Photon propagation in a material medium on a curved spacetime Amanda GuerrieriandM ario Novello

2025-05-02 0 0 496.39KB 21 页 10玖币
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Photon propagation in a material medium on a curved
spacetime
Amanda Guerrieri∗and M´ario Novello†
CBPF - Centro Brasileiro de Pesquisas F´ısicas
R. Dr. Xavier Sigaud, 150 - Urca, Rio de Janeiro - RJ, 22290-180
Abstract
We consider a nonlinear dielectric medium surrounding a static, charged and spherically symmet-
ric compact body which gravitational field is driven by General Relativity (GR). Considering the
propagating waves on the dielectric medium, we describe the trajectory of light as geodesics on an
effective geometry given by Hadamard’s discontinuities. We analyze some consequences of the effective
geometry in the propagation of light, with relation to the predictions of the background gravitational
field, that includes corrections on the geometrical redshift and on the gravitational deflection of light.
We show that the background electromagnetic field polarize the material medium, such that different
polarizations of light are distinguished by different corrections on these quantities. As a consequence,
we have two possible paths for the trajectory of light in such configuration, that coincide if we turn off
the electromagnetic field or if the permittivity is constant. We show that the effective metric associated
to the negative polarization, for a given dependence of the dielectric permittivity, is conformally flat.
∗amguerrieri@cbpf.br
†mnovello42@gmail.com
1
arXiv:2210.02634v1 [gr-qc] 6 Oct 2022
Contents
1 Introduction 3
2 Effective geometry 4
2.1 Propagating waves on a dielectric medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Dielectric medium on curved spacetimes 8
3.1 Effectivepotential ........................................ 9
3.2 Gravitational deflection of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Geometricalredshift ....................................... 10
3.4 Opticalhorizons ......................................... 11
4 Special cases 12
4.1 The class of material mediums with f=const. ........................ 12
4.1.1 TheGordoncase..................................... 12
4.1.2 The case f+=const. ................................... 12
4.2 A particular case for the permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Erasing the gravitational effect on photon propagation . . . . . . . . . . . . . . . . . . . . 16
5 Conclusions 17
2
1 Introduction
Dielectric mediums have been extensively studied in literature, both theoretically [1
–
5] and experimentally
[6
–
9]. In the past years, analogue models of gravity [10
–
20] considering a nonlinear dielectric medium
received attention in the scientific community [21
–
39]. These models predicted that nonlinear materials
can produce, in laboratory, an effective (optical) horizon with similar properties to the event horizon of
black holes [40]. At these horizons, light is trapped, such that a ring could be observed due to birefringence
phenomenon [41
–
44]. Mathematically, this is described by an effective geometry [21,24,33,34,36,38]
generated by the medium, such that light follows geodesics in this effective geometry, and it determines
the position of the optical horizons. In this paper, we are interested in applying this method to curved
spacetimes. For that, instead of considering the flat spacetime of a laboratory as the background geometry,
we are going to consider a curved background generated by a spherically symmetric object. This choice is
motivated by studies concerning a system formed by black holes surrounded by a plasma [45,46]. If this
medium has nonlinear properties, we can describe its effects on the propagation of light by means of an
effective geometry, in a similar manner to analogue models.
In order to apply this method to a system formed by a compact object surrounded by a nonlinear
electromagnetic material, we are going to consider a dielectric medium
µ
=
const.
,
=
(
E
) with negligible
mass, compared to the compact object, such that we can idealize a fixed background. In this paper, we
show that this simple model is enough to predict some important consequences in the propagation of light,
that includes corrections on the geometrical redshift and on the light deflection by a compact body. If the
medium surround a charged body, we can have two polarization modes for light, such that this model
gives a mechanism to evaluate the consequences of birefringence effects on curved spacetimes. Each mode
is associated to a different correction on these quantities, such that we can have two possible paths for the
light ray. These paths only coincide if the electrical field is turned off or if the permittivity is constant.
The vacuum appears as a particular case in this configuration.
In this paper, what we call event horizon depends only on the background metric, and what we call
killing horizon depends only on the effective metric, such that we are going to refer to it as an optical
horizon. Formally, only in the vacuum we could use the background metric to evaluate the optical horizon,
because in that case the effective metric and the background metric are equivalent. This is a consequence
of the definition of a killing horizon [47,48], which is described as the radius where the redshift diverges
and the time killing vector changes its sign, implying that we need to consider the effective metric to
determine it. We proved that the position of the optical horizons coincide with the position of the killing
horizons predicted by the background metric. This is a direct consequence of the form of the effective
metric.
The paper is organized as follows: In section 2we review Hadamard’s method of discontinuities and
construct the effective geometry for a nonlinear dielectric medium, in order to discuss birefringence effects
on flat spacetimes. In section 3, we apply this method to a spherically symmetric curved background
and show its consequences on the propagation of light. In section 4, we consider three particular cases
for our analyzes. The first, is a special case that doesn’t produce alterations on either the geometrical
redshift or the gravitational deflection of light. However, it changes the effective potential of light, which
coincides with the vacuum predictions only if
=
0
. The second, is a less restrict case that produce
alterations on the effective potential and on the deflection angle of light, but maintain the predictions for
the geometrical redshift of the positive polarization. The last, is a general case that account all possible
consequences on curved spacetimes. Both can be reduced to the first case in the particular situation where
the permittivity is constant. Finally, we end our analyzes showing that the effective metric associated to
the negative polarization can be written in a conformally flat form, for a given dependence of the dielectric
3
permittivity. In Section 5, we display our conclusions.
In this article, we consider the notation for the partial derivatives
¯
A
=
∂A
∂E
and
A0
=
∂A
∂H
. A
Minkowskian spacetime is used in subsection 2.1 as the background metric
γµν
and it has signature
(+
,−,−,−
). All quantities are refereed as measured by the observer
vµ
. In particular, we consider the
definition
˙
Xµ
=
Xµ
;νvν
. For an arbitrary vector
Xµ
= (0
,~
X
) we define its modulus by considering the
relation
X
= (
−XµXµ
)
1/2
and its associated unit vector as
ˆ
X
=
~
X/X
. For any two quantities X and
Y we denote its scalar product
XµYµ
following the notation (
X.Y
). The projection tensor is defined as
hµν
=
gµν −vµvν
and Kronecker tensor is represented by
δµ
ν
. Additionally, we consider Levi-civita tensor
to be defined as ηλγβσ =−1
√−gλγβσ, where gis the determinant of the background metric.
2 Effective geometry
In order to find an effective geometry associated with the propagation of light inside a dielectric medium,
we are going to consider Hadamard method [49,50]. Let us define a surface of discontinuity Σ(
xµ
) =
const.
which delimit locally two regions of the spacetime, represented by 1 and 2. Given a function
f
, we call
f(1)
and
f(2)
the values taken by the function on each domain. Hadamard’s discontinuity of the function
f, with relation to the surface Σ, is defined as
[f(x)] |Σ= lim
→0+f(1)(x+)−f(2)(x−),(2.1)
such that the point
x
belongs to the surface. We suppose that
f
is continuous on the surface Σ, but it’s
first derivatives f,α don’t
[f]|Σ= 0 ,(2.2)
[f,α]|Σ6= 0 .(2.3)
Considering the differentials of the function in both regions and knowing that the shift vector
dxα
belongs
to the surface Σ, we find
df(i)=∂αf(i)dxα,(2.4)
with
i
=
{
1
,
2
}
representing both regions. Hadamard showed that these differentials should exist and be
continuous on the surface, i.e., [
df
]
|Σ
= 0. That is, the discontinuity of the derivatives of
f
must be an
object orthogonal to the surface, as a consequence of
[df]|Σ= [∂αf]|Σdxα= 0.(2.5)
Thus, there exists a scalar
σ
(
x
)
6
= 0 such that [
f,α
]
|Σ
=
σ
(
x
)
kα
with
kα
= Σ
,α
. In the following subsection,
we are going to apply this method to the equations of electrodynamics on a material medium.
2.1 Propagating waves on a dielectric medium
The propagation of light inside a nonlinear medium is described by null geodesics in an effective geometry,
represented by an effective metric. This effective metric yields modifications on the background metric,
associated to the functions of the medium. The constitutive relations of a material medium are given by
Dα=β
α(Eµ, Hµ)Eβ(2.6)
Bα=µβ
α(Eµ, Hµ)Hβ,(2.7)
4
where
β
α
and
µβ
α
represents the permittivity and the permeability tensors of the medium. They relate the
electric field
Eµ
and the magnectic field
Bµ
with the displacement
Dµ
and the auxiliary
Hµ
fields. We
can decompose the tensors that represent the electromagnetic field, its dual and the polarization field, in
terms of an observer’s field vµ, as
Fµν =Eµvν−Eνvµ+ηρσ
µν vρBσ(2.8)
F∗
µν =Bµvν−Bνvµ−ηρσ
µν vρEσ(2.9)
Pµν =Dµvν−Dνvµ+ηρσ
µν vρHσ,(2.10)
in order to apply it to Maxwell’s equations in the absence of sources
∗Fµν;ν
= 0 and
Pµν;ν
= 0. For an
isotropic medium, one is able to write
β
α(Eµ, Hµ) = (E, H)(δβ
α−vβvα)
µβ
α(Eµ, Hµ) = µ(E, H)(δβ
α−vβvα),(2.11)
where
E
and
H
are the modulus of the electromagnetic field. By considering these relations, Maxwell’s
equations become
− Eµ˙vµ− Eα,α +Eα¯
EEλEλ,α +0
HHλHλ,α+ηµνρσ vρ,ν vµHσ= 0 (2.12)
−µ Hµ˙vµ−µ Hα,α +Hα¯µ
EEλEλ,α +µ0
HHλHλ,α−ηµνρσ vρ,ν vµEσ= 0 (2.13)
(˙
Eλ+Eλvν
,ν −Eνvλ
,ν )−Eλ¯
EEλEλ,α +0
HHλHλ,αvα+ηλνρσ (vρ,ν Hσ+vρHσ,ν ) = 0 (2.14)
µ(˙
Hλ+Hλvν
,ν −Hνvλ
,ν )−Hλ¯µ
EEλEλ,α +µ0
HHλHλ,αvα−ηλνρσ (vρ,ν Eσ+vρEσ,ν )=0.(2.15)
Hence, we can consider the definition of Hadamard’s discontinuities. It gives
[Eµ]Σ= 0 ,[Eµ,λ]Σ=eµkλ,
[Hµ]Σ= 0 ,[Hµ,λ]Σ=hµkλ,
where
eµ
and
hµ
represent the discontinuities of the fields on the surface Σ and
kλ
is the wave 4-vector.
When we apply the discontinuities to Maxwell’s equations (2.12-2.15) the terms proportional to
vµ,ν
disappear, since they are proportional to the discontinuities of the electromagnetic field. Therefore, for an
arbitrary observer, we have
 eαkα−Eα¯
EEλeλkα+0
HHλhλkα= 0 (2.16)
µ hαkα−Hα¯µ
EEλeλkα+µ0
HHλhλkα= 0 (2.17)
 eλkαvα−Eλ¯
EEβeβkαvα+0
HHβhβkαvα+ηλνρσ vρhσkν= 0 (2.18)
µ hλkαvα−Hλ¯µ
EEβeβkαvα+µ0
HHβhβkαvα−ηλνρσ vρeσkν= 0 ,(2.19)
where
˙
Aµ
=
Aµ
,ν vν
was written in terms of its discontinuities as [
˙
Aµ
]
Σ
=
aµkνvν
. In order to solve this
system of equations and find the effective geometry associated to the propagation of light inside a dielectric
5
摘要：

PhotonpropagationinamaterialmediumonacurvedspacetimeAmandaGuerrieri*andMarioNovello„CBPF-CentroBrasileirodePesquisasFsicasR.Dr.XavierSigaud,150-Urca,RiodeJaneiro-RJ,22290-180AbstractWeconsideranonlineardielectricmediumsurroundingastatic,chargedandsphericallysymmet-riccompactbodywhichgravitational...

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