gives much better constraints. With a detailed analysis and more extragalactic radio observations, (92) would

relation for *ω* for a plane-wave propagating in the z-direction is

Eq. (91) gives two constraints on the *χ*(1)*ijkl*'s (Ni, 1983a, 1984a, 1984b).

constraints, the constitutive tensor *χijkl* can be written in the following form

*χ*(0)*ijkl* = (1/2)*ηikηjl*- (1/2)*ηilηkj*, (86)

*ω*± = *k*{1+(1/4)[(*K*1+*K*2) ± [(*K*1-*K*2)2 + 4 *K2*]1/2]}, (87)

*χijkl*=(-*H*)1/2[(1/2)*Hik Hjl*-(1/2)*Hil Hkj*]*ψ + φeijkl*, (92)

where

where

= *ω*-, i.e.,

photon interaction.

Nevertheless, for long range propagation and large-scale phenomenon, curvature effects are important. For long range electromagnetic propagation, wavelength/frequency shift is important. From distant quasars, the redshift factor z exceeds 6, i.e., the wavelength changes by more than 6-fold. The gravitational redshift is given by

$$
\Delta \text{\tau}\_{\text{A}} / \Delta \text{\tau}\_{\text{B}} = g\_{00}(\text{B}) / g\_{00}(\text{A}) \,\tag{82}
$$

where Δ*τ*A and Δ*τ*B are the proper periods of a light signal emitted by a source A and received by B respectively. This formula applies equally well to the solar system, to galaxies and to cosmos. Its realm of practical application is in clock and frequency comparisons. In the weak gravitational field such as near earth or in the solar system, we have

$$g\_{00} = 1 \text{--} 2 \text{L/} c^2 \text{.} \tag{83}$$

in the first approximation, where *U* is the Newtonian potential. On the surface of earth, *U*/*c*<sup>2</sup> ≈ 0.7 x 10-9 and the redshift is a fraction of it. This redshift is measured in the laboratory and in space borne missions. It is regularly corrected for the satellite navigation systems such as GPS, GLONASS, Galileo and Beidou. Another effect of electromagnetic propagation in gravity is its deflection with important application to gravitational lensing effects in astrophysics.

## **6. Empirical tests of electromagnetism in gravity and the** *χ-g* **framework**

In section 1, we have discussed the constraints on Proca part of Lagrangian density, i.e., photon mass. In this section, we discuss the empirical foundation of the Maxwell (main) part of electromagnetism. First we need a framework to interpret experimental tests. A natural framework is to extend the GR constitutive tensor *χGRijkl* [equation (80)] into a general form, and look for experimental and observational evidences to test it to see how much it is constrained to the GR form. The general framework we adopt is the *χ-g* framework (Ni, 1983a, 1984a, 1984b, 2010).

The *χ-g* framework can be summarized in the following interaction Lagrangian density

$$L\_l = -\left(1/\left(1\text{fm}\right)\right)\chi^{\text{j}kl}\ F\_{\vec{\eta}}\left F\_{kl} - A\_k j^k \left(\text{-g}\right)^{\left(1/2\right)} - \Sigma\_l m\_l \left(d\mathbf{s}\_l\right) / \left(\text{df}\right)\delta(\mathbf{x}\cdot\mathbf{x}\_l),\tag{84}$$

where *χijkl* = *χklij* = -*χjikl* is a tensor density of the gravitational fields (e.g., *gij*, ϕ, etc.) or fields to be investigated and *Fij ≡ Aj,i - Ai,j* etc. have the usual meaning in classical electromagnetism. The gravitation constitutive tensor density *χijkl* dictates the behaviour of electromagnetism in a gravitational field and has 21 independent components in general. For general relativity or a metric theory (when EEP holds), *χijkl* is determined completely by the metric *gij* and equals (*-g*)1/2 [(1/2) *gik gjl -* (1/2) *gil gjk*].

In the following, we use experiments and observations to constrain the 21 degrees of freedom of *χijkl* to see how close we can reach general relativity. This procedure also serves to reinforce the empirical foundations of classical electromagnetism as EEP locally is based on special relativity including classical electromagnetism.

In the *χ-g* framework, for a weak gravitational field,

$$\mathcal{X}^{ijkl} = \mathcal{X}^{(0)ijkl} + \mathcal{X}^{(1)ijkl},\tag{85}$$

where

56 Trends in Electromagnetism – From Fundamentals to Applications

Nevertheless, for long range propagation and large-scale phenomenon, curvature effects are important. For long range electromagnetic propagation, wavelength/frequency shift is important. From distant quasars, the redshift factor z exceeds 6, i.e., the wavelength changes

where Δ*τ*A and Δ*τ*B are the proper periods of a light signal emitted by a source A and received by B respectively. This formula applies equally well to the solar system, to galaxies and to cosmos. Its realm of practical application is in clock and frequency comparisons. In

 *g00* = 1-2*U*/*c*2, (83) in the first approximation, where *U* is the Newtonian potential. On the surface of earth, *U*/*c*<sup>2</sup> ≈ 0.7 x 10-9 and the redshift is a fraction of it. This redshift is measured in the laboratory and in space borne missions. It is regularly corrected for the satellite navigation systems such as GPS, GLONASS, Galileo and Beidou. Another effect of electromagnetic propagation in gravity is its deflection with important application to gravitational lensing effects in

**6. Empirical tests of electromagnetism in gravity and the** *χ-g* **framework** 

The *χ-g* framework can be summarized in the following interaction Lagrangian density

where *χijkl* = *χklij* = -*χjikl* is a tensor density of the gravitational fields (e.g., *gij*, ϕ, etc.) or fields to be investigated and *Fij ≡ Aj,i - Ai,j* etc. have the usual meaning in classical electromagnetism. The gravitation constitutive tensor density *χijkl* dictates the behaviour of electromagnetism in a gravitational field and has 21 independent components in general. For general relativity or a metric theory (when EEP holds), *χijkl* is determined completely by

In the following, we use experiments and observations to constrain the 21 degrees of freedom of *χijkl* to see how close we can reach general relativity. This procedure also serves to reinforce the empirical foundations of classical electromagnetism as EEP locally is based

*LI = -* (1/(16π))*χijkl Fij Fkl - Ak jk* (*-g*)(1/2) *-* Σ*I mI* (*dsI*)/(*dt*) *δ*(*x*-*xI*), (84)

*χijkl* = *χ*(0)*ijkl* + *χ*(1)*ijkl*, (85)

In section 1, we have discussed the constraints on Proca part of Lagrangian density, i.e., photon mass. In this section, we discuss the empirical foundation of the Maxwell (main) part of electromagnetism. First we need a framework to interpret experimental tests. A natural framework is to extend the GR constitutive tensor *χGRijkl* [equation (80)] into a general form, and look for experimental and observational evidences to test it to see how much it is constrained to the GR form. The general framework we adopt is the *χ-g* framework (Ni,

the weak gravitational field such as near earth or in the solar system, we have

Δ*τ*A/Δ*τ*B = *g00*(B)/*g00*(A), (82)

by more than 6-fold. The gravitational redshift is given by

astrophysics.

1983a, 1984a, 1984b, 2010).

the metric *gij* and equals (*-g*)1/2 [(1/2) *gik gjl -* (1/2) *gil gjk*].

on special relativity including classical electromagnetism.

In the *χ-g* framework, for a weak gravitational field,

$$
\chi^{(0)ijkl} = (1/2)\eta^{ik}\eta^{jl\_{\omega}}(1/2)\eta^{il}\eta^{kj}.\tag{86}
$$

with *ηij* the Minkowski metric and |*χ*(1)*ijkl*| << 1 for all *i, j, k,* and *l*. The small special relativity violation (constant part), if any, is put into the *χ*(1)*ijkl*'s. In this field the dispersion relation for *ω* for a plane-wave propagating in the z-direction is

$$\alpha\_{\pm} = k \langle 1 + (1/4) \vert (K\_1 + K\_2) \pm \vert (K\_1 \therefore K\_2)^2 + 4 \vert K^2 \vert^{1/2} \vert \rangle \rangle. \tag{87}$$

where

$$\mathbf{K}\_1 = \mathbf{\overline{x}^{(1)1010}} \text{ - } \mathbf{\overline{z}^{(1)1013}} + \mathbf{\overline{x}^{(1)1313}},\tag{88}$$

$$K\_2 = \mathcal{X}^{(1)20230} + \mathcal{Z}\mathcal{X}^{(1)2023} + \mathcal{X}^{(1)2323},\tag{89}$$

$$K = \mathbb{X}^{(1)1020} \text{ - } \mathbb{X}^{(1)10223} \text{-} \mathbb{X}^{(1)1220} + \mathbb{X}^{(1)1223} \text{.} \tag{90}$$

Photons with two different polarizations propagate with different speeds *V*± = *ω*±/*k* and would split in 4-dimensional spacetime. The conditions for no splitting (no retardation) is *ω*<sup>+</sup> = *ω*-, i.e.,

$$K\_1 = K\_2, \quad K = 0. \tag{91}$$

Eq. (91) gives two constraints on the *χ*(1)*ijkl*'s (Ni, 1983a, 1984a, 1984b).

*Constraints from no birefringence.* The condition for no birefringence (no splitting, no retardation) for electromagnetic wave propagation in all directions in the weak field limit gives ten independent constraint equations on the constitutive tensor *χijkl*'s. With these ten constraints, the constitutive tensor *χijkl* can be written in the following form

$$\chi^{\circ \text{jkl}} = (\text{-H})^{1/2} [(1/2)\text{H}^{\text{ik}} \text{ H}^{\text{jl}} \text{-} (1/2)\text{H}^{\text{il}} \text{ } \text{H}^{\text{j}}] \text{q} + q \text{e}^{\text{jkl}},\tag{92}$$

where *H* = det (*Hij*) and *Hij* is a metric which generates the light cone for electromagnetic propagation (Ni, 1983a, 1984a,b). Note that (92) has an axion degree of freedom, *φeijkl*, and a 'dilaton' degree of freedom, *ψ.* Lämmerzahl and Hehl (2004) have shown that this nonbirefringence guarantees, without approximation, Riemannian light cone, i.e., Eq. (92) holds without the assumption of weak field also. To fully recover EEP, we need (i) good constraints from no birefringence, (ii) good constraints on no extra physical metric, (iii) good constraints on no *ψ* ('dilaton'), and (iv) good constraints on no *φ* (axion) or no pseudoscalarphoton interaction.

Eq. (92) is verified empirically to high accuracy from pulsar observations and from polarization measurements of extragalactic radio sources. With the null-birefringence observations of pulsar pulses and micropulses before 1980, the relations (92) for testing EEP are empirically verified to 10-14 – 10-16 (Ni, 1983a, 1984a, 1984b). With the present pulsar observations, these limits would be improved; a detailed such analysis is given by Huang (2002). Analyzing the data from polarization measurements of extragalactic radio sources, Haugan and Kauffmann (1995) inferred that the resolution for null-birefringence is 0.02 cycle at 5 GHz. This corresponds to a time resolution of 4 × 10-12 s and gives much better constraints. With a detailed analysis and more extragalactic radio observations, (92) would

Foundations of Electromagnetism, Equivalence Principles and Cosmic Interactions 59

If *φ* ≠ 0 in (96), the gravitational coupling to electromagnetism is not minimal and EEP is violated. Hence WEP I does not imply EEP and Schiff's conjecture (which states that WEP I implies EEP) is incorrect (Ni, 1973, 1974, 1977). However, WEP I does constrain the 21 degrees of freedom of *χ* to only one degree of freedom (*φ*), and Schiff's conjecture is largely

The theory with *φ* ≠ 0 is a pseudoscalar theory with important astrophysical and cosmological consequences (section 8). This is an example that investigations in fundamental physical laws lead to implications in cosmology. Investigations of CP problems in high energy physics leads to a theory with a similar piece of Lagrangian with *φ* the axion field for QCD [Quantum

In the nonmetric theory with *χijkl* (*φ* ≠ 0) given by Eq. (96) (Ni 1973, 1974, 1977), there are anomalous torques on electromagnetic-energy-polarized bodies so that different test bodies will change their rotation state differently, like magnets in magnetic fields. Since the motion of a macroscopic test body is determined not only by its trajectory but also by its rotation state, the motion of polarized test bodies will not be the same. We, therefore, have proposed the following stronger weak equivalence principle (WEP II) to be tested by experiments, which states that in a gravitational field, both the translational and rotational motion of a test body with a given initial motion state is independent of its internal structure and composition (universality of free-fall motion) (Ni 1974, Ni 1977). To put in another way, the behavior of motion including rotation is that in a local inertial frame for test-bodies. If WEP II is violated, then EEP is violated. Therefore from above, in the χ-g framework, the

WEP II state that the motion of all six degrees of freedom (3 translational and 3 rotational) must be the same for all test bodies as in a local inertial frame. There are two different scenarios that WEP II would be violated: (i) the translational motion is affected by the rotational state; (ii) the rotational state changes with angular momentum (rotational direction/speed) or species. Recent experimental results of Gravity Probe B experiment with rotating quartz balls in earth orbit (Everitt et al., 2011) not just verifies frame-dragging effect, but also verifies both aspects of WEP II for unpolarized-bodies to an ultimate precision (Ni,

In this section, we have shown that the empirical foundation of classical electromagnetism is solid except in the aspect of a pseudoscalar-photon interaction. This exception has important

In this section, we discuss the modified electromagnetism in gravity with the pseudoscalarphoton interaction which was reached in the last section, i.e., the theory with the

*LI=-*(1/(16π))(*-g*)*1/2*[(1/2)*gikgjl-*(1/2)*gilgkj*+*φ εijkl*]*FijFkl-Ak jk*(*-g*)(1/2) *-*Σ*I mI*(*dsI*)/(*dt*)*δ*(*x*-*xI*). (97)

ϕ

ϕ

is a scalar or pseudoscalar

ϕ


consequences in cosmology. In the following two sections, we address this issue.

constitutive tensor density (96). Its Lagrangian density is as follows

In the constitutive tensor density and the Lagrangian density,

function of relevant variables. If we assume that the

Chromodynamics] (Peccei and Quinn, 1977; Weinberg, 1978; Wilczek, 1978).

imposition of WEP II guarantees that EEP is valid.

**7. Pseudoscalar-photon interaction** 

right in spirit.

2011).

be tested down to 10-28-10-29 at cosmological distances. In 2002, Kostelecky and Mews (2002) used polarization measurements of light from cosmologically distant astrophysical sources to yield stringent constraints down to 2 × 10-32. For a review, see Ni (2010). In the remaining part of this subsection, we assume (92) to be correct.

*Constraints on one physical metric and no 'dilaton'* (*ψ*)*.* Let us now look into the empirical constraints for *Hij* and *ψ*. In Eq. (84), *ds* is the line element determined from the metric *gij.* From Eq. (92), the gravitational coupling to electromagnetism is determined by the metric *Hij* and two (pseudo)scalar fields *φ* 'axion' and *ψ* 'dilaton'. If *Hij* is not proportional to *gij*, then the hyperfine levels of the lithium atom, the beryllium atom, the mercury atom and other atoms will have additional shifts. But this is not observed to high accuracy in Hughes-Drever experiments (Hughes et al., 1960; Beltran-Lopez et al., 1961; Drever, 1961; Ellena et al., 1987; Chupp et al., 1989). Therefore *Hij* is proportional to *gij* to certain accuracy. Since a change of *Hik* to *λHij* does not affect *χijkl* in Eq. (92), we can define *H*11 = *g*11 to remove this scale freedom (Ni, 1983a, 1984a). For a review, see Ni (2010).

Eötvös-Dicke experiments (Eötvös, 1890; Eötvös et al., 1922; Roll et al., 1964; Braginsky and Panov, 1971; Schlamminger et al., 2008 and references therein) are performed on unpolarized test bodies. In essence, these experiments show that unpolarized electric and magnetic energies follow the same trajectories as other forms of energy to certain accuracy. The constraints on Eq. (92) are

$$\begin{array}{c} \mid \ 1\text{-}\psi \mid \ /\ \downarrow\!\!I \leq 10^{\text{-}10} \text{.} \tag{93} \end{array} \tag{93}$$

and

$$\mid H\_{\rm{IO}}\text{-}\;\text{go}\mid\mid\prime\;\mathsf{U}\leqslant\mathsf{10}\,\mathsf{6}\,\tag{94}$$

where *U* (~ 10-8) is the solar gravitational potential at the earth.

In 1976, Vessot et al. (1980) used an atomic hydrogen maser clock in a space probe to test and confirm the metric gravitational redshift to an accuracy of 1.4 × 10-4, i. e.,

$$\mid H\_{00} \text{ } g\_{00} \mid / \text{ } \mathcal{U} \le 1.4 \times 10^4 \text{ }\tag{95}$$

where *U* is the change of earth gravitational field that the maser clock experienced.

With constraints from (i) no birefringence, (ii) no extra physical metric, (iii) no *ψ* ('dilaton'), we arrive at the theory (84) with *χijkl* given by

$$\chi^{\rm jild} = (\cdot \text{g})^{1/2} \left[ \begin{pmatrix} 1/2 \end{pmatrix} \begin{smallmatrix} \mathbf{g}^{\rm i} \ \mathbf{g}^{\rm j} \end{smallmatrix} \begin{smallmatrix} \mathbf{1}/2 \end{smallmatrix} \begin{smallmatrix} \mathbf{g}^{\rm i} \ \mathbf{g}^{\rm j} \end{smallmatrix} + \begin{smallmatrix} \mathbf{q} \ \mathbf{e}^{\rm jild} \end{smallmatrix} \right] \tag{96}$$

i.e., an axion theory (Ni, 1983a, 1984a; Hehl and Obukhov 2008). Here *εijkl* is defined to be ( *g*)-1/2 *eijkl*. The current constraints on *φ* from astrophysical observations and CMB polarization observations will be discussed in section 8. Thus, from experiments and observations, only one degree of freedom of *χijkl* is not much constrained.

Now let's turn into more formal aspects of equivalence principles. We proved that for a system whose Lagrangian density is given by Eq. (84), the Galileo Equivalence Principle (UFF [Universality of Free Fall; WEP I [Weak Equivalence Principle I]) holds if and only if Eq. (96) holds (Ni, 1974, 1977).

58 Trends in Electromagnetism – From Fundamentals to Applications

be tested down to 10-28-10-29 at cosmological distances. In 2002, Kostelecky and Mews (2002) used polarization measurements of light from cosmologically distant astrophysical sources to yield stringent constraints down to 2 × 10-32. For a review, see Ni (2010). In the remaining

*Constraints on one physical metric and no 'dilaton'* (*ψ*)*.* Let us now look into the empirical constraints for *Hij* and *ψ*. In Eq. (84), *ds* is the line element determined from the metric *gij.* From Eq. (92), the gravitational coupling to electromagnetism is determined by the metric *Hij* and two (pseudo)scalar fields *φ* 'axion' and *ψ* 'dilaton'. If *Hij* is not proportional to *gij*, then the hyperfine levels of the lithium atom, the beryllium atom, the mercury atom and other atoms will have additional shifts. But this is not observed to high accuracy in Hughes-Drever experiments (Hughes et al., 1960; Beltran-Lopez et al., 1961; Drever, 1961; Ellena et al., 1987; Chupp et al., 1989). Therefore *Hij* is proportional to *gij* to certain accuracy. Since a change of *Hik* to *λHij* does not affect *χijkl* in Eq. (92), we can define *H*11 = *g*11 to remove this

Eötvös-Dicke experiments (Eötvös, 1890; Eötvös et al., 1922; Roll et al., 1964; Braginsky and Panov, 1971; Schlamminger et al., 2008 and references therein) are performed on unpolarized test bodies. In essence, these experiments show that unpolarized electric and magnetic energies follow the same trajectories as other forms of energy to certain accuracy.


In 1976, Vessot et al. (1980) used an atomic hydrogen maser clock in a space probe to test


With constraints from (i) no birefringence, (ii) no extra physical metric, (iii) no *ψ* ('dilaton'),

i.e., an axion theory (Ni, 1983a, 1984a; Hehl and Obukhov 2008). Here *εijkl* is defined to be ( *g*)-1/2 *eijkl*. The current constraints on *φ* from astrophysical observations and CMB polarization observations will be discussed in section 8. Thus, from experiments and

Now let's turn into more formal aspects of equivalence principles. We proved that for a system whose Lagrangian density is given by Eq. (84), the Galileo Equivalence Principle (UFF [Universality of Free Fall; WEP I [Weak Equivalence Principle I]) holds if and only if

*χijkl =* (*-g*)*1/2* [(1/2) *gik gjl -* (1/2) *gil gkj + φ εijkl*], (96)


part of this subsection, we assume (92) to be correct.

scale freedom (Ni, 1983a, 1984a). For a review, see Ni (2010).

where *U* (~ 10-8) is the solar gravitational potential at the earth.

we arrive at the theory (84) with *χijkl* given by

Eq. (96) holds (Ni, 1974, 1977).

and confirm the metric gravitational redshift to an accuracy of 1.4 × 10-4, i. e.,

observations, only one degree of freedom of *χijkl* is not much constrained.

where *U* is the change of earth gravitational field that the maser clock experienced.

The constraints on Eq. (92) are

and

If *φ* ≠ 0 in (96), the gravitational coupling to electromagnetism is not minimal and EEP is violated. Hence WEP I does not imply EEP and Schiff's conjecture (which states that WEP I implies EEP) is incorrect (Ni, 1973, 1974, 1977). However, WEP I does constrain the 21 degrees of freedom of *χ* to only one degree of freedom (*φ*), and Schiff's conjecture is largely right in spirit.

The theory with *φ* ≠ 0 is a pseudoscalar theory with important astrophysical and cosmological consequences (section 8). This is an example that investigations in fundamental physical laws lead to implications in cosmology. Investigations of CP problems in high energy physics leads to a theory with a similar piece of Lagrangian with *φ* the axion field for QCD [Quantum Chromodynamics] (Peccei and Quinn, 1977; Weinberg, 1978; Wilczek, 1978).

In the nonmetric theory with *χijkl* (*φ* ≠ 0) given by Eq. (96) (Ni 1973, 1974, 1977), there are anomalous torques on electromagnetic-energy-polarized bodies so that different test bodies will change their rotation state differently, like magnets in magnetic fields. Since the motion of a macroscopic test body is determined not only by its trajectory but also by its rotation state, the motion of polarized test bodies will not be the same. We, therefore, have proposed the following stronger weak equivalence principle (WEP II) to be tested by experiments, which states that in a gravitational field, both the translational and rotational motion of a test body with a given initial motion state is independent of its internal structure and composition (universality of free-fall motion) (Ni 1974, Ni 1977). To put in another way, the behavior of motion including rotation is that in a local inertial frame for test-bodies. If WEP II is violated, then EEP is violated. Therefore from above, in the χ-g framework, the imposition of WEP II guarantees that EEP is valid.

WEP II state that the motion of all six degrees of freedom (3 translational and 3 rotational) must be the same for all test bodies as in a local inertial frame. There are two different scenarios that WEP II would be violated: (i) the translational motion is affected by the rotational state; (ii) the rotational state changes with angular momentum (rotational direction/speed) or species. Recent experimental results of Gravity Probe B experiment with rotating quartz balls in earth orbit (Everitt et al., 2011) not just verifies frame-dragging effect, but also verifies both aspects of WEP II for unpolarized-bodies to an ultimate precision (Ni, 2011).

In this section, we have shown that the empirical foundation of classical electromagnetism is solid except in the aspect of a pseudoscalar-photon interaction. This exception has important consequences in cosmology. In the following two sections, we address this issue.
