Variation of the speed of light due to non-minimal coupling between electromagnetism and gr

发布于:2021-11-27 16:48:58

arXiv:gr-qc/0303081v3 11 Apr 2003

Variation of the speed of light due to non-minimal coupling between electromagnetism and gravity?
P. Teyssandier ? SYRTE, CNRS/UMR 8630, Observatoire de Paris, 61 av. de l’Observatoire, F-75014 Paris, France

Abstract We consider an Einstein-Maxwell action modi?ed by the addition of three terms coupling the electromagnetic strength to the curvature tensor. The corresponding generalized Maxwell equations imply a variation of the speed of light in a vacuum. We determine this variation in FriedmannRobertson-Walker spacetimes. We show that light propagates at a speed greater than c when a simple condition is satis?ed. PACS numbers: 04.40.Nr; 98.80.-k; 95.30.Sf

1

Introduction

Theories with a varying speed of light have been recently proposed to solve the initial value problems of standard cosmology (see, e.g., [1]–[6] and Refs. therein). However, these theories break local Lorentz invariance and su?er from quite unclear physical interpretations. So, it seems to us that it is necessary to explore less extremist ideas. The present paper is based on the observation that the invariant speed c involved in the local Lorentz transformations must be carefully distinguished from the speed of light considered as a signal propagating in a vacuum. In fact, it is possible both to maintain the basic principles of metric theories of gravity and to obtain a variable speed of light by modifying Maxwell’s equations [7]. For this reason, we suggest to call c the ”spacetime structure constant” instead of the ”speed of light”, which is a very misleading terminology. In what follows, we consider the modi?ed Einstein-Maxwell equations which arise from an electromagnetic ?eld non-minimally coupled to the gravitational
lecture given at the Meeting on Electromagnetism organized by the Fondation Louis de Broglie in Peyresq (France), September 2-7, 2002. ? E-mail: Pierre.Teyssandier@obspm.fr
? Invited

1

?eld. Several kinds of non-minimal couplings may be proposed [8], some of them violating gauge invariance [9]. In order to avoid such a radical consequence, we de?ne the action as I= ? √ c3 ?gd4 x , (R + 2Λ) + LEM ? j ? A? + Lmatter 16πG (1)

where G is the Newtonian gravitational constant, Λ is the cosmological constant, R is the scalar curvature, j ? is the current density vector, and A? is the vector potential. The non-minimally coupled electromagnetic ?eld Lagrangian is de?ned as 1 1 1 1 ν LEM = ? F?ν F ?ν + ξRF?ν F ?ν + ηR?ν F ?ρ F.ρ + ζR?νρσ F ?ν F ρσ , (2) 4 4 2 4 where F?ν = ?? Aν ? ?ν A? is the electromagnetic strength, R?νρσ is the curvature tensor, R?ν is the Ricci tensor, ξ, η, and ζ are constants having dimensions [length]2 . The ?eld variables are the metric g?ν , the vector potential A? and the variables describing matter in Lmatter . It has been previously noted that the generalized Maxwell equations deduced from Eqs. (1)–(2) imply a variable speed of light [10, 11, 12], except in the case where η = ζ = 0 [13]. However, as far as we know, the formula giving the speed of light in the Friedmann-Robertson-Walker (FRW) cosmological models has not been found in the general case. The main purpose of the present paper is to yield this formula. A detailed analysis of its cosmological implications will be developed elsewhere. The plan of the paper is as follows. In Sect. 2, we recall why a varying speed of light is compatible with the general axioms underlying special and general relativity. Section 3 is devoted to arguments in favour of the non-minimal coupling (NMC) examined here. In Sect. 4, we derive the equations of motion for the electromagnetic strength F?ν in any spacetime. In Sect. 5, we form the equations satis?ed by the wave vector in the limit of the geometric optics approximation, with a special emphasis on the case where the Einstein tensor has the structure corresponding to a perfect ?uid. In Sect. 6, we outline the theory of light rays in the FRW background. In Sect. 7, we get the explicit expression of the speed of light as a function of the energy and pressure content of the Universe when Einstein equations are satis?ed. Finally, we give some concluding remarks in Sect. 8. Conventions and notations.– The signature of the Lorentzian metric g is (+ ? ??). Greek indices run from 0 to 3. We put x0 = ct, t being a coordinate time. Given a vector or a tensor T , we often use the notations T,α = ?α T and T;α = ?α T . The curvature tensor R?νρσ is de?ned by w?;ρ;σ ? w?;σ;ρ = λ ?R?νρσ wν . For the Ricci tensor, we take R?ν = R.?λν . Given two 4-vectors v ? ν and w, we often use the notation (v·w) = g?ν v w . When there is no ambiguity, we write l2 = (l · l) = g?ν l? lν . We put κ = 8πG/c4 . 2

2

Variable speed of light and relativity

Let us give some additional arguments which justify the necessity to distinguish the spacetime structure constant c from the speed of light. Our analysis is inspired by [14]. The widely held idea that c must be identi?ed with the speed of light in relativistic theories is supported by the fact that the statement of invariance of the speed of light played a crucial role in the original Einstein’s paper [15]. However, this earliest approach is not the more logical one and can be criticized for several reasons. 1) If we state that the invariant quantity c is linked to a property of electromagnetic radiation, it is hard to understand why all interactions are governed, at least locally, by special relativity. 2) The invariance of the laws of physics under the Poincar? group allows the e existence of zero-mass particles but does not imply that any empirical interaction must be mediated by a zero-mass particle. As a consequence, if once a day a non-zero mass is found for the photon, only the usual presentations of special relativity based on the invariance of the speed of light will be disproved. 3) Shortly after the Einstein’s paper, it was pointed out in [16, 17] that it was not necessary to assume the constancy of the light velocity in order to derive the Lorentz transformations. Since these pioneering works, several other derivations of the correct transformation equations have been performed without imposing the existence of an invariant speed (see, e.g., [18] and Refs. therein). In [14], it was proved by elementary considerations that if i) the principle of relativity is valid, ii) spacetime is a four-dimensional manifold, iii) spacetime is homogeneous, iv) space is isotropic, v) the inertial transformations constitute a group and vi) causality is preserved, then the tranformation equations may be written as follows in the two-dimensional case: x′ = x ? vt 1? χv 2 , t′ = t ? χvx 1 ? χv 2 , (3)

where χ is a constant which must be ≥ 0 (χ < 0 would imply violations of causality). The Galileo transformations are recovered for χ = 0, while the Lorentz transformations are obtained for χ > 0. Clearly, Eq. (3) implies that √ the quantity c = 1/ χ is both an invariant and a limiting speed. The above-mentioned axioms are universal principles which do not require any de?nite, well-developed theory of some physical interaction. The existence of such axioms proves that the constant c appearing in the Lorentz transformations is not essentially the speed of light in a vacuum, but is in fact a structural constant which characterizes the four-dimensional continuum constituting the arena of physical events. It follows from this discussion that a variable speed of light cannot be forbidden by special or general relativity.

3

3

Arguments in favour of non-minimal coupling

Several arguments may be given in favour of the non-minimal coupling de?ned by Eqs. (1)-(2). 1) It may be argued from a theorem due to Horndeski [19, 20] that the most general electromagnetic equations which are i) derivable from a variational principle, ii) at most of second-order in the derivatives of both g?ν and A? , iii) consistent with the charge conservation, and iv) compatible with Maxwell’s equations in ?at space-time are given by a Lagrangian de?ned by Eq. (2) with η = ?2ξ , ζ = ξ, (4)

ξ being arbitrary. This beautiful theorem proves that a particular NMC is inevitable if one considers the generalization of Maxwell’s equations compatible with the currently accepted principles of electromagnetism. The interest of the Horndeski Lagrangian is enhanced by the fact that it can be recovered from the Gauss-Bonnet action in a ?ve-dimensional space with a Kaluza-Klein metric (see [21], [22] and Refs. therein). 2) It may also be argued that couplings to the curvature are induced by vacuum polarization in quantum electrodynamics(QED) [10, 23]. Indeed, vacuum polarization confers a size to the photon of the order of the Compton wavelength of the electron. So the motion of the photon is in?uenced by a tidal gravitational e?ect depending on the curvature. Working in the one-loop approximation, Drummond and Hathrell found an e?ective Lagrangian for QED given by Eq. (2) with α 2 13α 2 α 2 λc , η = ? λc , ζ = ? ? λc , ? (5) 36π 180π 90π where α is the ?ne-structure constant and λc is the Compton wavelength of the ? electron de?ned as λc = h/me c [24]. ? ? We think that these remarkable results show that the NMC introduced by Eqs. (1)–(2) deserves to be studied in detail. ξ=?

4

Equations of generalized electromagnetism

Varying the action I with respect to the vector potential A? leads to generalized Maxwell equations. It is easy to form these equations by using the following lemma. Let L be a scalar Lagrangian such that 1 ?νρσ E F?ν Fρσ ? j ? A? ≡ E ?νρσ A?;ν Aρ;σ ? j ? A? , (6) 4 where E ?νρσ is a 4-rank tensor which involves neither the vector potential A? nor its derivatives of any order and which satis?es the properties of symmetry and antisymmetry E ?νρσ = E ρσ?ν , (7) L= 4

Applying this lemma to the Lagrangian LEM de?ned by Eq. (2) and using Eq. (9), we get the equations of motion F ?ν;ν = ?j ? , where F ?ν is the 2-rank tensor
? ν F ?ν = (1 ? ξR) F ?ν ? η Rλ F λν ? Rλ F λ? ? ζR?νρσ Fρσ .

It is easily seen that the corresponding Euler-Lagrange equations may be written as ?ν (E ?νρσ Fρσ ) = j ? . (9)

E ?νρσ = ?E ν?ρσ = ?E ?νσρ .

(8)

(10)

(11) (12)

Of course, Eq. (10) must be complemented by the equations F?ν;ρ + Fρ?;ν + Fνρ;? = 0 .

Since the tensor F ?ν de?ned by Eq. (11) is obviously antisymmetric, the charge conservation equation ?α j α = 0 is a condition of integrability of Eqs. (11)-(12). So, charge conservation is embodied in the NMC theory deduced from the action I. It follows from Eq. (10) that F ?ν may be considered as the electromagnetic excitation [25]. Thus, the non-minimally coupled electromagnetic excitation in a vacuum must be distinguished from the electromagnetic strength F ?ν . Using the identities ? R?νρσ Fρσ ≡ 2Rρ;σ F ρσ , ;ν it is easily seen that Eq. (10) may be written as
? ν ?ν (1 ? ξR)F ?ν;ν ? η(Rλ F λν;ν ? Rλ F λ?;ν ) ? ζR. . ρσ F ρσ;ν 1 ? ? (2ξ + η)R,ν F ?ν ? (η + 2ζ)Rρ;σ F ρσ = ?j ? . (13) 2 It is worthy of note that these equations involve the third partial derivatives of the metric, except of course in the case where Eqs. (4) are satis?ed. Introducing now the Weyl tensor C ?νρσ de?ned by 1 C ?νρσ = R?νρσ ? (R?ρ g νσ + Rνσ g ?ρ ? R?σ g νρ ? Rνρ g ?σ ) 2 1 + R (g ?ρ g νσ ? g ?σ g νρ ) , (14) 6 Eq. (13) becomes

1 ? ν 1 ? (3ξ ? ζ)R F ?ν;ν ? (η + ζ)(Rλ F λν;ν ? Rλ F λ?;ν ) ? ζC.?νρσ F ρσ;ν . 3 1 ? (15) ? (2ξ + η)R,ν F ?ν ? (η + 2ζ)Rρ;σ F ρσ = ?j ? . 2 This last form of the equations of motion will be very useful to study the propagation of light in a gravitational wave or in the FRW background. 5

5

Geometric optics approximation

In order to determine the speed of light, we shall work in the limit of the geometric optics approximation. Treating the vector potential A? as the real part of a complex vector, we suppose that there exist solutions to Eq. (15) which admit a development of the form A? (x, ε) = ? [a? (x) + O(?)] exp i S(x) ε , (16)

where a? is a slowly varying, complex vector amplitude, S(x) is a real function and ε is a dimensionless parameter which tends to zero as the typical wavelength of the electromagnetic signal becomes shorter and shorter. Let us de?ne the wave vector l? as 1 (17) l? = S,? . ε We have i S(x) + ··· . (18) F?ν = ? i(l? aν ? lν a? ) exp ε Inserting Eq. (18) into Eq. (15), and then retaining only the leading terms of order ε?2 , we obtain the equations constraining the wave vector l? in the form: 1 ? 1 ? ξ ? ζ R l2 ? (η + ζ)Ric(l, l) a? ? (η + ζ)Rλ l2 aλ 3 1 ? 1 ? ξ ? ζ R (a · l) ? (η + ζ)Ric(a, l) l? 3
? +(η + ζ)Rλ (a · l) lλ ? 2ζC ?νρσ lν aρ lσ = 0 ,

(19)

where we use the notation Ric(v, w) = R?ν v ? wν . Equation (19) shows that the wave vector is generally not a null vector and that l2 will depend on the polarization vector f ? = a? /a, a being the scalar ? amplitude de?ned by a = | a? a? |. Thus, light rays are not null geodesics and a gravitational ?eld has properties of birefringence [10, 12, 26]. Moreover, Eq. (19) remain invariant under scaling of the wave vector l? . As a consequence, the photon trajectories are frequency independent: the gravitational ?eld is not dispersive. Let us now restrict our attention to the case where the Ricci tensor is of the form corresponding to a perfect ?uid in general relativity. This means that there exists a unit timelike vector u? such that R?ν may be written in the form R?ν = 1 1 (4U ? R)u? uν ? (U ? R)g?ν , 3 3 (20)

where U is a scalar function. This scalar function is such that U = Ric(u, u) . 6 (21)

Then, de?ning A and B as 1 2 A = ? (3ξ + 2η + ζ)R + (η + ζ)U , 3 3 Equation (19) reduces to (1 + A)l2 ? B(u · l)2 a? ? [(1 + A)(a · l) ? B(u · a)(u · l)] l? ?B (u · a)l2 ? (u · l)(a · l) u? ? 2ζC ?νρσ lν aρ lσ = 0 . Contracting Eq. (23) by u? yields the relation (1 + A ? B) (u · a)l2 ? (a · l)(u · l) = 2ζC(u, l, a, l) , where C(u, l, a, l) = C ?νρσ u? lν aρ lσ . Eliminating (u · a)l ? (a · l)(u · l) between Eq. (23) and Eq. (24), we ?nd (1 + A)l2 ? B(u · l)2 a? ? [(1 + A)(a · l) ? B(u · a)(u · l)] l? B C(u, l, a, l)u? ? 2ζC ?νρσ lν aρ lσ = 0 . ?2ζ 1+A?B It is easily seen that Eq. (26) are equivalent to Eq. (23) if the inequalities 1+A?B = 0, are satis?ed. 1+A=0 (27) (26)
2

B=

1 (η + ζ)(4U ? R) , 3

(22)

(23)

(24) (25)

6

Application to FRW cosmological models

In what follows, we assume that the ?eld F?ν is a test ?eld propagating in a Friedmann-Robertson-Walker (FRW) universe with a metric ds2 = (dx0 )2 ? a2 (x0 ) dr2 + r2 (dθ2 + sin2 θd?2 ) , 1 ? kr2 (28)

where a(x0 ) is the scale factor and k = 0, 1, ?1 for ?at, closed and open models, respectively. In these models, the Ricci tensor may be written in the form given by Eq. (20), where u? is the unit 4-velocity of a comoving observer (observer moving with the average ?ow of cosmic energy). Moreover, the metric g is conformally ?at, which means that C ?νρσ = 0 throughout spacetime. As a consequence, Eq. (26) reduces to (1 + A)l2 ? B(u · l)2 a? ? [(1 + A)(a · l) ? B(u · a)(u · l)] l? = 0 . 7 (30) (29)

Of course, we suppose that inequalities (27) hold. If it is assumed that (1 + A)l2 ? B(u · l)2 = 0, Eq. (30) yields a? = (1 + A)(a · l) ? B(u · a)(u · l) ? l , (1 + A)l2 ? B(u · l)2

which implies F?ν = 0 (see Eq. (18)). As a consequence, the wave vector must ful?ll the condition (1 + A)l2 ? B(u · l)2 = 0 . (31) Two cases are to be envisaged. 1. General case: B = 0 .– It results from Eq. (31) that l2 = 0 and (u · l) = 0 (indeed, l2 = 0 would imply (u · l) = 0, which is impossible if l = 0). Therefore, the phase velocity of light is not equal to the fundamental constant c if η + ζ = 0 and 4U ? R = 0. Since l2 = 0, it is possible to choose the gauge so that the Lorentz condition (a · l) = 0 (32) is satis?ed. With this choice, Eq. (30) gives (u · a) = 0 . (33)

The corresponding polarization vector is orthogonal to the unit 4-velocity u? and to the wave vector l? . 2. Case where B = 0 .– The wave vector l is then a null vector, as in the usual geometric optics approximation. It is worthy of note that Eq. (32) is now a consequence of Eq. (30) and does not result from a special choice of gauge. Nevertheless, the gauge may be chosen so that Eq. (33) is satis?ed. It follows from Eq. (31) that the phase speed of light with respect to a comoving observer has the same value cl in all directions and for all polarizations. Since the gravitational ?eld is not dispersive, cl is also the group speed of light. So we shall simply call cl the speed of light with respect to a comoving observer without any further speci?cation. Using the general theory of geometric optics exposed in [27], we deduce from Eq. (31) that the ratio cl /c is determined by 1+A (η + ζ)(4U ? R) c2 l = =1+ , c2 1+A?B 3 ? (3ξ + η)R ? 2(η + ζ)U (34)

where A and B are de?ned by Eq. (22). This formula shows that the speed of light in a FRW background generically di?ers from c except in the case where η + ζ = 0. We can suppose that at present time 1 + A ≈ 1 and 1 + A ? B ≈ 1. So, we shall henceforth restrict our attention to the part of the history of the Universe such that the two conditions 1+A > 0, 1+A?B >0 8 (35)

are satis?ed. Indeed, it is clear that the violation of at least one of these conditions can only occur in a domain of spacetime where the curvature is so great that the present theory is probably no longer realistic. Inequalities (35) are suf?cient to ensure that the quantity cl determined by Eq. (34) is real. Moreover, they imply the following equivalence: cl > c ?? (η + ζ)(4U ? R) > 0 . (36)

Equation (34) shows that the vacuum acts as a medium moving with the unit 4-velocity u? and having a refractive index n given by n= c = cl 1? B . 1+A (37)

Using a well-known result of the geometric optics approximation [27], we can state that the light rays are null geodesics with respect to the associated metric tensor g de?ned by g ?ν = g?ν ? 1 ? 1 n2 u? uν = g?ν + B u? uν . 1+A?B (38)

However, taking into account Eqs. (28) and (37), it is easily seen that the ? conformal metric d?2 = n2 ds2 is a FRW metric with the scale factor a = n a. s So, we can enunciate the following theorem: Theorem 1.– In a FRW background, the light rays are null geodesics of the new FRW metric de?ned as d?2 = (dx0 )2 ? a2 (x0 ) s ? where a(x0 ) = ? with c a(x0 ) = cl 1? B a(x0 ) , 1+A k a2 ˙ + 2 a2 a , , (40) dr2 + r2 (dθ2 + sin2 θd?2 ) , 1 ? kr2 (39)

a ¨ A = 2(3ξ + η) + 2(3ξ + 2η + ζ) a

(41) (42)

k ˙ a a2 ¨ B = 2(η + ζ) ? + 2 + 2 a a a

a denoting the derivative da/dx0 . ˙ It follows from this theorem that all geometrical properties of light rays can be obtained by substituting a(x0 ) for a(x0 ) into the usual de?nitions and ? relations (luminosity distances, number counts, gravitational lensing,...). In particular, the red-shift of an extragalactic comoving object will be given by 1+z = a0 n 0 a0 (cl )e = , ae n e ae (cl )0 9 (43)

where the subscripts e and 0 stand for the time of emission and for the present time, respectively. As a consequence, a non-minimal coupling between the electromagnetic ?eld and gravity is able to a?ect our informations concerning the evolution of the Universe.

7

Speed of light and energy content of FRW models

In order to connect the speed of light with the energy content of the Universe, we now suppose that the metric satis?es Einstein equations. In the FRW models, the r.h.s. of Einstein equations may always be considered as the energymomentum tensor T?ν of a perfect ?uid having an energy density ? and a pressure p, ? and p only depending on the cosmic time. So, we have 1 R?ν = κ (? + p)u? uν ? (? ? p)g?ν ? Λg?ν . 2 As a consequence, R and U are given by R = ?κ(? ? 3p) ? 4Λ , U= 1 κ(? + 3p) ? Λ . 2 (45) (44)

Substituting for R and U from Eq. (45) into Eq. (34), we obtain the second theorem of this paper. Theorem 2.– Given a FRW model, let ?, p and cl respectively denote the total energy density, the pressure and the speed of light with respect to a comoving observer. Then – if η + ζ = 0 , cl = c ; (46) – if η + ζ = 0 , cl = c where σ is de?ned as σ= and ?m = 1+ ?+p , 1 ?m + 3 (σ ? 2)? ? σp 3ξ + 2η + ζ η+ζ (47)

(48)

1 2 Λ + (2σ ? 1) . (η + ζ)κ 3 κ

(49)

From Eq. (45), it is easily seen that Eq. (36) may be written as cl > c ?? (η + ζ)(? + p) > 0 . 10 (50)

As a consequence, the speed of light is greater than c for any reasonable equation of state if and only if the condition η + ζ > 0 is satis?ed. Neglecting the pressure, Eq. (47) yields cl ?2 ? + O( 2 ) . =1+ c 2?m ?m (51)

Thus, up to the ?rst order in ?/?m , the variation of the speed of light when p is negligible is entirely governed by the value of ?m . Using the values of ξ, η and ζ given by Eq. (5), it may be seen that η + ζ > 0. So, the speed of light obtained from the Drummond-Hathrell Lagrangian is greater than c as long as ? + p > 0. Neglecting the contribution of the cosmological constant in Eq. (49), we ?nd ?m /c2 = 2.5 × 1051 g.cm?3 . Then, taking ?0 /c2 ≈ 2.5 × 10?30 g.cm?3 and neglecting the pressure, we see that at the present time cl ? 1 ≈ 5 × 10?82 . (52) c 0 As a consequence, the di?erence between cl and c predicted in the one-loop approximation of QED cannot be detected by local experiments. Finally, let us note that the speed of light cl = c in a de Sitter spacetime whatever the parameters ξ, η, and ζ, since one has ? + p = 0 in this case.

8

Conclusion

In this paper, we have outlined the theory of light rays propagating in a FRW background according to the NMC between electromagnetism and gravity de?ned by Eqs. (1)–(2). We have obtained the general expression of the speed of light cl as a function of the energy density and of the pressure of the Universe. We have found that light propagates at a speed greater than c if and only if η + ζ > 0, provided that ? + p > 0. This conclusion generalizes a result previously obtained by Drummond and Hathrell in the framework of QED. An application of these results to the horizon problem in cosmology is in preparation.

Acknowledgments
We are deeply grateful to B. Linet for indicating to us the existence of the Horndeski’s theorem and making several useful remarks.

References
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[2] J. D. Barrow, Phys. Rev. D 59, 043515 (1999). [3] A. Albrecht and J. Magueijo, Phys. Rev. D 59, 043516 (1999). [4] J. Magueijo, Phys. Rev. D 62, 103521 (2000). [5] J. Magueijo, J. D. Barrow and H. B. Sandvik, Phys. Lett. B 549, 284 (2002). [6] J. W. Mo?at, Int. J. Mod. Phys. D 12, 281 (2003). [7] Some other approaches have been developed. See, e.g., Ll. Bel, in Relativity and gravitation in general, ed. by J. Mart? E. Ruiz, F. Atrio and A. ?n, Molina, World Scienti?c (1999); gr-qc/9809051; gr-qc/9905016. [8] H. F. M. Goenner, Found. Phys. 14, 865 (1984). [9] M. Novello and J. M. Salim, Phys. Rev. D 20, 377 (1979). [10] I. T. Drummond and S. J. Hathrell, Phys. Rev. D 22, 343 (1980). [11] M. Novello and S. D. Jorda, Mod. Phys. Lett. A 4, 1809 (1989). [12] R. Lafrance and R. C. Myers, Phys. Rev. D 51, 2584 (1995). [13] M. Novello, L. A. R. Oliveira and J. M. Salim, Class. Quantum Grav. 7, 51 (1990). [14] J.-M. L?vy-Leblond, Am. J. Phys. 44, 271 (1976); see also A. R. Lee and e T. M. Kalotas, Am. J. Phys. 43, 434 (1975). [15] A. Einstein, Ann. Phys. 17, 891 (1905); English translation in The Principle of Relativity (Dover, New York). [16] W. v. Ignatowski, Phys. Z. 11, 972 (1910); Phys. Z. 12, 779 (1911). [17] P. Frank and H. Rothe, Ann. Phys. (Leipzig) 34, 825 (1911); Phys. Z. 13, 750 (1912). [18] V. Berzi and V. Gorini, J. Math. Phys. 10, 1518 (1969). [19] G. W. Horndeski, J. Math. Phys. 17, 1980 (1976). [20] G. W. Horndeski and J. Wainwright, Phys. Rev. D 16, 1691 (1977). [21] H. A. Buchdahl, J. Phys. A: Math. Gene. 12, 1037 (1979). [22] F. M¨ ller-Hoissen and R. Sippel, Class. Quantum Grav. 5, 1473 (1988). u [23] R. D. Daniels and G. M. Shore, Nucl. Phys. B 425, 634 (1994).

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[24] There are further terms in the e?ective Lagrangian of QED but they may be neglected in the discussion of the speed of light. [25] F. W. Hehl and Y. N. Obukhov, in Proc. 220th Heraeus-Seminar, Lect. Notes Phys. 562, 479 (2001); preprint gr-qc/0001010 (2000). [26] A. B. Balakin, Class. Quantum Grav. 14, 2881 (1997). [27] J. L. Synge, Relativity: The General Theory, chap. 11 (North-Holland Cy, Amsterdam, 1964).

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