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Author's personal copy Physics Letters A 374 (2010) 4175–4179
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Physics Letters A www.elsevier.com/locate/pla
Local nonlinear electrodynamics D.D. Pereira ∗ , R. Klippert Instituto de Ciências Exatas, Universidade Federal de Itajubá, Av. BPS 1303 Pinheirinho, 37500-903 Itajubá, MG, Brazil
a r t i c l e
i n f o
a b s t r a c t The propagation of electromagnetic waves in dielectric media characterized by the coefficients ε α β = ε α β ( E μ , B μ , ∂ν E μ , ∂ν B μ ) and μα β = μα β ( E μ , B μ , ∂ν E μ , ∂ν B μ ) is examined in the eikonal approximation of electrodynamics. Employing the techniques Hadamard–Papapetrou (HPD) and Spacetime Integration (STI), we derive the dispersion relation, the polarization modes and effective geometry associated to the model. © 2010 Elsevier B.V. All rights reserved.
Article history: Received 4 May 2010 Received in revised form 2 July 2010 Accepted 12 August 2010 Available online 17 August 2010 Communicated by P.R. Holland Keywords: Eikonal Shock waves Nonlinear electrodynamics
1. Introduction The propagation of electromagnetic waves in nonlinear theories of electromagnetism has, recently, attracted great interest in the scientific community. This problem can be investigated under two different aspects. We can examine this question in the regime of intense fields [1–3], and also in the context of material media [4–15]. In both cases, the field equations that govern the electromagnetic phenomena are nonlinear. In the regime of intense fields, the theory is analytically built from a nonlinear lagrangean, which generally is a function of both the Lorentz invariants F := F μν F μν ∗ of the electromagnetic field [16]. In the context and G := F μν F μν of materials media, the Maxwell equations must be supplemented with constitutive relations
D α = εα β E β , H α = μα β B β , where the coefficients ε α β and
(1)
μα
β represent the dielectric matrices. They are usually called electric permittivity and magnetic permeability, respectively. All information about the dielectric properties of the medium can be obtained from the constitutive relations [17–20]. In this case, the structure of the propagation of waves is dependent on the behavior of the medium acted upon by the external fields as encoded in terms of certain functions which in general are nonlinear. The electrodynamics in material media is also considered as a possible scenario for the investigation of analog models for gravitational phenomena [21].
This work aims to obtain and discuss the solutions to the Fresnel eigenvalue equation which describes the propagation of light rays in material media whose dielectric matrices have a functional dependence on the electromagnetic field and its first order derivatives. The analysis is restricted to local electrodynamics, where dispersive effects are neglected. Only monochromatic waves are considered, thus avoiding ambiguities with respect to the velocity of the waves. Section 2 presents two alternative techniques of propagation of field discontinuities: (i) Hadamard–Papapetrou (HPD), and (ii) Spacetime Integration (STI). Section 3 presents the eigenvalue equation for our model. Section 4 obtains the dispersion relation and the permitted polarization modes, as well as the optical metric for the model are analyzed. Section 5 summarizes our results. A Minkowskian spacetime described in an adapted Cartesian coordinate system is used throughout this work. The units are such that c = 1. The spacetime metric is denoted by ημν = diag(+1, −1, −1, −1). For an arbitrarily given function F (xμ ), its partial derivatives ∂μ F with respect to any given spacetime coordinate xμ is denoted by F ,μ . All quantities are referred to as meaμ μ sured by the geodetic observer V μ = δ0 , where δν denotes the μ μ Kronecker tensor and hν := δν − V μ V ν is the projector onto the three-dimensional rest space of this observer V μ . For any given ) and Y μ = (0, Y ), we have pair of space-like vectors X μ = (0, X · Y ) = − X μ Y μ . (X 2. The formalism of shock waves 2.1. Hadamard–Papapetrou (HPD)
*
Corresponding author. E-mail addresses:
[email protected] (D.D. Pereira),
[email protected] (R. Klippert). 0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.08.033
The technique used in the HPD formalism is to analyze the discontinuity of a function F by an orientable hypersurface in a
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differentiable manifold M [22,23]. In this work, we are interested in analyzing the discontinuity of a function F across a space-like (or possibly light-like) orientable borderless hypersurface. Let Σ be such a hypersurface, defined in terms of a given function Φ by Σ : Φ(xμ ) = 0. Such Σ then splits the spacetime into three consistently defined sets [4]. Let X − be the union of sets of spacetime points P − in the past of P for each P ∈ Σ , and similarly X + be the union of sets of spacetime points P + in the future of P for each P ∈ Σ . Causality of spacetime ensures that X + and X − are two disjoint sets. For each point P 0 ∈ Σ , any sufficiently small neighborhood U P 0 of P 0 is partitioned into three disjoint re+ − + and U 0 ⊂ Σ . Let r be the radius of gions: U − P0 P0 ⊂ X , U P0 ⊂ X
this neighborhood U P 0 , and let also P −
−
and P +
+
∈ U P0 ∈ U P 0 be any two neighbor points from P 0 arbitrarily chosen on opposite sides of Σ . Consider an arbitrary function F (xμ ) (or tensor field of arbitrary rank, whose indices being omitted for convenience of notation) defined in U P 0 . The discontinuity of F (xμ ) across Σ is then defined as
[ F ]Σ ( P 0 ) := lim F P
+
r →0+
− F P− .
(2)
Suppose that the function F (xμ ) has a vanishing discontinuity across Σ , i.e. that [ F ]Σ ( P ) = 0 at each point P ∈ Σ . Papapetrou has shown [23] that the discontinuity of F , μ across Σ has the form
[ F ,μ ]Σ ( P ) = F˜ K μ ,
(3)
where F˜ is a function defined on Σ , with the same rank and with the same algebraic symmetries of F , while K μ is the vector normal to Σ defined by K μ := ∂μ Φ . More generally, if the function F (xμ ) is such that all its derivatives F ,μ1 μ2 ...μi from order zero to order i have null discontinuities in Σ , then its derivative F ,μ1 μ2 ...μi μi+1 of order (i + 1) presents discontinuity through Σ as
[ F ,μ1 μ2 ...μi μi+1 ]Σ = F˜ K μ1 K μ2 . . . K μi K μi+1 ,
(4)
where F˜ is a function defined on Σ , with the same rank and with the same algebraic symmetries of F . 2.2. Spacetime Integration (STI) Maxwell equations constitute a system of coupled partial differential equations of first order, and must hold in the vicinity of any given point on which the fields are continuous. However, if there is any domain of spacetime for which the fields are discontinuous, then in this domain these equations are not well defined. Thus, it becomes necessary to replace this set of differential equations by a new set of equations to give us information about the discontinuity of the fields. Consider an open and connected limited domain G of the spacetime, and let Γ : φ(xμ ) = 0 be a hypersurface which is the boundary of G. Admit that Γ is continuous and orientable, and that it has continuous by parts tangent hyperplanes. Let also a function f be continuous in G¯ = G ∪ Γ and with continuous partial derivatives in G. Then, it holds the identity (Stokes’s Theorem)
(∂μ f ) dΩ = G
f nμ dS ,
(5)
Γ
where λ := ±(∂μ φ∂ μ φ)−1/2 and nμ := λ∂μ φ , while dΩ is an infinitesimal element of domain G and dS is an infinitesimal element of hipersurface Γ . Therefore, nμ is a normalized vector which is orthogonal to the hypersurface Γ . The signal of λ is chosen to ensure that the vector nμ is directed outwards from G. Consider a domain G of the spacetime over which the fields and B, the inductions D and H , and the sources ρ and J , are E
all continuous and have continuous partial derivatives. Integrating Maxwell equations
∂μ D μ = ρ ,
(6)
V ν ∂ν D μ + ημνα β V α ∂ν H β = J μ ,
(7)
∂ μ B μ = 0,
(8)
V ν ∂ν B μ − ημνα β V α ∂ν E β = 0,
(9)
over such G and using the identity equation (5), we have
Γ
Γ
J dΩ,
G
× E + B λ(∂t φ) dS = 0, n
Γ
−D λ(∂t φ) dS = ×H n
dS = ·D n
ρ dΩ, G
· B dS = 0, n
(10)
Γ
. If the vector fields ρ , J , E, B, D and H are := λ∇φ where n all continuous and have continuous partial derivatives in G, then Eqs. (10) are fully equivalent to Maxwell equations (6)–(9). However, Eqs. (10) hold also for discontinuous fields. We can therefore consider these last equations as a generalization of Maxwell equations. Discontinuous fields that solve these integral equations are called weak solutions of Maxwell equations [24,25]. Assume that Γ0 : Φ(xμ ) = 0 is a regular orientable hypersurface where the electromagnetic fields are discontinuous. This hypersurface Γ0 is a continuously differentiable submanifold that cuts an open connected limited domain G of spacetime in two open and disjoint subdomains G 1 and G 2 . The subdomain G 1 is enclosed by Γ1 and Γ0 , while the subdomain G 2 is enclosed by Γ2 and Γ0 . By means of a procedure similar to the one which leads to Eqs. (10), one can obtain two new sets of integral equations valid in subdomains G 1 and G 2 directly from Maxwell equations. The compatibility of these three sets of integral equations yields
× [H ]Γ0 − (∂t Φ)[ D ]Γ0 = 0, ∇Φ × [ E ]Γ0 + (∂t Φ)[ B ]Γ0 = 0, ∇Φ · [D ]Γ0 = 0, ∇Φ · [ B ]Γ0 = 0, ∇Φ
(11)
]Γ0 stands for the discontinuity of the quantity X across where [ X Γ0 making use of the notation of Eq. (2). Such a set of equations can be formally obtained from Maxwell equations (6)–(9) by replacing the differential operators ∂μ by the multiplicative operators ∂μ Φ . 3. Fresnel equation Various models of nonlinear electrodynamics where the dielectric matrices ε α β and μα β have a given functional dependence with respect to external electromagnetic fields applied to the medium are well known in the literature. We consider a model where such dielectric matrices depend not only on the external electromagnetic fields E μ and B μ , but also on their spatial derivatives. Accordingly, our analysis corresponds to local nonlinear electrodynamics. Consider the Maxwell equations (6)–(9) and the constitutive relations (1) of a given material medium is such that its dielectric parameters ε α β and μα β have the functional form
Author's personal copy D.D. Pereira, R. Klippert / Physics Letters A 374 (2010) 4175–4179
ε α β = ε α β ( 0) + ε α β E μ , B μ
+ ε( E ) ∂δ E α hδ β + ε( B ) ∂δ B α hδ β , μα β = μα β (0) + μα β E μ , B μ + μ( E ) ∂δ E α hδ β + μ( B ) ∂δ B α hδ β .
Applying the STI technique to Eq. (6), we find
(12)
Kν
ν μ ε μ Σ E = 0.
ε( E ) eν qν qμ E μ + ε( B ) bν qν qμ E μ = 0.
1 μνα β η V α K ν eβ ,
ω
(14) (15)
where ω := K μ V μ represents the angular frequency of electromagnetic wave, and qν := hμ ν K μ is the spatial part of the wave vector K ν .
(17)
Replacing Eq. (14) in Eq. (17), we obtain at each point where ε( E ) = = 0 that 0 and q · E
q ν e ν = 0.
(18)
Applying the latter technique to Eq. (7), we have
V ν Kν
μ λ ε λ Σ E + ημνα β V α K ν [μβλ ]Σ B λ = 0.
(19)
Recalling the model equations (12)–(13) in Eq. (19), we find
ωε( B ) bμ qλ E λ + ημνα β V α K ν μ( E ) eβ qλ B λ + ωε( E ) e μ qλ E λ + ημνα β V α K ν μ( B ) bβ qλ B λ = 0.
(20)
Substitution of Eq. (15) then yields
1
ωε( B ) qλ E λ ημνα β V α K ν eβ + ωε( E ) eμ qλ E λ ω + ημνα β V α K ν μ( B ) qλ B λ
1
ω
ηβ χ ξ γ V ξ K χ e γ
+ ημνα β V α K ν μ( E ) e β qλ B λ = 0.
(21)
It should be noted that by taking the discontinuity of dielectric matrices ε μ ν and μμ ν in Eqs. (16), (19) we are actually evaluating the discontinuity of the derivatives of the intensity fields E μ and B μ on the discontinuity hypersurface Σ . Thus, the polarization vectors e μ and bμ obtained from these expressions by the HPD method correspond to the same polarization vectors derived from Maxwell equations by the STI technique. This justifies the fact that we can substitute Eqs. (14), (15) in Eqs. (17), (20). Eq. (21) can be written as
Z μ σ e σ = 0,
(22)
where we define the generalized Fresnel matrix Z μ σ by
Z μ σ = Xhμ σ + Y q −1 ημνα σ V α qν ,
(23)
in terms of the auxiliary quantities
X := ε( E ) qλ E λ − μ( B ) Y :=
q 2
ω2
qλ B λ ,
ε( B ) qλ E λ + μ( E ) qλ B λ
q
ω
.
(24) (25)
Eq. (22) is an eigenvalue equation. The matrix Z μ σ has nontrivial kernel [27] if, and only if,
bμ =
(16)
Replacing Eq. (12) in Eq. (16), we have
(13)
In the coordinate system used, the dielectric matrices ε α β and μα β have vanishing temporal components, and their space components are time-independent. Thus, only derivatives with respect to spatial coordinates are considered in this model. In Eqs. (12)–(13) the quantities ε α β (0) and μα β (0) correspond to the values of electric permittivity and magnetic permeability of the medium in the absence of external fields. As the material may react to the action of external fields, we represent by ε α β ( E μ , B μ ) and μα β ( E μ , B μ ) the values that the parameters suffer due solely to the applied fields, and zero in the absence of fields. Finally, we denote by ε( E ) and ε( B ) two continuous functions (not necessarily constant) of the fields E μ and B μ that carry information about the dependence of ε α β in the spatial derivatives of electric and magnetic fields, respectively; while μ( E ) and μ( B ) carry information about the dependence of μα β in the spatial derivatives of electric and magnetic fields, respectively. Let Σ : Φ(xμ ) = 0 be an orientable hypersurface over which the fields E μ and B μ are continuous, but they possibly have discontinuous derivatives. Thus, the dielectric parameters ε α β and μα β may be discontinuous at Σ . This ensures that both D μ and H μ are bounded, even though the argument cannot be reversed in order to prove the continuity of the fields from that boundedness. Therefore, we assume such behavior of E μ and B μ with no justification but simplicity. In order to analyze the propagation of electromagnetic waves in the medium, we take the discontinuity of the field equations (6)–(9). The constitutive relations (1) immediately give discontin and H on Σ . Therefore, we cannot use a uous induction fields D single technique to deal with the discontinuities of the Maxwell and B are equations, since in Eqs. (8)–(9) the intensity fields E and H are continuous, while in Eqs. (6)–(7), the induction fields D discontinuous. When the material lacks dependence in the derivative of the fields, the coefficients of the matrices of electric permittivity and of magnetic permeability will be continuous quantities across the discontinuity hypersurface Σ . Thus, we can employ only the HPD technique in Maxwell equations. A detailed study of this last case can be found in recent literature [26]. In the present work, the inductions are discontinuous quantities, thus requiring the complementary use of two techniques in order to determine the propagation of the electromagnetic fields. Consider conditions of discontinuity for the fields directly from Maxwell equations. Eq. (3) then takes the form [∂ν E μ ]Σ = e μ K ν and [∂ν B μ ]Σ = bμ K ν , where e μ and bμ are space-like vectors and represent the vectors associated with the discontinuities of the derivatives of the fields E μ and B μ on the hypersurface Σ . Otherwise, e μ and bμ are the polarization vectors of the wave component of the fields E μ and B μ respectively, whose fronts are normal to the wave vector K ν . Applying the HPD technique to Eqs. (8)–(9), we have
b ν q ν = 0,
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det Z μ σ = 0,
(26)
where the determinant is calculated only on the three-dimensional structure of Z μ σ , because this matrix has all temporal components identically zero. This result is known in literature as the generalized Fresnel equation, and provides the effective dispersion equation, an essential tool to describe the propagation of light rays in material media.
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and B μ , one can use this technique. However, in broader situations, it is generally more convenient to use the covariant method.
4. Dispersion relation The standard method to solve Fresnel eigenvalue equation is to expand the eigenvectors on a basis of vectors of the threedimensional space [1,2,6,26]. This technique, although simple, can lead us to question wether the chosen set of vectors is effectively linearly independent. To avoid this subtlety, we employ here an alternative method that relies only on the algebraic structure of Eq. (22), and consists to making use of the covariant formula for the determinant of linear operators [28]. Accordingly, the threedimensional determinant of the matrix Z μ σ is given by
1 1 1 det Z μ σ = − ( Z 1 )3 + Z 1 Z 2 − Z 3 , 6 2 3
(27)
with Z 1 = Z μ μ , Z 2 = Z μ σ Z σ μ and Z 3 = Z μ σ Z σ τ Z τ μ . For the matrix Z μ σ given by Eq. (23), the traces Z 1 , Z 2 and Z 3 are given by
⎧ ⎨ Z1 = 3 X, Z = 3 X 2 − 2Y 2 , ⎩ 2 Z3 = 3 X 3 − 6 X Y 2.
(28)
Replacing Eqs. (27)–(28) in Eq. (26), we find
X X 2 + Y 2 = 0.
q
2
ω2
(q · B ) = 0,
+ μ( E ) B ) = 0. q · (ε( B ) E
(30)
(31)
Eq. (31) above is considered as a constraint, since it does not impose any condition to the frequency ω . From Eq. (30), we find the phase speed
(32)
where we define v 2 := ω2 / q 2 . It should be noted that Eq. (32) was obtained with the complementary use of the HPD and STI techniques from the set of Maxwell equations. From Eqs. (12)– (13), we see that the fields of intensity and of induction are all continuous when considering ε( E ) = ε( B ) = 0 = μ( E ) = μ( B ) , and therefore we can use the HPD technique in the whole set of field equations. Thus, we can analyze and recover the contribution of other terms of the continuous dependent fields. Due to the discontinuous nature of the dielectric parameters, the STI technique does not consider the contribution of the continuous part of the fields. It is interesting that the same result found in Eq. (29) can also be obtained by choosing the coordinate system appropriately. To see this, suppose the spatial coordinates of the wave vector qμ as qμ = (0, 0, q). Thus, the matrix Z μ σ defined in Eq. (23) takes the form
Y X 0
X ⎝ −Y 0
Y X 0
⎞⎛
⎞
⎛ ⎞
e1 0 0 0 ⎠ ⎝ e2 ⎠ = ⎝ 0 ⎠ . 0 X e3
(34)
From the compatibility condition derived from Gauss’ law for electricity, we obtain qμ e μ = 0. This result shows that the projection of the polarization mode along the direction defined by the vector qμ is zero. Thus, if qμ = (0, 0, q), then the polarization mode e μ = (0, 0, e 3 ) is not allowed. Based on these results, only the two remaining polarization modes are admissible. However, in order to allow room for some other polarization mode e μ different from e μ = (0, 0, e 3 ), the Fresnel matrix Z μ σ must be identically zero. This implies the validity of Eqs. (30) and (31). Thus, we would have, in principle, three admissible polarization modes. However, from Eq. (18), only two polarization modes are allowed, and they must be transverse to the direction of the wave vector qμ .
The conditions for the propagation of light rays given by Eq. (32) can be presented as μν
g e K μ K ν = 0,
(35) μν
where the symmetric tensor g e represents the effective optical geometry, also known as optical metric, associated with the propagation of electromagnetic waves in the material. This object characterizes the geometric structure of an effective spacetime in which the wave freely propagates. One can show [10] that K μ satisfies the geodesic equation
( D λ K μ ) K λ = 0,
μ( B ) (q · B ) ,
v 2 = ε( E ) (q · E )
X Z μ σ = ⎝ −Y 0
⎛
4.2. Effective geometry
and the second solution is to have Eq. (30) together with Y = 0, i.e.
⎛
To analyze the polarization modes allowed for this system, consider Eq. (22) written in the form
(29)
Eq. (29) immediately gives us two solutions. The first one is X = 0, i.e.
ε( E ) (q · E ) − μ( B )
4.1. Polarization
⎞
0 0 ⎠, X
(33)
from which it immediately follows Eq. (29). For situations where one can set the direction of the wave vector qμ and the fields E μ
(36)
where D μ represents the covariant derivative in the effective geμν ometry g e . This result ensures that the propagation vector K μ = μν g e K ν is a geodesic vector in the effective optical geometry [15]. μν Since K μ is a light-like vector in the effective geometry g e , it follows that its integral curves correspond to null geodesics in this geometry. Representing Eq. (32) as
ω2 ε( E ) (q · E ) − q 2 μ( B ) (q · B ) = 0, we find
ημν − 1 −
(37)
ε( E ) (q · E ) μ ν V V K μ K ν = 0. μ( B ) (q · B )
(38)
Comparing this result with Eq. (35), we can define the effective geometry for the light rays as μν
ge
ε( E ) (q · E ) μ ν = ημν − 1 − V V . μ( B ) (q · B )
(39)
5. Conclusion A tensor formalism has been presented for describing the propagation of monochromatic electromagnetic waves inside material media with nonlinear dielectric properties. The limit of geomet-
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rical optics was considered. The HPD and STI techniques were complementary applied to Maxwell equations of electromagnetism obtaining the Fresnel eigenvalue equation. This eigenvalue problem has been solved, resulting in the dispersion relation for the model. The direction of propagation and the magnitude of the phase velocity of electromagnetic waves are dependent on the direction of the applied external electromagnetic fields and also on the material properties of the medium. The functional form of the dispersion equation has qualitative behavior that differs from other dispersion relations for alternative models of nonlinear electrodynamics in the literature. Indeed, for other models previously discussed, the dependence on the direction of propagation represented by the vector q appears only in the numerator of the corresponding dispersion relation. In Eq. (32), however, this dependence appears both in the numerator and denominator of the expression. This distinct behavior may be operationally measured as follows. Let us suppose a material whose parameters μ( B ) and ε( E ) from Eqs. (12)–(13) are both nonzero and such that they do not depend on the direction of the fields (but possibly on their magnitudes). By imposing known external fields and B along the same direction, then Eq. (32) yields isotropic E phase velocities (with the mathematical exception of the propagation along the plane orthogonal to the electromagnetic fields, solvable by a limiting procedure) the measure of which determines the ratio μ( B ) /ε( E ) . Then, one may change the direction of the ex and B with no modification on their magnitudes. ternal fields E The phase velocity then becomes anisotropic, and can achieve values larger and smaller than the isotropic one depending on the direction q of the light ray. Mathematically, Eq. (32) allows room to infinity for any value of v from zero (for q orthogonal to B) all val For q along the direction of E × B, (for q orthogonal to E). ues of the phase velocity are allowed, each of which being found as a limiting process. The two physically admissible polarization modes are transverse to the direction of propagation. The optical metric is a deviation from the flat Minkowskian metric. Future works should be developed to find material media with dielectric properties such as those presented.
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Acknowledgements This work was partially supported by the Brazilian agencies CAPES (DDP) and CNPq (RK). References [1] Z. Bialynicka-Birula, I. Bialinicki-Birula, Phys. Rev. D 2 (1970) 2341. [2] V.A. De Lorenci, R. Klippert, M. Novello, J.M. Salim, Phys. Lett. B 482 (2000) 134. [3] M. Novello, V.A. De Lorenci, J.M. Salim, R. Klippert, Phys. Rev. D 61 (2000) 45001. [4] V.A. De Lorenci, R. Klippert, Phys. Lett. A 357 (2006) 61. [5] V.A. De Lorenci, R. Klippert, D.H. Teodoro, Phys. Rev. D 70 (2004) 124035. [6] V.A. De Lorenci, M.A. Souza, Phys. Lett. B 512 (2001) 417. [7] V.A. De Lorenci, J.M. Salim, Phys. Lett. A 360 (2006) 10. [8] M. Novello, J.M. Salim, Phys. Rev. D 63 (2001) 083511. [9] V.A. De Lorenci, Phys. Rev. E 65 (2002) 026612. [10] V.A. De Lorenci, R. Klippert, Phys. Rev. D 65 (2002) 064027. [11] Yu.N. Obukhov, G.F. Rubilar, Phys. Rev. D 66 (2002) 024042. [12] Yu.N. Obukhov, F.W. Hehl, Phys. Lett. B 458 (1999) 466. [13] Yu.N. Obukhov, T. Fukui, G.F. Rubilar, Phys. Rev. D 62 (2000) 44050. [14] G.F. Rubillar, Ann. Phys. (Leipzig) 11 (2002) 717. [15] G. Boillat, J. Math. Phys. 11 (1970) 941. [16] J. Plebanski, Lectures on Non-Linear Electrodynamics, Nordita, Copenhagen, 1968. [17] J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1965. [18] J.A. Stratton, Electromagnetic Theory, McGraw–Hill, New York, 1941. [19] L. Landau, E. Lifshitz, L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Mir, Moscow, 1969. [20] M. Born, E. Wolf, Principles of Optics, Academic Press, New York, 1970. [21] Workshop on Analog models of general relativity, http://www.cbpf.br/~bscg/ analog, Rio de Janeiro, Brazil, October 2000. [22] J. Hadamard, Leçons sur la propagation des ondes et les équations de l’hydrodynamique, Hermann, Paris, 1903. [23] A. Papapetrou, Lectures on General Relativity, D. Reidel, Dordrecht, Holland, 1974. [24] M. Kline, I.W. Kay, Electromagnetic Theory and Geometrical Optics, Interscience Publishers/John Wiley & Sons, New York, 1965. [25] R.K. Luneburg, Mathematical Theory of Optics, University of California Press, Los Angeles, 1966. [26] V.A. De Lorenci, G.P. Goulart, Phys. Rev. D 78 (2008) 045015. [27] S. Lang, Linear Algebra, Addison–Wesley, London, 1966. [28] R.R. Silva, J. Math. Phys. 39 (1998) 6206.
