arXiv:1211.5053v1 [gr-qc] 21 Nov 2012 gt sgoeisnti h akrudgeometry background the in not back- geodesics prop- the waves as electromagnetic of the agate moving that structure showed a He metric in ground. the waves modifying electromagnetic dielectric, of propagation the oini neetv emtyˆ geometry effective an in motion ilcrcmda hspoeuemmc h trajectory the mimics procedure This media. concerning arbitrary dielectric inside metric waves electromagnetic Gordon of propagation of the generalization the alyzing es htabtayaclrtdbd nagvnmet- given a in body substratum accelerated ric arbitrary that sense the and body the of map metric. acceleration kinematical background the this upon and only affirmative see depends is shall We answer metric? described the associated be that an would in forces di- motions of acceler- geodesic kind which moving as other in to in cases, due paths other waves ated for generalized electromagnetic be the electrics of scription medium. this paths the inside the waves to changes electromagnetic due which the is dielectric, of modification moving of this a null-cone of of the presence origin with The coincide the theory. to not the associated does is In which metric, cone, [3]. effective causal strengths the field cases, these the or all of [2] field functions electromagnetic complicated the more of intensity the on depend u nta nteeetv metric effective the in instead but where h ilcrcand dielectric the sdt ecienniersrcue vnwhen even structures be nonlinear could describe interpretation to this used that constant). recognized necessarily was not it Later, is (which consideration under ‡ † ∗ lcrncades [email protected] address: Electronic [email protected] address: Electronic .Nvloi eaeLte CAe Professor ICRANet Lattes Cesare is Novello M. n12 odn[]md eia ugsint treat to suggestion seminal a made [1] Gordon 1923 In hsmto losu ogoerz n oc nthe in force any geometrize to us allows method This de- particular such could question: the face then We ǫ and htcnen h lmnto faykn ffreatn nb on acting 04.20.-q force numbers: of PACS kind any meth of this metric. elimination consequence, backgro substratum the a the on As concerns only consideration. that depends under map bodies This the one. background the from ecieteptso ceeae oisi oigdielectr moving in bodies accelerated of paths the describe µ g eso htGro ercblnst agrcaso geomet of class larger a to belongs metric Gordon that show We r osatprmtr htcharacterize that parameters constant are ˆ g µν .INTRODUCTION I. µν v = µ seuvlnl ecie sgeodesic as described equivalently is η stefu-eoiyo h material the of four-velocity the is µν u r airSgu,10 E 29-8,Rod aer,Br Janeiro, de Rio 22290-180, CEP 150, Sigaud, Xavier Dr. Rua ( + nttt eCsooi eaiiaeAtosc CA-CBP - ICRA Astrofisica Relatividade Cosmologia de Instituto ǫµ − 1) q µν v µ esatb an- by start We . v ν .Novello M. , odnMti Revisited Metric Gordon Dtd oebr4 2018) 4, November (Dated: ǫ and ∗† η (1) µν µ n .Bittencourt E. and apn nteaao oeso rvtto htdeals that gravitation dynamically distinct. but of equivalence, repro- kinematically models what systems to analog exactly with possible is the procedure is in Such it GR. happens if of results post-Newtonian see some the to duce in order this metric in met- compare the approximation We space-time to a the approach. geometry on of Einstein’s effective particle to change a means according a of This ric, by acceleration (GR). of field relativity effects gravitational general the by describe done field to gravitational was the it of geometrization as the to led which ie background given hc h vector the which o e scntuta soitdmti esrˆ tensor metric associated an construct us let Now where and h uv.Tecnrvratcmoet ftevector ˆ the by of defined components are contravariant field The curve. the a esteult eowtotls fgenerality. of loss without zero to equal set be can omdfie nˆ in defined norm u 1 µ I EMTIIGACLRTDPATHS ACCELERATED GEOMETRIZING II. rdetad hrfr,a xc 1 exact an therefore, and, gradient h aua eoiyfil euet eeo hsscini g a is section this develop to use 1 we field velocity natural The osdravco field vector a Consider eeoigE.() eobtain we (2), Eq. Developing − sete rdeto omlzdvco field, vector normalized a or gradient a either is f omfield form ( p c sgoeisi ercˆ metric a in geodesics as ics || sa rirr ucino h parameter the of function arbitrary an is ) en oain eiaiewt epc oˆ to respect with derivative covariant means ddsrbsamr eea property general more a describes od de yasial hneo the of change suitable a by odies n ercado h oinof motion the on and metric und u µ ‡ 2 1 nGro prah h aevector wave the approach, Gordon In . is hc r epnil to responsible are which ries, q N ˆ µν u g ,µ µ u µν sgvnby given is µ u + aifisteequation the satisfies || h ceeainof acceleration The . a µ ν µ u u ≡ ˆ [ = µ,ν 1 ν u F q ˆ = u ] µν µ u ˆ azil µ ν f ; ihnorm with ν u ( = N − ν u p ˆ q ν form. ) µν n,cneunl,the consequently, and, f ≡ . u ( µ p different q ˆ , ) µν u µ u , µ N u u ν µ ≡ Whenever . sgvnby given is u α u p q k α µν along µ f na in iven in , q sa is (3) (2) ( µν p ) 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. In this case, the coefficients of the covariant and contravariant forms are related by gˆµν = ηµν + (ǫµ 1) vµ vν , (14) 1 β 0 − A = , B = , α −α(α + β) which depends only on the dielectric properties µ0, ǫ and µν vµ. The magnitude N of the wave vector in Minkowski where we set uµuν η = 1 and write the metric in the space-time (written in terms of dielectric properties) is form determined by Gordon relation qˆµν = αηµν + βuµ uν .

The associated covariant derivative is defined by gˆµν k k = (ηµν + (ǫµ 1) vµvν ) k k =0, = µ ν 0 − µ ν ⇒ (15) α α α ν = N = (1 µ ǫ)(k.v)2, u || µ = u , µ + Γµν u , ⇒ − 0 α where the corresponding Christoffel symbol is con- where k.v k v . µν b ≡ α structed usingq ˆ . The description of an accelerated The analysis of the wave propagation in material me- curve4 in the flat space-time as a geodesics in the metric dia and the study of effective geometry are particularly qˆ is possible if the following condition is satisfied interesting in the investigation of analog model [2, 6] for µν the understanding of kinematical properties at very small u Γǫ u uν =0, (19) scale of astrophysical objects (see details in [7, 8]). We µ,ν − µν ǫ quote Hawking radiation [9] and Unruh’s work on exper-   where we have used the metricq ˆµν to writeu ˆµ qˆµν u = imental black hole evaporation [10] which are systemati- b b ν (α + β)uµ. Therefore, ≡ cally studied and the modeling of specific dielectric media is developed in order to eventually detect these tiny ef- fects. A more complete discussion on this topic was given in [11, 12] and references therein. Here we shall point the 3 This is the case of general relativity as a field theory formulation. similarity between these geometries in a later section. The exact expression for metric tensor is set gµν ≡ ηµν + hµν .

IV. BINOMIAL METRICS A consequence is that its inverse, the covariant tensor gµν is an infinite series: − α − α β In recent years an intense activity concerning features gµν = ηµν hµν + hµα h ν hµα h β h ν + ... of Riemannian geometries similar to those described by This formulation was introduced by Feynman, Gupta and others Gordon approach has been done [6]. In particular, those (cf. [13]). 4 that allows a binomial form for both the metric and its Note that we are dealing with a collection of paths Γ that is inverse. That is, its covariant and the corresponding con- usually called a congruence of curves. It is understood that each element of this collection concern particles that have the same travariant expressions are characteristics. For instance, if the acceleration is due to an elec- tromagnetic field, all particles of Γ must have the same relation qµν = A ηµν + B Φµν , (16) between its charge and mass, to wit a bunch of electrons.

b 4

α only on the angle kαv between the wave vector kα and the dielectric four-vector vα. This is achieved by gen- ǫ ν uµ,ν Γ uǫ u =0. (20) eralizations of the last section. Let us list some examples: − µν   µ µν In order to preserve theb normu ˆ uˆµ along the curve we Case A: the metricq ˆ is given by µ assume β,µu = 0, without loss of generality (it corre- sponds to a simple re-parametrization along the curves). µν µν µ ν Once the acceleration in the background is defined by qˆ = αη + β k k , ν aµ = uµ,ν u , the condition of geodetic motion in the and its inverse is qˆµν -geometry takes the form 1 β qˆµν = ηµν kµ kν . ǫ ν α − α(α + βN) aµ = Γµν uǫ u . (21) The Christoffel symbol reduces to Once the wave vector kµ is a gradient of a given hyper- b surface Σ, then we have α + β Γǫ u uν = uα uν q . (22) µν ǫ 2 αν,µ k[µ,ν] =0.

Using the expressionb of qαβ in Eq. (22)b and substituting Substituting this result in Eq. (3), it follows that kµ must the result into the condition (21), it follows that satisfy b ˆ ˆ 1 ∂ (α + β) N(q),µ =0 = N(q) const., a = µ . ⇒ ≡ µ −2 (α + β) where we define Nˆ qˆµν k k . That is, in order to (q) ≡ µ ν It means that the acceleration vector aµ must be a gra- follow a geodesic motion inq ˆµν , kµ must have constant dient of a function Ψ, i.e., norm in the metricq ˆµν . The explicit expression for this constraint is

aµ ∂µΨ. (23) ≡ ˆ N(q) = (α + βN)N 1. (25) Thus, the expression of the coefficients α and β of the ≡ metricq ˆµν are given in terms of the potential Ψ of the Note that this approach transforms the non- acceleration by normalized wave vector kµ in Minkowski background in a normalized time-like vector in qµν . It does not vio- late Lorentz invariance, because everything happens in- α + β = e−2Ψ. (24) side the dielectric. We note thatb it is not possible to fix Nˆ equal to zero, otherwise the metric is ill-defined. This simple example gives a very useful formula, which Therefore, k is not a null-like vector in the Qˆ-metric. exhibits the connection between geometrical and mechan- µ Another feature is that the magnitude of the dielectric ical quantities. Later, in this paper, we shall analyze the four-vector case in which Ψ represents the gravitational potential. µν 2 qˆ vµvν = α + β(k.v) ,

B. Polynomial Metrics is not necessarily positive definite allowing observers with velocity great than speed of light inside the medium5. For Gordon approach depends explicitly on the velocity vα instance, if we set of the dielectric. Nevertheless such form of introducing an effective metric is not unique. Indeed it seems rea- α + β(k.v)2 =0, sonable to present another metricq ˆµν that describes the same results obtained by Gordon and besides reduces then, using Eq. (15), we obtain the dependence on the four-velocity vα. From practi- (µ ǫ 1)α cal reasons, it might be useful to weaken this constraint β = 0 − . of Gordon approach, once it is easier to determine the N shape of the electromagnetic wave packet in the labo- ratory than constructing nonlinear dielectric media with arbitrary tensorial parameters ǫαβ and µαβ—despite of 5 In the laboratory, the angle between these two vectors is easier the great advances in this research area recently [14, 15]. to manipulate than the dielectric velocity field only. We expect Let us now show that exists a class of geometries that this fact could be of reasonable utility in the research of which play the same role as Gordon metric depending analog models. 5

Substituting this result in Eq. (25), yields δ 1 Nˆ =(1+ βN)N + N˙ 2 =0. α = . (m) 4 µ0ǫN Note that this approach permits a null geodesic Therefore, the metricq ˆµν with these values of α and motion for the wave vector kµ. This is the simplest case β produces the following outcome: the wave vector k µ in which we regain the main Gordon result (kµ as a null becomes a normalized and time-like vector, while the geodesic). The sign of the norm of vµ is undetermined µ dielectric velocity v , which was a time-like vector in the and may be chosen equal to zero, as we saw in the Minkowski background, becomes a null geodesic inq ˆµν . previous case. Therefore, the causal structure is no more determined by kµ. Case C: the most general case involving first order derivatives of k occurs when the metric is expressed in µν µ Remark that the metricq ˆ presented in the precedent the form6 sections is not unique. We can enlarge the set of metrics that have the same properties showed above adding other terms toq ˆµν provided the condition (18) is valid. nˆµν = α ηµν + β kµ kν + δaµaν + λa(µkν) To exemplify these cases we consider: and its inverse is Case B: the polynomial metric is given by 1 nˆ = η + B k k + ∆a a +Λa k . µν α µν µ ν µ ν (µ ν) mˆ µν = ηµν + β kµ kν + δaµaν . The covariant metric coefficients are given by β(α δa2)+ λa2 It is straightforward to show that its inverse has an extra B = − , term − Z

2 mˆ µν = ηµν + B kµ kν + ∆aµaν +Λa(µkν), δ(α + βN) λ N ∆= − , − Z where ( ) means symmetrization. The coefficients of the inverse metric are and ˙ ˙ β(1 δa2) λ(2α + Nλ) 2βδN B = − , Λ= − , − X − 2Z where 2 δ(1 + βN) 2 2 2 N˙ 2 ∆= , Z = α α − αδa + αβN + αNλ˙ − (βδ − λ ) + Na . − X   4  and Once it involves more degrees of freedom, we can re- βδN˙ gain all outcomes presented before, but with different Λ= . algebraic relations. In particular, the magnitude of the 2X wave vector inn ˆµν is set 2 α µ Here, we defined a a aα, N˙ N,µk and ≡− ≡ δ Nˆ = (α + βN + λN˙ )N + N˙ 2. N˙ 2 (n) 4 X =1 δa2 + βN βδ + Na2 . − − 4 ! Remark that the metric and its inverse have the same number of polynomial terms as required from the begin- The appearance of an extra term also happens with ning. It did not happen in the case B where an extra term µ ν the inverse metric when we consider instead of a a a was necessary in the inverse metric expression. Following (µ ν) term of the form a k . In both cases an extra term this reasoning, in the next section we shall use only the is necessary breaking the polynomial symmetry between cases A and C which satisfy the conditions (16) and (17). the metric and its inverse. Nevertheless we will present Moreover, as an example, we will set the potential Ψ as the calculations for this case focusing only on the metric being the Newtonian potential. containing the term aµaν and indicating that the results are very similar when the other term is considered sepa- rately. 6 The geodetic motion condition for the wave vector If we use higher derivatives of kµ greater than that which appears leads to in the dynamics, then the metric tensorq ˆµν is ill-defined. 6

C. Application: identifying Ψ with the cases. One of them gets immediately a wrong linear gravitational potential regime compared to GR, while the other case, which is separated in two subcases, can give the expected weak In the weak field limit the description of Newton’s field regime. gravity can be formulated in terms of a geometric repre- sentation of the gravitational field. It can be done mak- Case I: consider theq ˆ-metric as follows ing use of effective potentials, which correspond to the well-known parameterized post-Newtonian approxima- qˆµν = ηµν + βuµ uν . (26) tion (PPN) [16]. In this section, we compare some PPN results—particularly, those concerning general relativity The inverse metric is (GR) predictions, which improve the Newtonian theory of gravity in the solar system—with someq ˆ metric given by the free-falling (geodesic) condition. We− stress that β qˆµν = ηµν uµ uν . (27) qˆµν just mimic the geodesic motion characteristic of the − 1+ βN solutions of GR. Note that there is no dynamical equa- The condition for uµ to follow a geodesic motion in tion forq ˆµν and we are not proposing such theory. It is this metric is provided by purely a kinematical analogy. This section shows that the analysis through geodesic paths are much more general than GR, once it is noth- ν 1 ˆ uµ||νuˆ = N(q),µ =0, (28) ing but a choice of the metric. According to Poincar´e’s 2 ideas upon geometrical descriptions of the world [17]: whereu ˆµ qˆµν u = (1+ βN)uµ and the magnitude of non-Euclidean geometry is as legitimate as our ordinary ν u inq ˆ -metric≡ is Euclidean space; the enunciation of the Physics in this µ µν modified geometry would become more complicated, but it ˆ µ N(q) uˆ uµ = (1+ βN)N, still would be possible. In other words, it is possible to ≡ choose the metric of the space-time, such that an accel- which is imposed by Eq. (27) to be constant different ˆ erated motion in a given geometry can be described as from zero. For convenience, we set N(q) = 1. geodesic in another one. A power law expansion in terms of ǫ rH /r of metric For convenience, consider Minkowski space-time in (27), which corresponds to the weak field∼ limit, gives the spherical coordinates following expressions

ds2 = dt2 dr2 r2dΩ2, − − qˆ 1 rH 1+ rH + (ǫ3), 00 ≈ − r r O and an observer field uµ = (1,f(r), 0, 0), which is a gra- 3/2 7 rH
rH 7/2 dient uµ ∂µΣ, where Σ = t F (r) . This vector has a qˆ01 r 1+ r + (ǫ ), ≡ ± ≈ − O (29) non-null acceleration given by 2 qˆ 1
rH 1+ rH
+ (ǫ4). 11 ≈ − − r r O

ν 1 ′

aµ = uµ;ν u = N,µ = (0, ff , 0, 0), We see that this metric does not correspond to linearized 2 − Schwarzschild solution (even in Painlev´e-Gullstrand where prime ′ means derivative with respect to radial coordinates due to the power 3/2 instead of 1/2 in µν coordinate r and N uµuν η . qˆ01). This metric is similar to some post-Newtonian We choose the scalar≡ function Ψ, which characterizes approximation if we consider a rectilinear moving source the acceleration aµ ∂µΨ, as identified with the New- for the gravitational field. The angular components of tonian potential. Then,≡ it follows a relation between N the metric are identical to Minkowski ones. and Ψ given by Case II: let us consider another geometryn ˆµν given by r N =1+2Ψ=1 H , − r nˆµν = ηµν + Buµ uν + ∆ aµ aν +Λ a(µ uν). (30) where rH 2M and M is the mass source of the The metric components are explicitly written as gravitational≡ field (G = c = 1). To go further in the cal- culations, we basically split the analysis in two distinct nˆ00 = 1+ B, nˆ = f(B Λf ′), 01 − (31) 7 2 ′2 ′ This situation can perfectly be adapted to describe wave vectors nˆ11 = 1+ f (B + ∆f 2Λf ). in material media. − − 7

The other spatial components are identical to Minkowski Remark that these calculations was basically done in metric in spherical coordinates. The condition which led order to illustrate some analogies between these geome- uµ to follow a geodesic motion in this metric is imposed tries and those studied in analog models of gravity. For on its magnitude this reason, the first order approximation is enough to show their strong correlation. If someone takes this ap- proach looking for a perfect kinematical analogy between N ∆(a2N + N˙ 2/4) Nˆ = − const., (32) Schwarzschild metric and somen ˆµν , then the introduc- (n) ≡ tion of a nonlinear scalar potential Ψ in Newtonian equa- Z tion of acceleration becomes necessary and, therefore, it ˆ µ ν where N(n) nˆµν u u . We also define can reproduce exactly the Schwarzschild geodesics, for ≡ instance. Notwithstanding, we will not enter into the 2 details of this generalization because it involves a com- N˙ Λ N˙ 2 plicated question about the physical meaning of the dy- = 1+ a2∆+NB+a2NΛ2 B∆ a2N + . Z 2 ! − − 4 ! namics of such non-linear potential whereas in this paper we want to discuss only kinematical properties of the par- ˆ ˆ ticle trajectory. In this case, we can set either N(n) =0 or N(n) = 1. Let us analyze both cases separately:

ˆ V. CONCLUSION Case II.a: if N(n) = 0, the assumption (32) implies that In this paper, we presented an extension of Gordon N metric constructing a larger class of geometries which ∆= . a2N + N˙ 2/4 describes accelerated motions in Minkowski space-time as geodesics in an effective geometryq ˆµν . This effec- Once uµ is null-like inn ˆ metric, we expect to regain tive metric depends only on the background metric and the metric component of the− PPN approximation respon- on the velocity vector of the accelerated body. In par- sible to describe correctly the light propagation deflec- ticular, we analyzed accelerated paths of light inside a tion, which is given by then ˆ11 component of the metric. moving dielectric and constructed a class of geometries So, using the considerations above, we set with peculiar properties in comparison to Gordon’s ap- proach, but with similar kinematical effects. Ultimately, we gather this new class of geometries, that we call Qˆ- r rH 2 nˆ00 1+ + (ǫ ), metrics, with the effective geometries of nonlinear elec- ≈ rH − r O tromagnetism, producing a collection of possible metric nˆ 0, 01 ≈ (33) structures of the space-time, which has several applica- 2 tions in the theory of analog models of gravity. We will nˆ 1 rH + rH + (ǫ3). 11 ≈ − − r r O come back to this in the future.
Note that a strange linear term appears inn ˆ00 due to our assumptions. If we try to avoid this term making some APPENDIX: Hadamard’s Method for Discontinuities coordinate transformation, then the asymptotically flat regime is lost by other metric components, in such way that the trouble persists. We analyze the discontinuities of the electromagnetic field according to the standard Hadamard method and ˆ obtain the dispersion relation for the wave vector kµ. Let Case II.b: in the case of N(n) = 1, we can correctly reproduce the linear approximation of Schwarzschild so- Σ be a surface of discontinuity of the field Aµ. The dis- lution. That is, continuity of an arbitrary function f is given by:

[f(x)]Σ = lim f(x + ǫ) f(x ǫ) . (35) ǫ→0+ − − rH rH 2 3 nˆ00 1 +2 + (ǫ ),
≈ − r r O The field Aµ and its first derivative ∂ν Aµ are continuous across Σ, while the second derivatives present a disconti- nˆ01 0,
≈ (34) nuity: rH 3 nˆ11 1 r + (ǫ ). ≈ − − O [Aµ]Σ = 0, (36)

[∂ν Aµ]Σ = 0, (37) Note thatn ˆ00 differs from PPN results already in second [∂α∂βAµ] = kαkβξµ(x), (38) order approximation in ǫ (whose the expected value in Σ 2 ǫ is 1/2 instead of 2). Therefore, it will surely produce where kµ ∂µΣ is the propagation vector and ξµ(x) is some discrepancy in high order terms of the expansion. the amplitude≡ of the discontinuity. Substituting these 8 discontinuity properties in the equation of motion VI. ACKNOWLEDGEMENTS

ηαν F ηαν A =0, µν;α ≡ [µ,ν];α it follows that: We acknowledge J. M. Salim, J. D. Toniato, E. αβ kαkβη =0. Goulart, V. de Lorenci and R. Klippert for their com- ments and the staff of ICRANet in Pescara where this This means that the discontinuities of the electromag- work was partially done. We would like to thank FINEP, netic field propagate as null geodesics in the Minkowski FAPERJ, CNPq and CFC/CBPF for their financial sup- metric ηµν . port.

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