Gordon Metric Revisited

Gordon Metric Revisited M. Novello∗† and E. Bittencourt‡ Instituto de Cosmologia Relatividade Astrofisica ICRA - CBPF Rua Dr. Xavier Sigaud, 150, CEP 22290-180, Rio de Janeiro, Brazil (Dated: November 4, 2018) We show that Gordon metric belongs to a larger class of geometries, which are responsible to describe the paths of accelerated bodies in moving dielectrics as geodesics in a metricq ˆµν different from the background one. This map depends only on the background metric and on the motion of the bodies under consideration. As a consequence, this method describes a more general property that concerns the elimination of any kind of force acting on bodies by a suitable change of the substratum metric. PACS numbers: 04.20.-q I. INTRODUCTION which led to the geometrization of the gravitational field as it was done by general relativity (GR). This means In 1923 Gordon [1] made a seminal suggestion to treat to describe the effects of acceleration of a particle on a the propagation of electromagnetic waves in a moving gravitational field by a change of the space-time met- dielectric, modifying the metric structure of the back- ric, according to Einstein’s approach. We compare this ground. He showed that the electromagnetic waves prop- effective geometry to the metric in the post-Newtonian agate as geodesics not in the background geometry ηµν approximation in order to see if it is possible to repro- but instead in the effective metric duce some results of GR. Such procedure is exactly what happens in the analog models of gravitation that deals gˆµν = ηµν + (ǫµ 1) vµ vν , (1) with systems kinematically equivalence, but dynamically − distinct. where ǫ and µ are constant parameters that characterize the dielectric and vµ is the four-velocity of the material under consideration (which is not necessarily constant). II. GEOMETRIZING ACCELERATED PATHS Later, it was recognized that this interpretation could be used to describe nonlinear structures even when ǫ and µ Consider a vector field1 u with norm N uαu in a depend on the intensity of the electromagnetic field [2] or µ α given background g . The acceleration of u≡is given by more complicated functions of the field strengths [3]. In µν µ all these cases, the causal cone, which is associated to the a = u uν . effective metric, does not coincide with the null-cone of µ µ;ν the theory. The origin of this modification is due to the Now let us construct an associated metric tensorq ˆµν , in presence of a moving dielectric, which changes the paths which the vector u satisfies the equation of the electromagnetic waves inside this medium. µ We then face the question: could such particular de- scription of the electromagnetic waves in moving di- u uˆν = f(p) u , (2) electrics be generalized for other cases, in which acceler- µ||ν µ ated paths due to other kind of forces would be described arXiv:1211.5053v1 [gr-qc] 21 Nov 2012 where means covariant derivative with respect toq ˆµν as geodesic motions in an associated metric? We shall see and f(p||) is an arbitrary function of the parameter p along that the answer is affirmative and this kinematical map the curve. The contravariant components of the vector depends only upon the acceleration of the body and the µ µν field are defined byu ˆ qˆ uν and, consequently, the background metric. norm defined inq ˆ is given≡ by Nˆ qˆµν u u . Whenever This method allows us to geometrize any force in the µν ≡ µ ν uµ is either a gradient or a normalized vector field, f(p) sense that arbitrary accelerated body in a given met- can be set equal to zero without loss of generality. ric substratum g is equivalently described as geodesic µν Developing Eq. (2), we obtain motion in an effective geometryq ˆµν . We start by an- alyzing the generalization of Gordon metric concerning the propagation of electromagnetic waves inside arbitrary 1 Nˆ + u uˆν = f(p) u , (3) dielectric media. This procedure mimics the trajectory 2 ,µ [µ,ν] µ ∗M. Novello is Cesare Lattes ICRANet Professor 1 The natural velocity field we use to develop this section is a given † Electronic address: [email protected] 1−form field uµ. In Gordon approach, the wave vector kµ is a ‡Electronic address: [email protected] gradient and, therefore, an exact 1−form. 2 where [] means skew-symmetrization. Choosingq ˆµν such that this equation is verified, then the accelerated path ′ α β ǫEα ǫ E E E =0, of uα in gµν becomes a geodesic motion in the associated ;α − E α;β qˆµν . Note that this is independent of its functional form. α µ0H ;α =0, We will show that Gordon metric is a particular exam- (6) ple of this procedure and that there is a class of metrics ′ λ α µ ǫE˙ λ ǫ E v E E + ηλβρσ v H =0, which play the same role in such “unforced-motion” pro- − E µ;α ρ σ;β cess exhibiting different geometrical properties for each µ H˙ λ ηλβρσv E =0. element of the class. 0 − ρ σ;β We define the unitary vector lµ by setting Eµ Elµ, µ α ≡ where l satisfies lαl = 1. III. LIGHT PATHS ON MOVING DIELECTRIC: We use Hadamard conditions− [5] to obtain the prop- GORDON APPROACH agation waves through the characteristics surface Σ (for details, see appendix). The symbol [X]Σ represents the Let us define two skew-symmetric tensors Fµν and Pµν discontinuity of X through this surface. Then, the dis- representing the electromagnetic field inside the material continuities of Eqs. (6) become medium. These tensors are expressed in terms of the field strengths Eµ and Hν and field excitations Dµ and Bµ as [Eµ,λ] = eµkλ, [Hµ,λ] = hµkλ, (7) follows Σ Σ where eµ(x) and hµ(x) are the amplitudes of the disconti- nuities and kµ ∂µΣ is the wave vector. Thus, it follows αβ ≡ Fµν Eµvν Eν vµ + ηµν vαBβ, that ≡ − αβ Pµν Dµvν Dν vµ + ηµν vαHβ, ′ ≡ − α ǫ α β ǫk eα E E eαE kβ =0, where vµ is a given four-vector comoving with the di- − αβ α electric and ηµν is the Levi-Civita tensor. We assume µ0h kα =0, (8) that the electromagnetic properties of the medium are ′ α µ ǫ λ α µ µναβ characterized by the constitutive relations ǫk vαe E eλv kαE + η kν vαhβ =0, − E α λ λβρσ µ0kαv h η kβvρeσ =0, ν ν − Dα = ǫα (E,H)Eν , Bα = µα (E,H)Hν , where ǫ′ is the derivative of ǫ with respect to E. Combin- ν ν ing these equations we obtain the following intermediary where ǫα (E,H) and µα (E,H) are arbitrary tensors. relation Consider Maxwell equations on dielectric media [4] with permittivity ǫ and permeability µ that characterize the µ β e ν ν 2 k eβ µ α µ dielectric: α [k kν (k vν ) ] α k + ǫk vαe + µ0kαv − − µ0kαv ′ ǫ λ α µ µν E eλv kαE =0, P ;ν =0, −E (4) (9) ∗ µν F ;ν =0. which multiplying by Eµ yields the dispersion relation From now on, we take the background metric as flat ′ µν ′ µ ν ǫ µ ν Minkowski space-time and assume that µ µ0 is a con- η + (µ0ǫ 1+ µ0ǫ E)v v E E kµkν =0. ≡α α − − ǫE stant and ǫ = ǫ(E), where E √ EαE and E is (10) the electric field. It is straightforward≡ − to generalize these We see that the envelop of discontinuity propagates dif- equations to arbitrary curved space-time. Indeed, sup- ferently from Minkowski light-cone of the linear Maxwell pose an observer with velocity vµ comoving with the di- µ theory. In this case, the causal structure is given by an electric and such that v ;ν = 0. Then, Eqs. (4) written 2 µν µ µ effective Riemannian geometry gˆ . From this point of in terms of the displacement vectors D and B become µν view, kµ is null-like ing ˆ , namely, gˆµν k k =0. (11) µ ν ν µ µναβ µ ν D ;ν v D ;ν v + η vαHβ;ν =0, − (5) Bµ vν Bν vµ ηµναβ v E =0. ;ν − ;ν − α β;ν 2 µ Mathematically, the metric tensor is a covariant tensor of rank The projection with respect to v yields the four in- 2. However, in this paper, we sometimes shall call “metric” a dependent non-linear equations of motion describing the contravariant tensor of rank 2. In particular, that is the way the electromagnetic field inside the dielectric medium: Gordon metric appears naturally. 3 The expression of the effective geometry is given by and qµν = α ηµν + β Φµν . (17) ǫ′E gˆµν = ηµν + (µ ǫ 1+ µ ǫ′E)vµvν lµlν . (12) µν 0 − 0 − ǫ This form of the metric requires that the tensor Φ must satisfy the conditionb A simple calculation show that its inverse is νλ λ λ Φµν Φ = mδµ + n Φµ. (18) 1 ξ Such feature allows us to write the inverse metric simi- gˆµν = ηµν 1 vµvν + lµlν , (13) larly to the binomial form of the metric, avoiding diffi- − − µ ǫ(1 + ξ) 1+ ξ 0 culties with infinite series3. Two remarkable examples where of this property are the scalar field (in which Φµν = ∂µφ∂ν φ) and the electromagnetic field (in which Φµν = ǫ′E F αF ). ξ . µ αν ≡ ǫ In particular, when ǫ is a constant, this formula re- A. Special case duces to Gordon pioneer work, in which was shown that the waves propagate as geodesics not in the background In this section, we limit our analysis to the simplest geometry ηµν but instead in the effective metric form by setting Φµν = uµ uν.

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