Fresnel analysis of wave propagation in nonlinear electrodynamics

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Fresnel analysis of wave propagation in nonlinear electrodynamics

Yuri N. Obukhov*

Instituto de Fı´sica Teo´rica, UNESP, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil Guillermo F. Rubilar

Institute for Theoretical Physics, University of Cologne, 50923 Ko¨ln, Germany

共Received 11 April 2002; published 31 July 2002兲

We study wave propagation in local nonlinear electrodynamical models. Particular attention is paid to the derivation and the analysis of the Fresnel equation for the wave covectors. For the class of local nonlinear Lagrangian nondispersive models, we demonstrate how the originally quartic Fresnel equation factorizes, yielding the generic birefringence effect. We show that the closure of the effective constitutive共or jump兲 tensor is necessary and sufficient for the absence of birefringence, i.e., for the existence of a unique light cone structure. As another application of the Fresnel approach, we analyze the light propagation in a moving isotropic nonlinear medium. The corresponding effective constitutive tensor contains nontrivial skewon and axion pieces. For nonmagnetic matter, we find that birefringence is induced by the nonlinearity, and derive the corresponding optical metrics.

DOI: 10.1103/PhysRevD.66.024042 PACS number共s兲: 04.20.Cv, 04.30.Nk, 11.10.Lm

I. INTRODUCTION

Wave phenomena belong to the most interesting and im-portant processes in physics. Among other field theories, nonlinear electrodynamics attracts a lot of attention in con-nection with the prominent role played by light in the experi-mental and theoretical studies of the structure of spacetime and matter.

Nonlinearities in electrodynamical models can arise in different ways in classical and quantum field theories. For example, the old Born-Infeld theory 关1兴 was a fundamental theory alternative to classical Maxwell electrodynamics which provided a model of a classical electron. On the other hand, quantum Maxwell electrodynamics predicts nonlinear effects which arise due to radiative corrections, see 关2–4兴. Finally, in modern string theories a generalized Born-Infeld action naturally arises as the leading part of the effective string action, see 关5–7兴, for example.

Wave propagation in the various nonlinear electrodynami-cal theories was studied previously in 关8–13兴 and also in 关14–19兴. A general feature revealed in these studies is the existence of birefringence. In crystal optics, the notion of birefringence means the emergence of two rays共ordinary and extraordinary兲 with different velocities inside the material medium. We will use the expression ‘‘birefringence’’ in a similar sense, associating it with the situation when two dif-ferent light cones exist for the wave normal covectors. How-ever, the earlier results are incomplete in the sense that the full Fresnel equation, governing the wave normals, was never derived explicitly. Moreover, it was not demonstrated how it happens that the original quartic surface of wave nor-mals reduces to the light cone. That is the primary interest in our study, and we will try to clarify this aspect for local nonlinear electrodynamical models, using and expanding our

earlier results obtained within the framework of linear elec-trodynamics关20,21兴.

We restrict our attention to local electrodynamical mod-els, thus assuming that, at each point, the corresponding 共ef-fective兲 constitutive tensor of the medium under consider-ation depends only on the field strength itself and not on its derivatives. In other words, dispersive effects are neglected. This approximation is valid in general when the wavelength of the propagating electromagnetic field is larger than the length scales related to the microscopic共sub兲structure of the medium. For instance, for the Heisenberg-Euler effective La-grangian, which describes nonlinearities due to radiative cor-rections in QED, this means that the wavelength of the pho-tons has to be much larger than the Compton length of the electron. For the isotropic media considered in Sec. XI, this means that the wavelength of the photons is assumed to be much larger than the length scales related to the resonance frequencies of the medium.

Our basic tool will be the general formula for the Fresnel equation derived earlier within linear electrodynamics 关20,21兴. Now we observe that the analysis of the wave propa-gation in a general local nonlinear model reduces to the lin-ear case because the jumps of the derivatives of the excita-tion and of the field strength are in all cases related by a linear law. We can then make use of our master formula for the Fresnel tensor and derive the Fresnel equation for any local nonlinear model, and thereby explain the reduction to the light cones.

II. ELECTROMAGNETIC WAVES AND FRESNEL TENSOR

Quite generally, Maxwell’s equations for the excitation 2-form H⫽(D,H) and the field strength 2-form F⫽(E,B) read

dH⫽J, dF⫽0. 共2.1兲

*On leave from Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia.

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Here J is the electric current 3-form. These equations must be supplemented by a constitutive law H⫽H(F). The latter relation contains the crucial information about the underlying physical continuum 共i.e., spacetime and/or material me-dium兲. Mathematically, this constitutive law arises either from a suitable phenomenological theory of a medium or from the electromagnetic field Lagrangian. It can be a non-linear or even nonlocal relation between the electromagnetic excitation and the field strength.

If local coordinates xi are given, with i, j , . . .⫽0,1,2,3, we can decompose the excitation and field strength 2-forms into their components according to

H⫽1 2Hi jdx

i_⵩dxj_, _F_⫽1 2Fi jdx

i_⵩dxj_. _共2.2兲

There are several approaches to the study of the wave propagation in nonlinear electrodynamics. For example, one can consider the geometric optics approximation scheme when a field strength is split into a sum of a background field plus a wave term of the form fi jei⌽ where the amplitude f varies slowly compared to the phase⌽. Under the condition that the scale over which the background electromagnetic field varies is much larger than that of variations of⌽, one then derives the algebraic system for the amplitude f which ultimately determines the characteristics of the wave solu-tions in terms of the background field. The advantage of this approach is its clear physical interpretation.

An alternative approach is represented by the Hadamard theory of weak discontinuities. It yields the same results, whereas it is more mathematically transparent, in our opin-ion. Correspondingly, here we will study the propagation of a discontinuity of the electromagnetic field using the Had-amard approach following the lines of Ref. 关20兴, see also Ref.关21兴. The surface of discontinuity S is defined locally by a function⌽ such that ⌽⫽const on S. Across S, the geomet-ric Hadamard conditions are satisfied:

关Fi j兴⫽0, 关⳵iFjk兴⫽qifjk, 共2.3兲 关Hi j兴⫽0, 关⳵iHjk兴⫽qihjk. 共2.4兲 Here关F兴(x) denotes the discontinuity of a function F across S, and q_iª⳵_i⌽ is the wave covector. Given the constitutive law H(F), which determines the excitation in terms of the field strength, the corresponding tensors fi jand hi j, describ-ing the jumps of the derivatives of field strength and excita-tion, are related by关22兴

hi j⫽ 1 2␬i j kl_f kl, with ␬i jklª ⳵Hi j ⳵F_kl. 共2.5兲 We will call␬i jklthe jump tensor. In linear electrodynamics, its components coincide with the components of the consti-tutive tensor 共which describes the linear law Hi j ⫽1

2␬i jklFkl), and they are independent of the electromag-netic field. However, in general the jump tensor ␬i jkl is a function of the electromagnetic field, the velocity of matter, the temperature, and other physical and geometrical vari-ables. Quite remarkably, all the earlier results obtained for

linear electrodynamics remain also valid in the general case because whatever local relation H(F) may exist, the relation between the jumps of the field derivatives, according to Eq. 共2.5兲, is always linear.

If we use Maxwell’s equations 共2.1兲, then Eqs. 共2.3兲 and 共2.4兲 yield

⑀i jkl_q

jhkl⫽0, ⑀i jklqjfkl⫽0. 共2.6兲 Here ⑀i jkl is the Levi-Civita tensor density of weight ⫹1, with ⑀0123⫽1. We will also use the Levi-Civita tensor den-sity of weight ⫺1, which we denote by ⑀ˆ_{i jkl}, with ⑀ˆ₀₁₂₃ ⫽1.

Let us introduce the analogue of the conventional consti-tutive matrix ␹i jkl_ª1 2⑀ i jmn_␬ mnkl⫽ ⳵Hi j ⳵Fkl , 共2.7兲 where we denote Hi j_ª1 2⑀ i jmn_H mn. Similarly to ␬i jkl, we will often call the tensor ␹i jkl the jump tensor density.

Now, making use of Eqs. 共2.5兲 and 共2.7兲, we rewrite the system共2.6兲 as

␹i jkl_q

jfkl⫽0, ⑀i jklqjfkl⫽0. 共2.8兲 Solving the last equation by fi j⫽qiaj⫺qjai, we finally re-duce the first part of Eq.共2.8兲 to

␹i jkl_q

jqkal⫽0. 共2.9兲 This algebraic system has a nontrivial solution for a_i only when the wave covectors satisfy a certain condition. The latter gives rise to our covariant Fresnel equation 关20,21兴

Gi jkl_共_␹_兲q

iqjqkql⫽0, 共2.10兲 with the fourth order Fresnel tensor densityG of weight ⫹1 defined by

Gi jkl_共_␹_兲ª1

4!⑀ˆmn pq⑀ˆrstu␹

mnr(i_␹j兩ps兩k_␹l)qtu_. _共2.11兲 It is totally symmetric, Gi jkl(␹)⫽G(i jkl)(␹), and thus has 35 independent components. A particular case of that construc-tion has been introduced by Tamm 关23兴.

III. LOCAL NONLINEAR ELECTRODYNAMICS

Let us denote the two independent electromagnetic invari-ants as

I1ªFi jFi j, I2ªFi jF˜i j, 共3.1兲 where F˜i jª12␩

i jkl_F

kl and␩i jklª(⫺g)⫺1/2⑀i jkl. The Hodge operator for the exterior forms is denoted by as asterisk (*), as usual; and we use a tilde to denote the dual 2-tensors.

We will study the class of nonlinear electrodynamics models which are described by the Lagrangian 4-form

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Here, as usual,␩ is the 4-form of the spacetime volume. Our analysis is generally covariant, i.e., we do not restrict our-selves to Cartesian coordinates for which gi j⫽diag(1,⫺1, ⫺1,⫺1). We instead consider gi j as an arbitrary Lorentzian spacetime metric, although we do not take into account pos-sible nonminimal coupling of the electromagnetic field to the spacetime curvature. In other words, we exclude the La-grangian models with the curvature terms arising from the quantum corrections on a curved spacetime which were stud-ied by Drummond and Hathrell关18兴, e.g. As a matter of fact, such terms can be straightforwardly included in our general approach using the general quartic Fresnel equation 共2.10兲, 共2.11兲 but this goes beyond the scope of the current paper. The corresponding generalization will be discussed else-where.

The electromagnetic excitation 2-form, which enters the Maxwell equation 共2.1兲, is derived as the derivative of the Lagrangian form, H⫽⫺⳵V/⳵F. Explicitly, we then have the nonlinear constitutive law

H⫽4共⫺L₁*F_⫹L₂F兲. 共3.3兲 We denote the partial derivatives of the Lagrangian function with respect to its arguments as

L_aª⳵L ⳵Ia , L_abª ⳵ 2_L ⳵Ia⳵Ib , a,b⫽1,2. 共3.4兲

In accordance with Eqs.共2.5兲 and 共2.7兲, the direct differen-tiation of Eq.共3.3兲 yields the jump tensor density

␹i jkl_⫽

冑

_⫺g关k

1gi[kgl] j⫹k2Fi jFkl⫹k3F˜i jFkl

⫹k4Fi jF˜kl⫹k5˜Fi jF˜kl⫹k6␩i jkl兴. 共3.5兲

The coefficients kA,A⫽1, . . . , 6, are functions of the elec-tromagnetic fields:

k1⫽4L1, k2⫽8L11, k3⫽k4⫽8L12,

k5⫽8L22, k6⫽2L2. 共3.6兲

The identifications 共3.6兲 are derived for the nonlinear La-grangian 共3.2兲 from the constitutive law 共3.3兲. However, in most computations below we will consider the most general case with unspecified arbitrary coefficients kA. This may be useful if we want to study nonlinear electrodynamics of a more general type, for instance, with dissipation effects and/or in moving media.

In general, the untwisted tensor density␹i jkl(x) of weight ⫹1 has 36 independent components. We can decompose it into irreducible pieces关21兴 with respect to the 6-dimensional 共‘‘bivector’’兲 linear group as follows:

␹i jkl_⫽(1)_␹i jkl_⫹(2)_␹i jkl_⫹(3)_␹i jkl_. _共3.7兲 The irreducible pieces of␹ are defined by

(2)_␹i jkl_:_⫽1 2共␹

i jkl_⫺_␹kli j_兲⫽⫺(2)_␹kli j_,

(3)_␹i jkl_ª_␹[i jkl]_, _共3.8兲 (1)_␹i jkl_:_⫽_␹i jkl_⫺(2)_␹i jkl_⫺(3)_␹i jkl_⫽(1)_␹kli j_. _共3.9兲 The irreducible pieces (1)␹, (2)␹, and (3)␹ have 20, 15, and 1 independent components, respectively. The possible pres-ence of an axion piece (3)␹ was first studied by Ni 关24兴, whereas a constitutive law with an isotropic skewon (2)␹ was discussed by Nieves and Pal关25兴.

In the Lagrangian models 共3.2兲, the effective constitutive tensor is automatically symmetric, i.e., (2)␹⫽0, which fol-lows from Eq. 共3.6兲, since k₃⫽k₄. However, in general the jump tensor density共3.5兲 has all three irreducible pieces:

Our analysis in the framework of the Hadamard formal-ism yields the Fresnel equation in the generally covariant form 共2.10兲 with the Fresnel tensor density 共2.11兲. For the explicit jump tensor density共3.5兲, it thus remains to substi-tute its components into Eq.共2.11兲. A straightforward calcu-lation yields the result:

Gi jkl_⫽⫺k1 8

冑

⫺g共Xg (i j_gkl)_⫹2Yg(i j_tkl)_⫹Zt(i j_tkl)_兲. 共4.1兲 Here we denote X⫽k1 2_⫹k1 2共k3⫹k4兲I2⫺k1k5I1⫹ 1 4共k3k4⫺k2k5兲I2 2 , 共4.2兲 Y⫽k1共k2⫹k5兲⫹共k3k4⫺k2k5兲I1, 共4.3兲 Z⫽4共k2k5⫺k3k4兲, 共4.4兲 and ti jªFikFjk. 共4.5兲

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The latter is closely related to the Minkowski energy-momentum tensor

Ti jª⫺ti j⫹1 4I1g

i j_. _共4.6兲

The most remarkable property of Eq. 共4.1兲 is that it is obviously factorizable into a product of two second order tensors. Correspondingly, the quartic Fresnel surface of the wave normals reduces to the product of two second order surfaces: Gi jkl_共_␹_兲q iqjqkql⫽ ⫺k1 8X 共g1 i j_q iqj兲共g2 kl_q kql兲 ⫽⫺k₈_{Z 共}1 ¯g1 i j qiqj兲共g¯2 kl qkql兲. 共4.7兲 In other words, the wave normals lie not on the quartic sur-face but on one of the two cones which are determined by the pair of optical metric tensors:

g₁i j:⫽Xgi j⫹共Y⫹

冑

Y2⫺XZ兲ti j, 共4.8兲 g2

i j

Let us discuss the results obtained in the previous section. The following general observations are in order.

The Fresnel equation is trivially satisfied for all wave covectors when k1⫽0, see Eq. 共4.1兲. Thus, in order to have

waves, every electrodynamical Lagrangian L should neces-sarily depend on the invariant I₁⫽F_{i j}Fi j _{共thus providing} k1⫽0). Accordingly, we will always assume that k1⫽0.

In order to have a decent light propagation, the optical metrics should be real and with Lorentzian signature. How can one be a priori sure that for every L an optical metric necessarily has these properties?

Using Eqs.共4.2兲–共4.4兲 we find an explicit expression for the quantity under the square root in the above formulas:

Y2_⫺XZ⫽N 1 2_⫹N

2N3, 共5.1兲

where we have denoted

N₁:⫽k₁共k₂⫺k₅兲⫹共k₃k₄⫺k₂k₅兲I₁, 共5.2兲 N2:⫽2k1k3⫹共k3k4⫺k2k5兲I2, 共5.3兲

N3:⫽2k1k4⫹共k3k4⫺k2k5兲I2. 共5.4兲

The expression共5.1兲 is always non-negative in every nonlin-ear theory共3.2兲 because N2⫽N3 when we take into account

that k3⫽k4, see Eq. 共3.6兲.

The signature of a four-dimensional metric is Lorentzian if and only if its determinant is negative. Straightforward computation yields, for the optical metrics共4.8兲, 共4.9兲;

共detga i j_{兲⫽共detg}i j_兲

冋

_␣2_⫹␣␤a 2 I1⫺ ␤a 2 16I2 2

册

2 , a⫽1,2. 共5.5兲

Here ␣⫽X and ␤1⫽Y⫹

冑

Y2⫺XZ, ␤2⫽Y⫺

冑

Y2⫺XZ. As

we see, both optical metrics have Lorentzian signature as soon as the spacetime metric gi j is Lorentzian.

Summarizing, we have demonstrated that Eqs. 共4.8兲 and 共4.9兲 indeed describe the generic effect of a birefringent light propagation for all local nonlinear Lagrangians 共3.2兲.

Recently, the emergence of the two ‘‘effective geom-etries’’ has been described in 关14,15兴 without using the Fresnel approach. These results agree with those obtained in our analysis, since one can show that our optical metrics are conformally equivalent to the effective metrics of 关14,15兴. Moreover, one can recast the optical metrics 共4.8兲–共4.11兲 into the form of the so-called Boillat metrics of关16兴.

VI. SPECIAL LAGRANGIANS

It is worthwhile to discuss certain particular nonlinear models which are potentially of physical interest.

A. Lagrangian LÄL„I2…

When the Lagrangian depends only on the second electro-magnetic invariant, we have k1⫽0, and there are no waves

in such models.

B. Lagrangian LÄL„I1…

For the Lagrangian which, on the contrary, depends on the first invariant only, we find k3⫽k4⫽k5⫽0. Accordingly,

from Eqs. 共4.2兲–共4.4兲 we find X⫽k₁2,Y⫽k1k2, and Z⫽0,

and thus Eqs.共4.8兲 and 共4.9兲 yield g₁i j⫽k1共k1gi j⫹2k2ti j兲, g2

i j_⫽k

1 2

gi j. 共6.1兲 Correspondingly, we still have birefringence with some pho-tons moving along the standard null rays of the spacetime metric gi j, whereas other photons choosing the rays null with respect to the optical metric L1gi j⫹4L11ti j, cf.关14兴.

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C. Lagrangian LÄU„I1…¿␣I2

This is a simple generalization of the previous case. Here

␣ does not depend on the electromagnetic field, although it is not a constant, in general. When it depends on the spacetime coordinates, ␣⫽␣(x), one can identify it with the axion field, cf.关24兴.

Here we again have k₃⫽k₄⫽k₅⫽0 and we recover the same light cone structure 共6.1兲. To put it differently, the ax-ion field does not disturb the light cones which are solely determined by the spacetime metric and by the dependence of the Lagrangian on the invariant I₁.

D. Lagrangian LÄaI1¿V„I2…

Then k1⫽4a, k2⫽k3⫽k4⫽0, and k5⫽8V

⬙

, which yields

X⫽16a2_⫺32V

_⬙

_I

1, Y⫽32aV

⬙

, and Z⫽0. Consequently,

g₁i j⫽16a关共a⫺2V

⬙

I₁兲gi j_⫹4V

_⬙

_ti j_兴,

g₂i j⫽16a共a⫺2V

⬙

I₁兲gi j, 共6.2兲 i.e., there is again birefringence with one cone determined by the standard spacetime metric.

E. Heisenberg-Euler Lagrangian

In the one-loop approximation, polarization of vacuum in quantum electrodynamics is described by the Heisenberg-Euler effective Lagrangian关2–4兴. The critical magnetic field Bk⫽(m2c2/eប) ⬇4.4⫻109 T 共for the electron mass m and charge e) essentially determines the physical structure of the nonlinear model. For the weak fields which satisfy I1

ⰆBk

2_,I 2ⰆBk

2 _{the Heisenberg-Euler Lagrangian reads}

L⫽⫺1 4I1⫹ ␣ 90B_k2

冋

I1 2_⫹7 4I2 2_⫺ I1 B_k2

冉

2 7I1 2_⫹13 28I2 2

冊

册

. 共6.3兲

Here␣⫽ (e2/4␲បc) is the fine structure constant. From this Lagrangian we find, keeping only the leading order terms,

. 共6.6兲 As a result, we obtain the optical metrics

g₁i j⫽

冉

1⫹6␣I1 45B_k2

冊

g i j_⫺ 16␣ 45B_k2t i j_, _共6.7兲 g₂i j⫽

冉

1⫹6␣I1 45B_k2

冊

g i j_⫺ 28␣ 45B_k2t i j_. _共6.8兲

Thus we have birefringence and both light cones differ from the one determined by the spacetime metric, cf. also关14兴.

1⫹ 1 2b2I1⫺ 1 16b4I2 2

冊

2. 共6.12兲

As a result, we find Y2⫺XZ⫽0, and thus the two optical metrics 共4.8兲 and 共4.9兲 coincide,

g₁i j⫽g₂i j⫽

冉

1⫹ 1 2b2I1

冊

冉

1⫹ 1 2b2I1⫺ 1 16b4I2 2

冊

2 ⫻

冋冉

1⫹ 1 2b2I1

冊

g i j_⫺ 1 b2t i j

册

_. _共6.13兲 Birefringence disappears, and the photons propagate along a single light cone determined by the ‘‘quasi-metric’’ of Ple-banski 关11兴

冉

1⫹ 1 2b2I1

冊

g i j_⫺1 b2t i j_. _共6.14兲

VII. VELOCITY OF LIGHT

The knowledge of the optical metric is sufficient for de-termining the velocity of light. The phase velocityv of wave propagation is defined by the wave covector components by means of the relation

qa⫽ q0

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where ka are the components of the unit 3-covector. Al-though we confine our attention to the phase velocity, it is straightforward to show that for the models under consider-ation the group velocity is always greater or equal to the phase velocity. Substituting Eq. 共7.1兲 into the light cone equations g₁i jqiqj⫽0 and g2

i j

qiqj⫽0, one can find the veloci-ties of light explicitly. Usually, of interest is the mean veloc-ity value which is obtained after averaging over the direc-tions of propagation and polarizadirec-tions. A direct computation using the general form of the optical metrics 共4.8兲 and 共4.9兲 yields, in Cartesian coordinates,

具

v2

典

⫽

冉

1⫹4 3T 00 Y⫹Zt 00 X⫹2Yt00_⫹Z共t00_兲2

冊

⫹2 3␦abt 0a_t0b2Y 2_{⫺XZ⫹Z共t}00_兲2_⫹2YZt00 关X⫹2Yt00_⫹Z共t00_兲2_兴2 . 共7.2兲 In accordance with Eq. 共4.6兲, T00⫽⫺t00⫹I1/4⫽(B2

⫹E2_{)/2 is the energy density of the Maxwell field, whereas}

␦abt0at0b⫽S2 where S⫽E⫻B is the energy flux density.

A. Velocity shift in a polarized vacuum

As an application of the general formula共7.2兲, let us ob-tain the velocity of light in a polarized vacuum. For the weak field limit of the Heisenberg-Euler electrodynamics, substi-tuting Eqs.共6.4兲–共6.6兲 into Eq. 共7.2兲, we find, in the leading order,

具

_典

_{is 1/3 for E}2

⫽b2 _{and B zero or}_{共anti-兲parallel to E.}

VIII. NO BIREFRINGENCE CONDITION

In this section, we will restrict our attention to the La-grangian theories for which the constitutive tensor is Eq. 共3.5兲, and the coefficients are derived as Eq. 共3.6兲. It is im-portant that the Fresnel analysis reveals that k₁⫽0, other-wise there is no decent wave propagation at all.

As it is clear from Eq.共5.1兲, the necessary and sufficient condition of the absence of birefringence is provided by the pair of equations:

N1⫽k1共k2⫺k5兲⫹共k3k4⫺k2k5兲I1⫽0, 共8.1兲

N2⫽N3⫽2k1k3⫹共k3k4⫺k2k5兲I2⫽0. 共8.2兲

Here the property k₃⫽k₄ of the Lagrangian models is used. Taking into account that all k’s are the partial derivatives of the Lagrangian L with respect to I₁and/or I₂ as displayed in Eq. 共3.6兲, we can view the above system as a pair of partial differential equations, the solution L⫽L(I1,I2) of

which describes a model without birefringence 共i.e., with a single light cone兲. At least two such particular solutions are already known: one is rather simple, namely, the standard Maxwell theory with L⫽⫺I1/4. Another is more nontrivial,

this is the Born-Infeld theory with the Lagrangian共6.9兲. One may ask the question: Are these the only solutions of the system共8.1兲, 共8.2兲? The immediate inspection of the system 共8.1兲, 共8.2兲 shows that the answer is negative. For example, the Lagrangian function L(I₁,I₂)⫽aI₁/I₂, with constant a, satisfies the equations 共8.1兲, 共8.2兲. Such a nonlinear 共and nonpolynomial兲 model thus also has no birefringence. It is an open problem to find the complete set of solutions of Eqs. 共8.1兲, 共8.2兲, leading then to a single light cone.

IX. CLOSURE CONDITION

⫺ 2k4 3k1 I1

冊

. 共9.7兲

The third lines in Eqs.共9.4兲–共9.7兲 give the a’s in terms of the combinations 共8.1兲–共8.2兲. Certainly, we use the assumption that k1⫽0.

Like the constitutive tensor of linear electrodynamics, the jump tensor␬i jklª

1

2⑀ˆi jmn ␹mnkl determines a linear map in the six-dimensional space of 2-forms. When the action of this map, repeated twice, brings us 共up to a factor兲 back to the identity map, we speak of the closure property of ␬_{i j}kl_. The importance of the closure property is related to the fact that ultimately␬i jkl turns out to be a duality operator which determines a unique conformal Lorentzian metric on the spacetime.

In local nonlinear electrodynamics, the jump tensor den-sity共3.5兲 has the closure property when

a₀⫽0, a₁⫽0, a₂⫽0, a₃⫽0, a₄⫽0, 共9.8兲 as is evident from Eq.共9.2兲.

X. EQUIVALENCE OF CLOSURE AND NO BIREFRINGENCE CONDITIONS

In linear electrodynamics, there is much evidence 共al-though the final rigorous proof is still missing兲 that the quar-tic Fresnel surface of wave covectors reduces to a unique light cone if and only if the constitutive tensor has the clo-sure property.

For the local nonlinear Lagrangian theories共3.2兲 it is pos-sible to make some progress in solving the equivalence prob-lem. In a certain sense, the situation here is simpler because we have discovered that the quartic Fresnel surface is always reduced to the product of the light cones共birefringence兲. The next step is thus to study under which conditions the bire-fringence disappears and, correspondingly, a unique light cone arises.

Theorem. For the nonlinear electrodynamical models de-scribed by the Lagrangian共3.2兲, the Fresnel equation implies a single light cone共no birefringence兲 if and only if the trace-less part of the jump tensor density satisfies the closure prop-erty.

Proof. As a preliminary remark, we note that for the La-grangian models共3.2兲 the jump tensor density 共3.5兲 is sym-metric because k3⫽k4. As a result, N2⫽N3.

The necessary condition is evident. The birefringence is absent when N1⫽N2(⫽N3)⫽0, see Eq. 共8.2兲. Then we

im-mediately read from Eqs. 共9.3兲–共9.7兲 that a₀⫽a₁⫽a₂⫽a₃ ⫽a4⫽0, and thus the closure is recovered from Eq. 共9.2兲.

The sufficient condition is also proved straightforwardly. The closure condition共9.8兲 has the unique solution

N1⫽0,

N2⫹N3

2 ⫽0, 共10.1兲

when we analyze Eqs. 共9.3兲–共9.7兲. Since for the Lagrangian models 共3.2兲 we have N2⫽N3, then Eq. 共10.1兲 yields N1

⫽N2⫽N3⫽0. Thus, there is no birefringence.

To put it differently, we have proven that the closure of the traceless jump tensor density is the necessary and suffi-cient condition for the reduction of the fourth order Fresnel wave surface to a single light cone. This is true for all non-linear Lagrangian electrodynamical theories共3.2兲. Returning to our studies of the general linear electrodynamics, we ex-pect that a similar result holds true there.

On asymmetric jump tensors

Symmetry of the jump tensor density is very important in the equivalence proof above. In order to clarify this point, let us consider an arbitrary jump tensor density 共3.5兲 without assuming the explicit form of the coefficients 共3.6兲. When k3⫽k4, the jump tensor density has the nontrivial skewon

part 共3.11兲. Also, N2⫽N3 关in fact, as we can see from Eqs.

共5.3兲 and 共5.4兲, N2⫺N3⫽2k1(k3⫺k4)兴.

We can easily verify that the closure of an asymmetric operator is not equivalent to the no-birefringence property. Indeed, take, for instance, k1⫽0, k2⫽k3⫽0, k4⫽0, and

k₅⫽0. Then the jump tensor density is asymmetric and N₁ ⫽N2⫽0, but N3⫽0. We then obtain a unique light cone

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condition 共9.8兲 is not satisfied, since Eqs. 共9.3兲–共9.7兲 are nontrivial for N1⫽N2⫽0 and N3⫽0. This also means that

the requirement of a unique light cone does not necessarily imply that ␹ must be symmetric.

The opposite is also true: Suppose an asymmetric jump tensor density共3.5兲 has the closure property, i.e., Eq. 共9.8兲 is fulfilled. Then we find Eq.共10.1兲 again. However, the second part of Eq. 共10.1兲 yields N2⫽⫺N3, and consequently Eq.

共5.1兲 together with Eqs. 共4.8兲 and 共4.9兲 describe the case of a birefringent and dissipative wave propagation.

These examples show that the closure of an asymmetric jump共or constitutive兲 tensor density does not guarantee the absence of birefringence, and, vice versa, no-birefringence is not accompanied by the closure property for an asymmetric operator.

XI. MOVING ISOTROPIC NONLINEAR MEDIA

Recently, there has been some interest in the light propa-gating in moving media with nontrivial dielectric and mag-netic properties. The first covariant analysis of the Fresnel equation for this case was done by Kremer关26兴. An isotropic medium is characterized by the constitutive law

Hi j_⫽

冑

_⫺g

冋

1

␮Fi j⫹2

冉

1

␮⫺␧

冊

ukFk[iuj]

册

, 共11.1兲 where ui is the 4-velocity of the moving matter共normalized as usual by u_iui_{⫽1), and ␧ and}_␮ _{are the permeability and} permittivity functions of the isotropic medium. The case when they do not depend on the electromagnetic field strength共being constant in space and time, for example兲 was investigated in关26兴.

More recently, the nonlinear case when ␧⫽␧(F) and ␮ ⫽␮(F) are functions of the electromagnetic field has been studied by De Lorenci et al.关15兴. However, the attention was restricted to certain special cases, and the general result is still missing.

We can perform a fairly complete analysis of the wave propagation in a nonlinear moving media on the basis of our covariant Fresnel equation共2.10兲. By differentiation, we eas-ily find the jump tensor density共2.7兲:

␹i jkl_⫽

冑

_⫺g

冋

2 ␮gi[kgl] j⫹4

冉

1 ␮⫺␧

冊

u[kgl][iuj] ⫹mkl_Fi j_⫹2共mkl_⫺ekl_兲u mFm[iuj]

册

. 共11.2兲 The second line is absent in the linear theory, and the tensors

mi j_ª⳵共1/␮兲

⳵Fi j

, ei j_ª ⳵␧

⳵Fi j

, 共11.3兲

are responsible for the nonlinear electrodynamical effects. Inspection immediately reveals that the jump tensor den-sity共11.2兲 contains both an axion and a skewon part. There are claims in the literature that axion and skewon, in general, do not have physical sense. However, here we encounter a

simple and a physically sound example with both pieces be-ing nontrivial. The irreducible pieces 共3.8兲, 共3.9兲 read:

(1)_␹i jkl_⫽

冑

_⫺g

冋

2 ␮gi[kgl] j⫹4

冉

1 ␮⫺␧

冊

u[kgl][iuj] ⫹1 2共m i j_Fkl_⫹mkl_Fi j_兲⫹共mkl_⫺ekl_兲P[i_uj] ⫹共mi j_⫺ei j_兲P[k_ul] ⫹ 1 12␩ i jkl_关Fmn_m_˜ mn⫹2共m˜mn⫺e˜mn兲P m_un_兴

册

_, 共11.4兲 (2)_␹i jkl_⫽

冑

_⫺g

冋

1 2共F i j_mkl_⫺Fkl_mi j_兲 ⫹共mkl_⫺ekl_兲P[i_uj]_⫺共mi j_⫺ei j_兲P[k_ul]

册

_, _共11.5兲 and for the axion piece, we have

⑀ˆ

i jkl␹i jkl⫽

冑

⫺g⑀ˆi jkl关Fi jmkl⫹2共mkl⫺ei j兲Piuj兴. 共11.6兲 Here we have denoted Pj_ªu

iFi j. Thus, nonlinear isotropic matter does have axion and skewon induced by nonlinearity.

Nonmagnetic matter

Let us consider the case when the magnetic constant is independent of the electromagnetic field, that is mi j⫽0. 关In the simplest case, we can restrict the attention to the purely dielectric medium with ␮⫽1. However, we will formally keep ␮⫽1, for the sake of generality.兴

It is convenient to make the evident split of Eq.共11.2兲 into the sum ␹i jkl⫽␾i jkl⫹␺i jkl, with

␾i jkl_⫽

冑

_⫺g

冋

2 ␮gi[kgl] j⫹4

冉

1 ␮⫺␧

冊

u[kgl][iuj]

册

冑

␧

␮

冑

兩g°兩g°i[kg°l] j. 共11.12兲

Now, for␹⫽␾⫹␺, using a compact notation by omitting the indices, we have

G共␹兲⫽G共␾兲⫹G共␺兲⫹_4!1 共O1⫹O2⫹O3⫹T1⫹T2⫹T3兲.

共11.13兲 Here the mixed terms O_a contain one␺ factor and the T_a’s two␺ factors. Postponing the symmetrization over i, j,k,l to the very last moment, these terms read explicitly as follows: O₁i jkl共␾,␺,␾兲⫽⑀ˆmn pq⑀ˆrstu␾mnri␺j psk␾lqtu, 共11.14兲 O₂i jkl共␺,␾,␾兲⫽⑀ˆmn pq⑀ˆrstu␺mnri␾j psk␾lqtu, 共11.15兲 O₃i jkl共␾,␾,␺兲⫽⑀ˆmn pq⑀ˆrstu␾mnri␾j psk␺lqtu, 共11.16兲

T1

i jkl_共_␺

,␾,␺兲⫽⑀ˆmn pq⑀ˆrstu␺mnri␾j psk␺lqtu, 共11.17兲 T₂i jkl共␺,␺,␾兲⫽⑀ˆmn pq⑀ˆrstu␺mnri␺j psk␾lqtu, 共11.18兲 T₃i jkl共␾,␺,␺兲⫽⑀ˆmn pq⑀ˆrstu␾mnri␺j psk␺lqtu. 共11.19兲 An important observation is that all T’s are vanishing for Eq. 共11.8兲. Indeed, since

␺i jkl_⫽⫺2

冑

_⫺gP[i_uj]_ekl_, _共11.20兲 we straightforwardly find, for example,

T₁i jkl共␺,␹,␺兲⫽⫺4g⑀ˆ_{mn pq}⑀ˆ_rstuP[m_un]_eri_P[l_uq]_etu_␹j psk ⫽⫺2g⑀ˆ_{mn pq}_⑀ˆ_rstu_␹j psk_Pm_un

⫻共Pl_uq_⫺Pq_ul_兲eri_esk

⫽0. 共11.21兲

This is zero because either the symmetric unuq or symmetric PmPq is contracted with the antisymmetric⑀ˆmn pq. Note that we on purpose write ␹j psk as the second argument, because its form is arbitrary, not necessarily equal to Eq. 共11.7兲. Analogously, we find:

T₂i jkl共␺,␺,␹兲⫽⫺4g⑀ˆ_{mn pq}⑀ˆ_rstuP[mun]eriP[ jup]esk␹lqtu ⫽⫺2g⑀ˆ_{mn pq}_⑀ˆ_rstu_␹lqtu_Pm_un

⫻共Pj_up_⫺Pp_uj_兲eri_esk

⫽0. 共11.22兲

It is a little bit more nontrivial to prove that T3also vanishes.

We have, explicitly: T3 i jkl_共_␹ ,␺,␺兲⫽⫺4g⑀ˆmn pq⑀ˆrstuP[ jup]eskP[luq]etu␹mnri ⫽⫺g⑀ˆ_{mn pq}_⑀ˆ_rstu_␹mnri_共Pj_up_Pl_uq ⫺Pp_uj_Pl_uq_⫺Pj_up_Pq_ul_⫹Pp_uj_Pq_ul_兲esk_etu ⫽0. 共11.23兲

The first and the last terms in the parentheses contain the symmetric combinations upuqand PpPqwhich are vanishing when contracted with the antisymmetric⑀ˆmn pq. The two re-maining terms in the parentheses are reduced, by means of a relabeling of indices, to⫺Ppuq( Pluj⫺Pjul). Recalling that at the end we impose the symmetrization over the free indi-ces (i, j ,k,l), we thus prove that T₃(␹,␺,␺)⫽0.

Since in all the three formulas 共11.21兲–共11.23兲, the argu-ment␹j pskis completely arbitrary, we can put␹j psk⫽␺j psk, in particular. Then Eqs.共11.21兲–共11.23兲 yield that G(␺)⫽0. As the next choice, we put ␹j psk⫽␾j psk. Then Eq. 共11.13兲 combined with Eqs.共11.21兲–共11.23兲, yields

G共␹兲⫽G共␾兲⫹_4!1 共O1⫹O2⫹O3兲. 共11.24兲

It thus remains to compute the O terms. The corresponding calculation is straightforward and simple if we use the rep-resentation共11.12兲. Then we find

O₁i jkl共␾,␺,␾兲⫽O₂i jkl共␺,␾,␾兲 ⫽O3

i jkl_共_␾ ,␾,␺兲

⫽8_␮␧g°(i j␺k兩mn兩l)g°_mn. 共11.25兲 Note that this result is valid for all possible tensors ␺, not only Eq. 共11.8兲 which means that we can further use Eq. 共11.25兲 for the future calculations involving more general nonlinear pieces共in particular, for the case with a nontrivial mi j). Using then Eq.共11.25兲 in Eq. 共11.24兲, we get

Gi jkl_共_␾_⫹_␺_兲⫽⫺ ␧ ␮2

冑

⫺gg° (i j_g 1 kl)_. _共11.26兲 Here we denoted g 1 i j_ªg°i j_⫺ ␮

冑

⫺g␺ (i兩mn兩 j)_g° mn. 共11.27兲 Hence, a purely dielectric nonlinear moving medium will in general exhibit the birefringence effect: the light will propagate in such a medium along the cone of the original optical metric g°i j 共one may call it ordinary ray兲, and along the second cone determined by the metric g1i j 共‘‘extraordi-nary’’ ray兲.

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The explicit form of the extraordinary optical metric is obtained when we substitute Eq.共11.20兲 into Eq. 共11.27兲:

g 1 i j_⫽g°i j_⫺_␮_P kek(iuj)⫹ 1 ␧ukek(iPj), 共11.28兲 ⫽g°i j_⫹_␮_ek(i_uj)_F klul⫺ 1 ␧ukek(iFj)lul. 共11.29兲 These results generalize the special cases considered in关15兴. As an example, let us consider the case of double refrac-tion caused by an electric field. We consider the medium in its rest frame and use adapted coordinates such that ui⫽␦₀i ⫽(1,0ជ). We assume that the dielectric permittivity is given by

␧⫽␧¯⫹aE2_. _共11.30兲

Here␧¯ and a are constant parameters. The components of the electric 3-vector E are defined as usual, Ea⫽⫺F0a. Then

we find

e0a_⫽⫺ea0_⫽⫺2aEa_, _and _Pj_⫽u

iFi j⫽共0,Eb兲. 共11.31兲 The spatial indices are lowered and raised with the help of the 3-metric ␦ab 共we neglect gravity兲. As a result, from Eq. 共11.31兲 we obtain

Pkeki⫽共⫺2aE2,0兲, ukeki⫽共0,⫺2aEb兲. 共11.32兲 Finally, using all this in Eq.共11.28兲 we find the components of the extraordinary optical metric:

g00⫽n2

冉

1⫹2a

␧ E2

冊

, gab⫽⫺␦ab⫺ 2a

␧ EaEb. 共11.33兲 Here nª

冑

␧␮ is the refraction index of the medium. In this way we obtain a natural description of the optical Kerr effect 共see 关28兴, for example兲 when birefringence is induced by the applied electric field.

XII. DISCUSSION AND CONCLUSION

In this paper, we have performed a systematic analysis of the light propagation for two classes of local nonlinear elec-trodynamics models on the basis of the Fresnel approach. We studied 共a兲 general local nonlinear Lagrangian theories, and 共b兲 moving isotropic nonlinear matter. In the former case, the Lagrangian 共3.2兲 is an arbitrary function of the two electro-magnetic invariants, whereas in the latter case, the perme-ability and permittivity functions of the medium 共11.1兲 de-pend arbitrarily on the electromagnetic field.

The study of the first class of models reveals the generic nature of the birefringence effect: The quartic Fresnel surface reduces to the two light cones for all nonlinear Lagrangians 共3.2兲. We show that the resulting optical metrics are always

real and have the correct Lorentzian signature. In this way, we confirm and extend the recent results of关14,15兴. Further-more, we are able to demonstrate the validity of the so-called closure–no birefringence conjecture in the context of nonlin-ear electrodynamics: Birefringence is absent 共and thus the quartic Fresnel surface reduces to a unique light cone兲 if and only if the effective constitutive共or jump兲 tensor satisfies the closure property.

The nonlinear isotropic moving matter with the constitu-tive law 共11.1兲 gives a sound example of a model in which the effective constitutive tensor naturally has nontrivial axion and skewon contributions. Accordingly, one should then ex-pect that the Fresnel surface remains quartic, in general关21兴. However, for nonmagnetic material media, we show that bi-refringence is again the generic effect. The Fresnel surface factorizes into two light cones, one of which corresponds to the Gordon optical metric 共independent of nonlinearities兲, whereas the other关Eq. 共11.28兲兴 manifests the nonlinear prop-erties of the model. The optical Kerr effect represents a par-ticular example of our general derivations. It is worthwhile to mention other natural applications of our formalism and, in particular, of the general Fresnel equations 共2.10兲 and 共2.11兲: nonlinear electrodynamics for the case of finite tem-perature关19兴 or Casimir 关29兴 quantum vacuum. We will dis-cuss these cases elsewhere.

As a final remark, we feel it is necessary to comment on the recent paper关30兴 which also discusses wave propagation in local Lagrangian nonlinear electrodynamics. Our results basically agree within the more narrow class of models con-sidered in 关30兴. However, in our opinion, the concept of bi-refringence is misused in 关30兴 in the sense that it is applied there also to the case when the Fresnel quartic surface does not factorize into the two light cones. We find this unfortu-nate because it contradicts the standard terminology of the classical crystal optics. Namely, as it is well known共see 关28兴, for example兲, an electromagnetic wave falling on a surface of a uniaxial crystal is refracted, giving rise to the two waves, ordinary and extraordinary. Correspondingly, such an effect is called birefringence. The mathematical explanation of this phenomenon is based precisely on the fact that the quartic Fresnel surface decomposes into a product of the two light cones for uniaxial crystal. In contrast, for a biaxial crystal the Fresnel surface does not factorize. As a result, an electro-magnetic wave refracts on the surface of a biaxial crystal in a more complicated way giving rise to what is known as the conical refraction which is markedly distinct from the bire-fringence in a uniaxial crystal. In conformity with this stan-dard terminology, we thus prefer to speak of birefringence, in the broader context of the general local nonlinear models, only when the Fresnel wave surface factorizes into the prod-uct of two light cones. This makes the introdprod-uction of a notion of ‘‘bimetricity’’关30兴 redundant and unnecessary.

ACKNOWLEDGMENTS

The work of Y.N.O. was partially supported by FAPESP, and G.F.R. would like to thank the German Academic Ex-change Service 共DAAD兲 for financial support.

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关1兴 M. Born and L. Infeld, Proc. R. Soc. London A144, 425 共1934兲.

关2兴 W. Heisenberg and H. Euler, Z. Phys. 98, 714 共1936兲共in

Ger-man兲.

关3兴 C. Itzykson and J.-B. Zuber, Quantum Field Theory

共McGraw-Hill, New York, 1985兲.

关4兴 J.S. Heyl and L. Hernquist, J. Phys. A 30, 6485 共1997兲. 关5兴 A.A. Tseytlin, in The Many Faces of the Superworld, Yuri

Golfand Memorial Volume, edited by M. Shifman共World Sci-entific, Singapore, 2000兲, pp. 417–452.

关6兴 G.W. Gibbons and D.A. Rasheed, Nucl. Phys. B454, 185 共1995兲.

关7兴 R. Kerner, A.L. Barbosa, and D.V. Gal’tsov, in 37th Karpacz

Winter School Of Theoretical Physics: New Developments In Fundamental Interactions Theories 共Karpacz, Poland, 2001兲, pp. 377–389.

关8兴 D.I. Blokhintsev and V.V. Orlov, Zh. Eksp. Teor. Fiz. 25, 513 共1953兲.

关9兴 M. Lutzky and J.S. Toll, Phys. Rev. 113, 1649 共1959兲. 关10兴 G. Boillat, Ann. Inst. Henri Poincare, Sect. A 5, 217 共1966兲; G.

Boillat, J. Math. Phys. 11, 941共1970兲.

关11兴 J. Pleban´ski, Lectures on Non-Linear Electrodynamics

共Nor-dita, Copenhagen, 1970兲.

关12兴 S.A. Gutie´rrez, A.L. Dudley, and J.F. Plebanski, J. Math. Phys. 22, 2835 共1981兲; H.S. Ibarguen, A. Garcia, and J. Plebanski,

ibid. 30, 2689共1989兲.

关13兴 S.L. Adler, Ann. Phys. 共N.Y.兲 67, 599 共1971兲; Z.

Bialynicka-Birula and I. Bialynicki-Bialynicka-Birula, Phys. Rev. D 2, 2341共1970兲; W.-Y. Tsai and T. Erber, ibid. 12, 1132共1975兲.

关14兴 M. Novello, V.A. De Lorenci, J.M. Salim, and R. Klippert,

Phys. Rev. D 61, 045001共2000兲; V.A. De Lorenci, R. Klippert, M. Novello, and J.M. Salim, Phys. Lett. B 482, 134共2000兲.

关15兴 V.A. De Lorenci, and M.A. Souza, Phys. Lett. B 512, 417 共2001兲; V.A. De Lorenci and R. Klippert, Phys. Rev. D 65,

064027共2002兲.

关16兴 G.W. Gibbons and C.A.R. Herdeiro, Phys. Rev. D 63, 064006 共2001兲.

关17兴 J.P.S. Lemos and R. Kerner, Gravitation Cosmol. 6, 49 共2000兲. 关18兴 I.T. Drummond and S.J. Hathrell, Phys. Rev. D 22, 343 共1980兲;

J.I. Latorre, P. Pascual, and R. Tarrach, Nucl. Phys. B437, 60

共1995兲; G.M. Shore, ibid. B460, 379 共1996兲.

关19兴 W. Dittrich and H. Gies, Phys. Rev. D 58, 025004 共1998兲; H.

Gies, ibid. 60, 105033共1999兲.

关20兴 Yu.N. Obukhov, T. Fukui, and G.F. Rubilar, Phys. Rev. D 62,

044050共2000兲.

关21兴 G.F. Rubilar, Yu.N. Obukhov, and F.W. Hehl, Int. J. Mod.

Phys. D共to be published兲; G.F. Rubilar, Ph.D. thesis, Univer-sity of Cologne, 2002.

关22兴 F.W. Hehl, Yu.N. Obukhov, and G.F. Rubilar, Int. J. Mod.

Phys. A共to be published兲.

关23兴 I.E. Tamm, J. Russ. Phys.-Chem. Soc. 57, 209 共1925兲共in

Rus-sian兲; 关Reprinted in I.E. Tamm, Collected Papers 共Nauka,

Moscow, 1975兲, Vol. 1, pp. 33–61 共in Russian兲兴.

关24兴 W.-T. Ni, ‘‘A non-metric theory of gravity,’’ Dept. Physics,

Montana State University, Bozeman Report, 1973. The paper is available via http://gravity5.phys. nthu.edu.tw/; W.-T. Ni, Phys. Rev. Lett. 38, 301共1977兲.

关25兴 J.F. Nieves, and P.B. Pal, Phys. Rev. D 39, 652 共1989兲; J.F.

Nieves and P.B. Pal, Am. J. Phys. 62, 207共1994兲.

关26兴 H.F. Kremer, Phys. Rev. 139, B254 共1965兲; H.F. Kremer, J.

Math. Phys. 8, 1197共1967兲.

关27兴 W. Gordon, Ann. Phys. 共Leipzig兲 4, 421 共1923兲.

关28兴 L.D. Landau and E.M. Lifshitz, Electrodynamics of

Continu-ous Media 共Pergamon, Oxford, 1960兲, p. 327, Sec. 80; L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed.共Pergamon, Oxford, 1984兲, p. 347, Sec. 100.

关29兴 K. Scharnhorst, Phys. Lett. B 236, 354 共1990兲; G. Barton, ibid. 237, 559共1990兲; G. Barton and K. Scharnhorst, J. Phys. A 26,

2037共1993兲.

关30兴 M. Visser, C. Barcelo, and S. Liberati, in the Festschrift of M.