# Spherical symmetry and nonmagnetic dielectrics in analogue models of gravity

Eduardo Bittencourt<EMAIL_ADDRESS>Renato Klippert  Retired
Federal University of Itajubá, Itajubá, Minas Gerais 37500-903, Brazil  Diego
Renan da Silva  São Paulo State University, Guaratinguetá, São Paulo
12516-410, Brazil  Érico Goulart  Federal University of São João d’El Rei,
C.A.P., Ouro Branco, Minas Gerais 36420-000, Brazil

###### Abstract

Nonmagnetic and spherically symmetric dielectric material media are
investigated in their generality as analogue models, in the domains of
geometrical optics, for a few relevant static and spherically symmetric
solutions of gravitation, including black holes and wormholes. Typic and
desirable gravitational features are systematically linked to the nonlinear
properties of the phenomenological dielectric permittivity of the medium. From
this, the limitations of such analogies are explored in terms of their
corresponding geometric flexibility, dielectric configurations and causal
structure.

analogue models of gravity; nonlinear electromagnetism; geometric optics.

##  I Introduction

Analogue models of gravity are the easiest way for testing experimentally the
kinematical aspects of any geometric model of gravity in laboratory. The first
example of these models was developed by Gordon [1] in the optical context,
with further developments by De Felice [2] and Plebanski [3]. However, the
turning point came with Unruh’s model on black hole evaporation in the context
of fluid mechanics [4], where the quantum analysis became possible. Since
then, several new models have been explored in the realm of condensed matter
systems [5, 6]. More recently, it has been emphasized that the optical
properties of material media can be fairly easily designed nearly at will by
the present day technology on materials [7, 8, 9]. This suggests that the
optical analogue models could be favoured as natural candidates for mimicking
the behavior of gravitational systems from the experimental of view.

The present work aims to draw the basic lines to guide material designers,
from the perspective that each kind of matter can be used to produce the
desired effective gravitational potential. In order to obtain definite results
of broad significance, some simplification is needed; the discussion applies
to nonmagnetic isotropic dielectric media, which are used to model some
properties of static and spherically symmetric solutions of the theory of
gravitation.

For completeness, Sec. II reviews the theory of optical analogue models
through the generalized Fresnel eigenvalue problem up to obtaining the
ordinary and extraordinary modes of light propagation and the two
corresponding optical analogue metrics. In Sec. III the general analogue
models are obtained and the corresponding electromagnetic quantities are
provided and compared with two paradigmatic gravitational models: the black
hole and the wormhole models. Section IV discusses the causality issues
related to Cauchy developments. The effective light cones were constructed in
order to provide one possible physical interpretation for the optical analogue
metrics.

Notation is such that Latin indices $i,j,l,\ldots$ run from $1$ to $3$ and
repeated indices are summed upon. Partial derivatives are denoted as
$\partial_{i}=\partial/\partial x^{i}$, $\partial_{t}=\partial/\partial t$,
and $\eta_{ijl}$ is the completely skew-symmetric Levi-Civita symbol with
$\eta_{123}=1$.

##  II Geometric optics inside a nonlinear dielectric

To begin with, recall that Maxwell equations inside a material medium can be
written, in a Cartesian coordinate system, as

$\begin{array}[]{l}\partial_{i}D_{i}=\rho,\qquad\eta_{ijl}\partial_{j}E_{l}=-\partial_{t}B_{i},\\\\[4.30554pt]
\partial_{i}B_{i}=0,\qquad\eta_{ijl}\partial_{j}H_{l}=J_{i}+\partial_{t}D_{i},\end{array}$
(1)

where $D_{i}$ and $H_{i}$ describe the field excitations and $E_{i}$ and
$B_{i}$ stand for the corresponding electromagnetic field strengths. The
nonlinear behavior of the medium is taken into account by replacing the usual
linear constitutive relations by the following nonlinear ones:

$\displaystyle D_{i}=\epsilon_{0}E_{i}+P_{i},$
$\displaystyle\quad\longrightarrow\quad$ $\displaystyle
D_{i}=\epsilon_{ij}E_{j},$ (2) $\displaystyle
H_{i}=\frac{1}{\mu_{0}}B_{i}-M_{i},$ $\displaystyle\quad\longrightarrow\quad$
$\displaystyle H_{i}=\mu^{{}_{-1}}_{ij}B_{j}.$ (3)

Here, $P_{i}$ is the polarization, $M_{i}$ is the magnetization, and
$\epsilon_{0}$ and $\mu_{0}$ are the vacuum dielectric constants. In general
cases, the coefficients of permittivity $\epsilon_{ij}$ and the inverse
permeability $\mu^{{}_{-1}}_{ij}$ both can depend on the fields $E_{i}$ and
$B_{i}$. Note that extra terms accounting for the electric/magnetic excitation
due to magnetic/electric field strength can both be included, but we shall not
deal with this here (see Ref. [10] for details).

In the regime of geometric optics, the field strengths, the charge $\rho$ and
the current $J_{i}$ are continuous, but their derivatives may have a finite
step through the wavefront $\Sigma_{t}(x_{i})=\mbox{const.}$, for a given
instant of time $t$. According to Hadamard’s theorem [11, 12],
$\left[E_{i}\right]_{\Sigma}=0=\left[B_{i}\right]_{\Sigma}$ implies

$\begin{array}[]{l}\left[\partial_{t}E_{i}\right]_{\Sigma_{t}}=-\omega\,e_{i},\quad\left[\partial_{j}E_{i}\right]_{\Sigma_{t}}=e_{i}\,k_{j},\\\\[4.30554pt]
\left[\partial_{t}B_{i}\right]_{\Sigma_{t}}=-\omega\,b_{i},\quad\left[\partial_{j}B_{i}\right]_{\Sigma_{t}}=b_{i}\,k_{j},\\\\[4.30554pt]
\end{array}$ (4)

where the symbol $\left[f\right]_{\Sigma_{t}}(p)\equiv\lim_{\delta\to
0^{+}}[f(p_{+})-f(p_{-})]$ indicates how the step of the electromagnetic
quantities are evaluated on $\Sigma_{t}$, with $p\in\Sigma_{t}$ and
$p_{\pm}\in\Sigma_{t\pm\delta}$ such that $\lim_{\delta\to 0^{+}}p_{\pm}=p$.
We also denote the wave frequency by $\omega$ and the wave vector by $k_{i}$.
The polarization modes of the electric and magnetic components of the light
rays are represented by $e_{i}$ and $b_{i}$, respectively.

Once applied to the Maxwell equations (1), the procedure described above
yields a linearly polarized wave with the magnetic polarization being given by
$\omega b_{i}=\eta_{ijl}k_{j}e_{l}$ , while the electric polarization $e_{i}$
satisfies the eigenvalue problem [13]

$Z_{ij}e_{j}=0,$ (5)

where $Z_{ij}$ represent the components of the Fresnel matrix [14]. The
existence of nontrivial solutions for this eigenvalue problem is equivalent to
the requirement

$\det[Z_{ij}]=0.$ (6)

From this equation, the dispersion relation associated with the light rays can
then be obtained and the nonlinear behavior of the system is encoded in the
dielectric parameters.

It is worth to mention that the vectors $E_{i}$ and $B_{i}$ represent the
total field, that is, the composition of an external field plus a wave field.
As long the limit of geometric optics is valid, the fields associated with the
propagating waves are much weaker than the controllable fields—those produced
by a given source distribution in the medium or by imposed external fields.
Thus, wave fields were consistently neglected in these sums.

The majority of the optical media respond linearly to external magnetic
fields. This means that the permeability can be considered as
$\mu^{{}_{-1}}_{ij}=(1/\mu_{0})\delta_{ij}$, where $\mu_{0}$ is the vacuum
permeability and $\delta_{ij}$ is the Kronecker delta, which is either $1$ for
$i=j$ or $0$ otherwise. Besides, we shall consider the simplest case of an
isotropic nonlinear electric medium such that the permittivity matrix is the
same for all spatial directions and depends only on the norm
$E=\sqrt{E_{i}E_{i}}$ of the electric field, that is,
$\epsilon_{ij}=\epsilon(E)\delta_{ij}$. In this case, the Fresnel matrix is
given by [15]

$Z_{ij}=\epsilon\delta_{ij}+\frac{\epsilon^{\prime}}{E}E_{i}E_{j}-\frac{1}{\mu_{0}v^{2}}(\delta_{ij}-\hat{k}_{i}\hat{k}_{j}),$
(7)

where $\epsilon^{\prime}=d\epsilon/dE$, $v=\omega/k$ is the phase velocity and
$\hat{k}_{i}=k_{i}/k$, with $k=\sqrt{k_{i}k_{i}}$.

Straightforward calculations show that Eq. (6) can be written as a polynomial
equation for the phase velocity and a simple manipulation of such equation
casts it as a pair of equations of the form $g^{\mu\nu}k_{\mu}k_{\nu}=0$, from
which we can read out the effective optical metrics [16, 17]. With Greek
indices running from $0$ to $3$ and the Minkowski background metric in
Cartesian coordinates taking the form $[\eta_{\mu\nu}]={\rm
diag}(-1,+1,+1,+1)$, these effective metric matrices can be written as

$\displaystyle
g^{(+)}_{\alpha\beta}=\gamma_{\alpha\beta}+\left(1-\frac{\textstyle{1}}{\textstyle{\mu_{0}\epsilon}}\right)v_{\alpha}v_{\beta},$
(8) $\displaystyle
g^{(-)}_{\alpha\beta}=\gamma_{\alpha\beta}+\left[1-\frac{\textstyle{1}}{\textstyle{\mu_{0}(\epsilon+\epsilon^{\prime}E)}}\right]v_{\alpha}v_{\beta}-\frac{\textstyle{\epsilon^{\prime}E}}{\textstyle{\epsilon+\epsilon^{\prime}E}}\hat{l}{}_{\alpha}\hat{l}{}_{\beta},$
(9)

where $\gamma_{\alpha\beta}$ denotes the Minkowski metric in an arbitrary
coordinate system, $v^{\mu}=\delta^{\mu}_{0}$ represents the class of
observers comoving with the laboratory, that is, it measures the electric and
magnetic components of the electromagnetic fields, and $\hat{l}{}^{\alpha}$ is
a unit vector pointing along the direction of the electric field. The
effective metric $g^{(+)}_{\alpha\beta}$ is the well-known Gordon metric [1,
16] and it easily is recognizable by its isotropicity. This metric is
responsible for describing the propagation of the ordinary modes. On the other
hand, the optical metric $g^{(-)}_{\alpha\beta}$ does depend on the nonlinear
behavior of the dielectric and it shall be responsible for the propagation of
the extra-ordinary modes. This is the metric we are interested in.

It is easy to see that, in the particular case of vacuum, where the dielectric
coefficients are $\epsilon_{0}$ and $\mu_{0}$, these two metrics degenerate to
a unique optical metric $g_{\mu\nu}$ given by the Minkowski one. The relevance
of these optical metrics lies on the fact that the integral curves of the wave
vector $k_{\mu}$ correspond to null geodesic curves in the artificial
spacetime endowed with a geometry given by the effective optical metric [18,
19]. As far as solely kinematic aspects of the gravitation are considered, the
effective geometry could be compared with the metric of the curved spacetime.
Thus, the sort of phenomena predicted in gravitational systems could be
scrutinized in the realm of optics inside material media. There are many
applications of this analogy including tests of bending of light, cosmological
models in terrestrial laboratories, or the probe of predictions in quantum
gravity phenomenology [5].

##  III Static and spherically symmetric optical metrics

Let us consider a dielectric material with the features described above,
subjected to an external radial electric field and without external magnetic
fields. For convenience, we choose coordinates adapted to the observers such
that the four-velocity is $v^{\alpha}=\delta^{\alpha}_{0}$. Then, the static
and spherically symmetric situation we are dealing with admits solely a non-
vanishing charge density $\rho$. Therefore, Maxwell equations (1) essentially
reduce to

$\frac{\partial_{r}(r^{2}\epsilon E)}{r^{2}}=\rho.$ (10)

Thus, the effective geometry Eq. (9) expressed in spherical-like coordinates
$(t,r,\theta,\phi)$ reads

$g_{\alpha\beta}={\rm
diag}\left(-\frac{\textstyle{1}}{\textstyle{\mu_{0}(\epsilon+\epsilon^{\prime}E)}},\,\frac{\epsilon}{\epsilon+\epsilon^{\prime}E},\,r^{2},\,r^{2}\sin^{2}\theta\right).$
(11)

This expression allows one to seek for analogue static and spherically
symmetric metrics of the following form

$g_{\alpha\beta}={\rm
diag}\left(-e^{2\Phi(r)},\,\frac{\textstyle{1}}{\textstyle{1-\frac{b(r)}{r}}},\,r^{2},\,r^{2}\sin^{2}\theta\right).$
(12)

Inspired by the Morris-Thorne Ansatz concerning wormholes [20], the redshift
function $\Phi(r)$ and the shape function $b(r)$ are enough to describe any
static and spherically symmetric spacetime in a unified form. In particular,
the well-known solutions of Einstein equations such as Schwarzschild,
Reissner-Nordström, de-Sitter as well as the class of Morris-Thorne wormholes.
The matching of Eqs. (11) and (12) gives the two independent equations

$\displaystyle\mu_{0}(\epsilon+\epsilon^{\prime}E)=e^{-2\Phi},$ (13)
$\displaystyle\frac{\epsilon+\epsilon^{\prime}E}{\epsilon}=1-\frac{b}{r}.$
(14)

Once combined, these conditions provide the radial dependence of the
dielectric permittivity, that is

$\epsilon(r)=\epsilon_{0}\,e^{-2\Phi}\left(1-\frac{b}{r}\right)^{-1},$ (15)

where we used the identification $\mu_{0}^{-1}=\epsilon_{0}$, since the speed
of light in vacuum is set to unity. Substitution of Eq. (15) back in Eq. (13)
yields the electric field in terms of the radial coordinate, as follows

$E(r)=E_{0}\left|\frac{r}{b}-1\right|\exp\left(2\int\frac{r}{b}\,\frac{d\Phi}{dr}dr\right),$
(16)

where $E_{0}$ is an integration constant. This equation is valid for any signs
of $r-b$ and $b$. However, we are mainly interested in the case $r-b\geq 0$,
in order to preserve the Lorentzian signature of metric in Eq. (12).

Finally, the expressions of the electric displacement $D=\epsilon E$ and the
charge density $\rho$ in terms of $r$ read as

$\displaystyle D$ $\displaystyle=$
$\displaystyle\epsilon_{0}E_{0}\,\frac{\textstyle{r}}{\textstyle{b}}\exp{\left[2\int\left(\frac{\textstyle{r}}{\textstyle{b}}-1\right)\,\frac{\textstyle{d\Phi}}{\textstyle{dr}}dr\right]},$
(17) $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\epsilon_{0}E_{0}\,\left[\frac{\textstyle{3}}{\textstyle{b}}-\frac{\textstyle{r}}{\textstyle{b^{2}}}\frac{\textstyle{db}}{\textstyle{dr}}+2\,\frac{\textstyle{d\Phi}}{\textstyle{dr}}\left(\frac{\textstyle{r}}{\textstyle{b}}-1\right)\right]\times$
(18)
$\displaystyle\times\exp{\left[2\int\left(\frac{\textstyle{r}}{\textstyle{b}}-1\right)\,\frac{\textstyle{d\Phi}}{\textstyle{dr}}dr\right]}.$

Therefore, for any choice of the functions $\Phi(r)$ and $b(r)$ describing the
gravitational metric to be mimicked, the corresponding analogue model can be
constructed, at least theoretically, via equations (15)-(18). In what follows,
two important cases of static and spherically symmetric geometries in the
context of general relativity are considered. Some possible experimental
limitations for reproducing such metrics in laboratory are also discussed
latter, by assuming a nonmagnetic and nonlinear isotropic material.

###  III.1 Black hole analogue models

First, consider the case in which the functions $\Phi(r)$ and $b(r)$ correlate
as

$\Phi(r)=\frac{1}{2}\ln\left(1-\frac{b(r)}{r}\right).$ (19)

This choice leads to a gravitational metric of the form given in Eq. (12)
where the condition $g_{00}=-1/g_{11}$ holds. This condition includes all
black hole metrics like Schwarzschild, Kottler and Reissner-Nordström (De
Sitter and anti-De Sitter spacetimes are also solutions of this, but without
event horizons).

Accordingly, when Eqs. (15)-(18) are restricted to the case of Eq. (19), the
corresponding analogue model presents the following electromagnetic
configuration

$\begin{array}[]{lcl}\epsilon(r)&=&\epsilon_{0}\left(1-\frac{\textstyle{b}}{\textstyle{r}}\right)^{-2},\\\\[4.30554pt]
E(r)&=&E_{0}\left(\frac{\textstyle{r}}{\textstyle{b}}-1\right)^{2},\\\\[4.30554pt]
D(r)&=&\frac{\textstyle{\epsilon_{0}E_{0}r^{2}}}{\textstyle{b^{2}}},\\\\[4.30554pt]
\rho(r)&=&\frac{\textstyle{2\epsilon_{0}E_{0}r^{2}}}{\textstyle{b^{2}}}\left(\frac{\textstyle{2}}{\textstyle{r}}-\frac{\textstyle{1}}{\textstyle{b}}\frac{\textstyle{db}}{\textstyle{dr}}\right).\end{array}$
(20)

In particular, for this case one has

$\epsilon(E)=\epsilon_{0}\left(\sqrt{\frac{E_{0}}{E}}\pm 1\right)^{2}.$ (21)

Thus, the permittivity diverges as $E$ approaches zero. Therefore, we have a
large permittivity for small values of $E$, maintaining always the electric
displacement finite (see Eqs. 20). This cannot be achieved by using materials
one can find in nature, but possibly with short-band human made media like
metamaterials. Besides, the physically reasonable expression for $\epsilon(E)$
is the one with the plus sign, avoiding the situation $\epsilon|_{E_{0}}=0$.

###  III.2 Wormhole analogue models

The (intuitive) conditions for the existence of a wormhole without an event
horizon are that $\Phi(r)$ is finite everywhere and $b(r_{0})=r_{0}$ for some
$r=r_{0}>0$ [20]. This implies that the permittivity from Eq. (15) diverges at
the throat, while the electric field (16) goes to zero, at least as far as the
argument of the exponential remains finite. For the sake of illustration, take
the special class of Morris-Thorne wormholes with zero redshift function.
Thus,

$\begin{array}[]{lcl}\epsilon(r)&=&\epsilon_{0}\left(1-\frac{\textstyle{b}}{\textstyle{r}}\right)^{-1},\\\\[4.30554pt]
E(r)&=&E_{0}\left(\frac{\textstyle{r}}{\textstyle{b}}-1\right),\\\\[4.30554pt]
D(r)&=&\frac{\textstyle{\epsilon_{0}E_{0}r}}{\textstyle{b}},\\\\[4.30554pt]
\rho(r)&=&\epsilon_{0}E_{0}\,\left(\frac{\textstyle{3}}{\textstyle{b}}-\frac{\textstyle{r}}{\textstyle{b^{2}}}\frac{\textstyle{db}}{\textstyle{dr}}\right).\end{array}$
(22)

The elimination of the variable $r$ from the first two relations above yields

$\epsilon(E)=\epsilon_{0}\left(\frac{E_{0}}{E}+1\right).$ (23)

Although the equations are slightly different from the previous case, the
conclusion is quite similar: a divergent permittivity and a finite electric
displacement as the electric field goes to zero, which happens at the throat
at least. Again, metamaterials would be required in order to reproduce this
sort of geometries.

###  III.3 Power-law media

The previous subsections have shown that any gravitational black-hole or
wormhole (with zero redshift function) cannot be reproduced in laboratory by
conventional materials - those media whose electric permittivity is
proportional to a positive power of the magnitude of the electric field, i.e.
$\epsilon\propto E^{\alpha}$ with $\alpha>0$. Then, the natural question
raises: what kind of metrics can be produced with conventional media? Thus,
consider a power law relation between the electric permittivity and the
electric field. With this assumption we cover the well-known Pockels and Kerr
effects, first and second order corrections in $E$, respectively, to the
electric susceptibility of a given medium.

Thus, from the substitution of

$\epsilon(E)=\epsilon_{0}\left(\frac{\textstyle{E}}{\textstyle{E_{0}}}\right)^{\alpha},$
(24)

with $\alpha>0$ in Eqs. (15) and (16), it follows that

$\Phi=-\frac{1}{2}\ln\left(1-\frac{b}{r}\right),$ (25)

for any $\alpha>0$. Please, note a sign on comparing this with Eq. (19).
Returning back to Eqs. (15) and (16), one finds out that $\epsilon$ and $E$
are actually constant and that the effective metric reduces to the flat
Minkowski spacetime. Once again, nontrivial effective metrics that are static
and spherically symmetric demand a more ellaborated relationship between
$\epsilon$ and $E$.

###  III.4 Neutral dielectric media

Finally, assume that the charge density vanishes identically, such that the
profile of the electric displacement is directly obtained from Gauss law, that
is

$D=\frac{k}{r^{2}},$ (26)

where $k$ is an integration constant. The comparison of this pattern with the
product $\epsilon E$ given by Eqs. (15) and (16) yields a first order
differential equation involving $\Phi(r)$ and $b(r)$, which can be solved,
leading to

$\Phi(r)=-\frac{1}{2}\ln\left(1-\frac{b}{r}\right)+\ln\frac{r}{r_{0}}-\int\frac{dr}{(r-b)}.$
(27)

The last two terms on the right-hand side of this relation distort the
solution in Eq. (25), yielding a nontrivial effective metric. The dielectric
parameter and the electric field strength are, respectively, given by

$\begin{array}[]{lcl}\epsilon(r)&=&\epsilon_{0}\left(\frac{\textstyle{r_{0}}}{\textstyle{r}}\right)^{2}\exp\left(2\int\frac{dr}{(r-b)}\right),\\\\[8.61108pt]
E(r)&=&E_{0}\exp\left(-2\int\frac{dr}{(r-b)}\right).\end{array}$ (28)

From this, it is clear that $k=\epsilon_{0}E_{0}r_{0}^{2}$. Moreover, the
permittivity is $\epsilon=\epsilon_{0}E_{0}r_{0}^{2}\,r^{-2}E^{-1}$. The
actual dependence of $\epsilon$ on $E$ requires the explicit behavior of
$E(r)$ to be algebraically inverted, which in turns demands the specification
of the shape function $b(r)$.

##  IV Hyperbolicity and causal structure

In this Section, the interconnected issues of hyperbolicity and causal
structure for the models described so far are briefly discussed. Hyperbolicity
lies at the roots of physics, since it amounts to the predictable power of the
theory by asserting that solutions exist, are unique and depend continuously
on the initial data. For physical models described by quasi-linear partial
differential equations with constraints, as is the case under investigation,
additional care is needed since the principal part of the equations will
depend explicitly on the electromagnetic fields. Roughly speaking, this means
that the Cauchy problem may be well-posed for some field intensities while
ill-posed for others. Needless to say, if hyperbolicity was to be violated,
one could have dispersion relations endowed with pathologies, which would
compromise the propagation of waves assumed at the very beginning of our
analysis. These pathologies are often associated with non-real eigenvalues of
the characteristic polynomial (evanescent modes) or cones of influence with
non-convex topologies.

The hyperbolicity of Maxwell equations with local, but otherwise arbitrary,
constitutive laws has been analyzed in [21, 10]. In particular, one
inescapable condition for well-posedness to hold for Eqs. (1)-(3) is that, at
each spacetime point, the homogeneous multivariate polynomial of fourth order

$P(\xi)\equiv(g_{\alpha\beta}^{(+)}\xi^{\alpha}\xi^{\beta})(g_{\alpha\beta}^{(-)}\xi^{\alpha}\xi^{\beta})$
(29)

is hyperbolic with respect to a direction in the tangent space [22, 23]. In
other words, $\exists\ w^{\mu}\in T_{p}M$ such that $P(w)>0$ and the map
$\lambda\mapsto P(u+\lambda w)$, itself a univariate polynomial of fourth
order, has only real roots $\lambda_{i}$, for all $u^{\mu}\in T_{p}M$.
Geometrically, hyperbolicity in the direction of $w^{\mu}$ is the requirement
that every line parallel to $w^{\mu}$ in $T_{p}M$ intersects the null cones of
$g_{\alpha\beta}^{(+)}$ and $g_{\alpha\beta}^{(-)}$ at exactly four points
(counting multiplicities). Therefore, one obtains the conditions

  1. 1.

the effective metrics $g_{\alpha\beta}^{(+)}$ and $g_{\alpha\beta}^{(-)}$ both
must have a Lorentzian signature equal to $(-,+,+,+)$;

  2. 2.

the set of future-directed time-like vectors with respect to both metrics must
form a unique connected, open, convex set.

Needless to say, if the above requirements are satisfied, the causal structure
of the theory will be well-behaved: wave excitations or discontinuities over
any background configuration will propagate in a predictable way with a finite
velocity of propagation.

Now, a direct calculation using Eqs. (9) shows that the effective metrics will
be Lorentzian, whenever the following inequalities are satisfied

$g^{(+)}=\frac{\gamma}{\mu_{0}\epsilon}<0,\qquad
g^{(-)}=\frac{\gamma}{\mu_{0}\epsilon(1+\frac{\epsilon^{\prime}E}{\epsilon})^{2}}<0.$
(30)

where $g^{(\pm)}=\det\left(g_{\alpha\beta}^{(\pm)}\right)$ and
$\gamma\equiv\mbox{det}(\gamma_{\alpha\beta})$. The latter imply that, in
order to preserve the correct signature, the following two conditions must be
fulfilled:

$\epsilon>0,\quad\quad\quad\epsilon+\epsilon^{\prime}E>0.$ (31)

These two conditions are automatically satisfied by all spherically symmetric
models described so far. That condition 2 above is also fulfilled is
immediate. Since the class of observers comoving with the laboratory,
$v^{\mu}$, is time-like with respect to both effective metrics then, by
continuity, there will be an open set of time-like vectors containing
$v^{\mu}$ which is necessarily connected, open and convex, as required. Hence,
these models are not plagued by mathematical inconsistencies.

Figure 1: Null cones of $g^{(+)}_{\mu\nu}$ are inside the Minkowski light
cones, shown as a function of $r$. The vertical plane on left represents the
worldsheet of the horizon, i.e. $r=r_{H}$, for which the effective light cones
degenerate. As the $r$-coordinate grows, the vertices of the other light cones
are placed at $2r_{H},3r_{H}$ and $4r_{H}$, respectively. Also,
$\mu_{0}^{-1}=\epsilon_{0}=1=E_{0}$ were adopted for simplicity. Notice the
isotropic pattern of the propagation. Figure 2: Null cones of the effective
metric $g^{(-)}_{\mu\nu}$ also are inside the Minkowski light cones, here
shown as a function of $r$. All numerical choices are the same as in Fig. 1.
The anisotropic pattern is due to the term $\epsilon^{\prime}E$ in the
effective metric. Since this term vanishes for large values of $r$, the
effective cones gradually approach Minkowski light cones from inside.

Another interesting feature of the effective metrics, Eqs. (8)-(9), is that
either their null cones coincide or they share two (and only two) common
directions. Indeed, in Cartesian coordinates adapted to the laboratory frame
i.e., $v^{\mu}=\delta^{\mu}_{0}$ and $\gamma_{\mu\nu}=\eta_{\mu\nu}$, we may
write $\xi^{\mu}=(\xi^{0},\vec{\xi})$ and the corresponding null cone
conditions read as

$(\xi^{0})^{2}-\mu_{0}\epsilon(\vec{\xi}\cdot\vec{\xi})=0,\qquad(\xi^{0})^{2}-\mu_{0}(\epsilon+\epsilon^{\prime}E\mbox{sin}^{2}\psi)(\vec{\xi}\cdot\vec{\xi})=0$
(32)

with $\psi$ denoting the angle between $\vec{\xi}$ and the electric field
$\vec{E}$. Therefore, if $\epsilon^{\prime}E=0$, the cones are precisely the
same, whereas if $\epsilon^{\prime}E\neq 0$, these cones agree along the
directions of $\pm\vec{E}$. This means that, in a static and spherically
symmetric situation, it would be impossible to distinguish between the two
effective geometries, and consequently between the ordinary and the extra-
ordinary light rays, by solely studying the behavior of radial light
propagation.

For a concrete example, consider the causal structure in the simple case of a
Schwarzschild geometry, for which $b(r)=r_{H}$. A direct computation using
Eqs. (20) gives the following electromagnetic quantities

$\begin{array}[]{lcl}\epsilon(r)&=&\epsilon_{0}\left(1-r_{H}/r\right)^{-2},\\\\[4.30554pt]
E(r)&=&E_{0}\left(r/r_{H}-1\right)^{2},\\\\[4.30554pt]
D(r)&=&\epsilon_{0}E_{0}\left(r/r_{H}\right)^{2},\\\\[4.30554pt]
\rho(r)&=&4\epsilon_{0}E_{0}(r/r^{2}_{H}),\end{array}$ (33)

which are finite in the regime $r_{H}<r<\infty$. Thus, one concludes that the
electric permittivity is a positive definite, monotonically decreasing
function, diverging as $r\rightarrow r_{H}$ and approaching $\epsilon_{0}$ as
$r\rightarrow\infty$. Conversely, the electric field intensity vanishes as
$r\rightarrow r_{H}$ and diverges as $r\rightarrow\infty$. Also, by noticing
that

$\epsilon^{\prime}E=\left(\frac{d\epsilon}{dr}\right)\left(\frac{dE}{dr}\right)^{-1}E=\epsilon_{0}\frac{r_{H}}{r}\left(1-\frac{\textstyle{r_{H}}}{\textstyle{r}}\right)^{-2}$
(34)

one easily shows that the anisotropic term, given by $\epsilon^{\prime}E$,
vanishes as $r\rightarrow\infty$, as expected. Therefore, the effective null
cones, Eqs. (32), coincide at spatial infinity and increasingly disagree as
the radial coordinate is gradually decreased. Nevertheless, the two cones
always intersect for rays propagating along the radial direction, as discussed
before. Clearly, both cones coincide with the Minkowski light cone for $r\gg
r_{H}$, and they degenerate at $r=r_{H}$, which is avoided here by considering
only the exterior region.

The qualitative behavior of the light cones for both effective metrics are
depicted in Figs. (1)-(2) for different values of the radii, that is,
$r/r_{H}=1,2,3,4$. Figure (1) displays the effectively causal cones defined by
$g_{\mu}^{(+)}$. The effective light cone lies inside the Minkowski background
causal cone, and approaches it as $r$ goes to infinity. The isotropic feature
of the effective cones is evident. In the case of $g_{\mu}^{(-)}$ in Fig. (2),
the same qualitative behavior for the effective light cones of
$g_{\mu\nu}^{(-)}$ for the same values of the parameter, despite their
anisotropy.

##  V Concluding Remarks

Most of the literature on analogue models present some interesting properties
of a certain kind of material medium, without actually providing what is the
specific feature of the material which allows the obtained behavior. The
present work is a first step to construct a global picture, by discussing what
it can be obtained, and what cannot, from these properties. In particular, it
was shown that most of the non standard optical properties of non magnetic
isotropic dielectrics demand a medium whose electric permittivity diverges in
the limit of $E\rightarrow 0$. The discussion of the causal structure provides
the basic requirements on the effective metrics in order to mimic some
reliable spacetime geometry.

###### Acknowledgements.

DRS would like to thank Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES) for the financial support.

## References

  * Gordon [1923] W. Gordon, Zur lichtfortpflanzung nach der relativitätstheorie, Annalen der Physik 72, 421 (1923).
  * de Felice [1971] F. de Felice, On the gravitational field acting as an optical medium, Gen. Rel. and Grav. 2, 1396 (1971).
  * Plebanski [1960] J. Plebanski, Electromagnetic waves in gravitational fields, Phys. Rev. 118, 1396 (1960).
  * Unruh [1981] W. G. Unruh, Experimental black-hole evaporation?, Phys. Rev. Lett. 46, 1351 (1981).
  * Barceló  _et al._ [2005] C. Barceló, S. Liberati, and M. Visser, Analogue gravity, Living Reviews in Relativity 8, 12 (2005).
  * Novello  _et al._ [2002] M. Novello, M. Visser, and G. Volovik,  _Artificial Black Holes_ (World Scientific, 2002).
  * Pendry  _et al._ [2006] J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312, 1780 (2006).
  * Smith  _et al._ [2000] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84, 4184 (2000).
  * Fleischhauer  _et al._ [2005] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys. 77, 633 (2005).
  * Perlick [2011] V. Perlick, On the hyperbolicity of maxwell’s equations with a local constitutive law, Journal of Mathematical Physics 52, 042903 (2011).
  * Hadamard [1903] J. Hadamard,  _Leçons sur la propagation des ondes et les équations de hydrodynamique_ (Hermann, Paris, France, 1903).
  * Zakharov [1973] V. D. Zakharov,  _Gravitational waves in Einstein’s theory_ (John Wiley & Sons, New York, USA, 1973).
  * De Lorenci and Klippert [2006] V. A. De Lorenci and R. Klippert, Electromagnetic light rays in local dielectrics, Physics Letters A 357, 61 (2006).
  * De Lorenci and Goulart [2008] V. A. De Lorenci and G. P. Goulart, Magnetoelectric birefringence revisited, Physics Review D 78, 045015 (2008).
  * Bittencourt  _et al._ [2017] E. Bittencourt, G. H. S. Camargo, V. A. De Lorenci, and R. Klippert, Controlled opacity in a class of nonlinear dielectric media, Phys. Rev. A 95, 033826 (2017).
  * Novello and Bittencourt [2012] M. Novello and E. Bittencourt, Gordon metric revisited, Physical Review D 86, 124024 (2012).
  * Bittencourt  _et al._ [2016] E. Bittencourt, J. P. Pereira, I. I. Smolyaninov, and V. N. Smolyaninova, The flexibility of optical metrics, Classical and Quantum Gravity 33, 165008 (2016).
  * De Lorenci and Klippert [2002] V. A. De Lorenci and R. Klippert, Analogue gravity from electrodynamics in nonlinear media, Physical Review D 65, 064027 (2002).
  * De Lorenci [2002] V. A. De Lorenci, Effective geometry for light traveling in material media, Physical Review E 65, 026612 (2002).
  * Morris and Thorne [1988] M. S. Morris and K. S. Thorne, Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity, American Journal of Physics 56, 395 (1988).
  * Rätzel  _et al._ [2011] D. Rätzel, S. Rivera, and F. P. Schuller, Geometry of physical dispersion relations, Phys. Rev. D 83, 044047 (2011).
  * GÅRDING [1959] L. GÅRDING, An inequality for hyperbolic polynomials, Journal of Mathematics and Mechanics 8, 957 (1959).
  * Beig [2006] R. Beig, Concepts of hyperbolicity and relativistic continuum mechanics, in  _Analytical and Numerical Approaches to Mathematical Relativity_, edited by J. Frauendiener, D. J. Giulini, and V. Perlick (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp. 101–116.