pp波模式在电磁有源动态醚中引起的双折射资源-CSDN文库

Open

Access

11 浏览量 2020-04-10 21:06:53 上传评论收藏 551KB PDF 举报

资源推荐

资源详情

资源评论

Eur. Phys. J. C (2017) 77:699

DOI 10.1140/epjc/s10052-017-5299-2

Regular Article - Theoretical Physics

Birefringence induced by pp-wave modes in an

electromagnetically active dynamic aether

Timur Yu. Alpin

, Alexander B. Balakin

Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya street 18, Kazan 420008, Russia

Received: 5 June 2017 / Accepted: 10 October 2017 / Published online: 23 October 2017

Abstract In the framework of the Einstein–Maxwell-

aether theory we study the birefringence effect, which

can occur in the pp-wave symmetric dynamic aether. The

dynamic aether is considered to be a latently birefringent

quasi-medium, which displays this hidden property if and

only if the aether motion is non-uniform, i.e., when the aether

ﬂow is characterized by the non-vanishing expansion, shear,

vorticity or acceleration. In accordance with the dynamo-

optical scheme of description of the interaction between elec-

tromagnetic waves and the dynamic aether, we shall model

the susceptibility tensors by the terms linear in the covari-

ant derivative of the aether velocity four-vector. When the

pp-wave modes appear in the dynamic aether, we deal with

a gravitationally induced degeneracy removal with respect

to hidden susceptibility parameters. As a consequence, the

phase velocities of electromagnetic waves possessing orthog-

onal polarizations do not coincide, thus displaying the bire-

fringence effect. Two electromagnetic ﬁeld conﬁgurations

are studied in detail: longitudinal and transversal with respect

to the aether pp-wave front. For both cases the solutions

are found, which reveal anomalies in the electromagnetic

response on the action of the pp-wave aether mode.

1 Introduction

The effect of birefringence is well documented in the elec-

trodynamics of continuous media [1–5]. This effect reveals

itself, in particular, when the electromagnetic waves possess-

ing two orthogonal polarizations are forced to move with dif-

ferent phase velocities, thus being converted to the so-called

ordinary and extraordinary waves. The medium behaves as

the birefringent one, when the electric and magnetic suscep-

tibility tensors of the medium are anisotropic, i.e., when these

tensors possess non-coinciding eigen-values (the medium is

e-mail: Timur.Alpin@kpfu.ru

e-mail: Alexander.Balakin@kpfu.ru

called bi-axial, if all three eigen-values are different, and

uni-axial, when only two of them coincide). The birefringent

property of the medium can be the intrinsic one (e.g., in the

spatially anisotropic crystals [6], in moving uni-axial media

[7]), or they can be induced by external inﬂuences (e.g., by an

external electric ﬁeld [4,5], a magnetic ﬁeld [8,9], stresses,

anisotropic heating, etc., [6]). When we deal with electro-

magnetic waves propagating under the inﬂuence of the grav-

itational ﬁeld, various versions of the gravitation theory pre-

dict different results. For instance, the pre-metric axiomatic

theory guarantees (see, e.g., [10]) that there is no intrinsic

birefringence. Similarly, the minimal Einstein version of the

theory of gravity excludes birefringence. However, in the

framework of the modiﬁed theories of gravity the effect of

birefringence was predicted by many authors. For instance,

the nonminimal Einstein–Maxwell theory admits the bire-

fringence effect since the coupling of photons to the curva-

ture makes the nonminimal susceptibility tensor anisotropic

(see, e.g., [11–13]). Violation of the Lorentz invariance of

the model [14–18], a torsion nonminimally coupled to pho-

tons [19], interactions with strings [20], also can be the origin

of the birefringence effect. These predictions have attracted

the attention to the problem of cosmic birefringence and its

observations [21–25].

Our goal is to study the birefringence induced by the

dynamic aether. We assume that when the motion of the

aether is uniform, the aether is not birefringent, i.e., the effect

we search for is hidden. In other words, when the motion

of the aether is uniform, the test electromagnetic waves do

not display the dependence of the phase on the polariza-

tion; however, when the aether ﬂow is characterized by non-

vanishing acceleration, shear, rotation or expansion, we deal

with the so-called degeneracy removal with respect to the

hidden parameters in analogy with effects described in [26].

The idea of a mathematical description of this degeneracy

removal was disclosed in Ref. [1]; there the corresponding

term dynamo-optical phenomena was introduced. In order

123

699 Page 2 of 14 Eur. Phys. J. C (2017) 77 :699

to describe this effect the authors of [1] have introduced

the terms with derivatives of the medium velocity into the

permittivity tensors, thus rendering these tensors spatially

anisotropic.

Our consideration is based on the Einstein-aether theory

[27–36] and its extension, the Einstein–Maxwell-aether the-

ory [37]. The macroscopic velocity four-vector U

appears

in the Einstein-aether and Einstein–Maxwell-aether theo-

ries as a dynamic time-like vector ﬁeld normalized by unity

= 1). The covariant derivative ∇

enters the

basic Lagrangian of the Einstein-aether theory [27], and it

appears in the interaction terms in the Einstein–Maxwell-

aether theory [37]. In this sense, we can indicate our approach

as an extension of the idea of dynamo-optical interactions,

fulﬁlled in the framework of the Einstein-aether theory. The

Einstein-aether and the Einstein–Maxwell-aether theories

realize the idea of a preferred frame of reference [38–40]

associated with a world-line congruence for which the cor-

responding time-like velocity four-vector U

is the tangent

vector. In this sense they are characterized by a violation of

Lorentz invariance (see, e.g., [16]). There is also an alter-

native approach to introduce dynamo-optical interactions,

which is based on the analysis of the time-like unit eigen

four-vector of the stress-energy tensor of the cosmic substra-

tum (the vacuum, the aether, the dark ﬂuid and so on) (see,

e.g., [41–43]). Such a velocity ﬁeld appears algebraically

as an intrinsic vectorial quantity; the velocity ﬁeld which we

consider now is related to the additional dynamic vector ﬁeld.

In this paper we consider the birefringence effect, which

is dynamo-optically induced by the aether pp-wave modes.

What does this mean? First, we consider the pp-wave back-

ground formed by the gravitationally self-interacting aether

and ﬁx the constraints on the Jacobson coupling parameters,

which guarantee that the so-called pp-wave modes can exist

in the dynamic aether. Second, we study the propagation of

test electromagnetic waves dynamo-optically coupled to the

pp-wave symmetric background. The modeling of the sus-

ceptibility tensors of such a potentially birefringent aether is

based on the introduction of two coupling constants; the phe-

nomenologically constructed susceptibility tensors describe

some effective bi-axial quasi-medium. Then we analyze the

master equations for the longitudinal and transversal electro-

magnetic ﬁeld conﬁgurations, and we prove that this aether

behaves as a birefringent medium.

The paper is organized as follows. In Sect. 2 we consider

the basic elements of the Einstein–Maxwell-aether theory,

and we describe the background state possessing the pp-

wave symmetry and introduce a speciﬁc background state

indicated as pp-wave aether mode. In Sect. 3 we study solu-

tions for electromagnetic waves in the aether with excited

pp-wave modes. In Sect. 4 we discuss the magnitudes of

the birefringence effect and demonstrate that anomalies can

exist in the electromagnetic response on the action of the pp-

wave aether modes. Brieﬂy our conclusions are presented in

Sect. 5.

2 The formalism

2.1 Action functional of the Einstein-aether theory

The Einstein-aether theory [27–34] uses the action functional

(0)



√

−g

2κ

[R + λ(g

− 1)

+ K

abmn

(∇

)(∇

)], (1)

which describes the interaction between the gravitational

ﬁeld and the unit vector ﬁeld U

attributed to the velocity of

some hypothetic medium, the dynamic aether. In the func-

tional (1), the quantity g = det(g

) is the determinant of

the metric; R is the Ricci scalar; κ is the Einstein constant.

The term λ

(

−1

)

ensures that the U

is normalized

to one; the function λ is the Lagrange multiplier. The term

abmn

∇

is quadratic in the covariant derivative

∇

of the vector ﬁeld U

. The tensor K

abmn

is constructed

using the metric tensor g

and the velocity four-vector U

only (see, e.g., [27]):

abmn

= C

+ C

. (2)

Here C

, C

, and C

are phenomenologically intro-

duced coupling constants [27–29]. In order to interpret the

coupling constants C

, C

, one uses the standard

decomposition of the tensor ∇

into the sum

∇

= U

+ σ

+ ω



. (3)

The acceleration four-vector DU

, symmetric trace-free

shear tensor σ

, anti-symmetric vorticity tensor ω

, and the

expansion scalar  are given by the formulas

≡ U

∇

≡



(

∇

+∇

)

−



,

≡



(∇

−∇

 ≡∇

,

= δ

−U

. (4)

In these terms the scalar

K ≡ K

abmn

(∇

)(∇

) (5)

in the action functional (1) can be rewritten as follows:

K = C

+ C



. (6)

123

Eur. Phys. J. C (2017) 77 :699 Page 3 of 14 699

Here we used the notations

= C

+ C

, C

= C

− C

= C

+ C

, C



= C

+ 3C

+ C

. (7)

As shown in [28], the Einstein-aether theory admits waves of

three types, which can be classiﬁed formally as scalar, vecto-

rial, and tensorial; respectively, one can speak of waves with

spin zero, spin one, and spin two. The parameters C

, C

and C



are connected with velocities of the corresponding

waves, denoted as S

(0)

, S

(1)

, and S

(2)

. For weak waves on

the Minkowski background these velocities of the waves are

found to be (compare with [28])

(0)



+ 2C

)(2 −C

)

(1 − C

)(2 +C



)

, (8)

(1)

+ C

(1 − C

)

(1 − C

)

, S

(2)

(1 − C

)

. (9)

Our ansatz is that the tensorial mode propagates with the

velocity coinciding with the speed of light in vacuum, i.e.,

(2)

= 1; this quantity also coincides with the standard veloc-

ity of propagation of the weak gravitational waves on the

Minkowski background [44]. According to (9) this means

that C

= 0. Also, we assume that the {g, U } model is

pure vectorial-tensorial, and the scalar modes cannot prop-

agate at all, S

(0)

= 0; then according to (8) we obtain



= 0. The velocity S

(1)

is free of restrictions; now we

obtain S

(1)



, and the coupling constants C

and C

are assumed to be of the same signs. Below we will show

that these phenomenological motives lead to the same result

as the strict deﬁnition of the pp-wave aether modes.

2.2 Master equations describing the background state

2.2.1 Equations for the unit dynamic vector ﬁeld

The aether dynamic equations are known to be found by

varying the action (1) with respect to the Lagrange multiplier

λ and to the unit vector ﬁeld U

. The variation with respect

to λ gives the equation

= 1, (10)

which is the normalization condition of the time-like vector

ﬁeld U

. Variation of the functional (1) with respect to U

shows that U

itself satisﬁes the standard balance equation

∇

= I

+ λ U

, (11)

where the auxiliary quantities J

and I

are deﬁned as

follows:

≡ K

ab j n

(∇

), I

= C

(DU

)(∇

). (12)

The Lagrange multiplier λ can be obtained by convolution

of (11) with U

; it has the following form:

λ = U

[∇

− I

]. (13)

In more detail, using the constitutive tensor (2) and the sym-

bols 

≡∇

,  ≡∇

, we obtain

= C



+ C

 + C



+ C

= C

(DU

)

. (14)

2.2.2 Equations for the gravitational ﬁeld

The variation of the action (1) with respect to the metric g

yields the gravitational ﬁeld equations:

−

= λU

+ T

(U)

. (15)

The term T

(U)

describes the stress-energy tensor associated

with the self-gravitation of the vector ﬁeld U

(U)

∇

+∇

k)m

]−∇

m(i

]−∇

(ik)

]

[(∇

)(∇

) − (∇

∇

)]

∇

)(U

∇

). (16)

As usual, the symbol p

≡

( p

+ p

) denotes the

procedure of symmetrization. As will be shown below, the

trace of this tensor,

(U)

= 2[C



+ C



]

(

+ C

− C

)

∇

−

(

+ 4C

+ C

)

D

−

(

+ 2C

+ C

)



+ 3C

, (17)

has to be equal to zero for the model with the pp-wave sym-

metry.

2.3 Master equations reduced to the case with pp-wave

symmetry

2.3.1 Metric and Killing’s vectors

We consider space-times, which possess the G

group of

isometries [45], with ﬁve Killing vectors {ξ

(1)

,ξ

(2)

,ξ

(3)

,ξ

(4)

(5)

}, three of which, {ξ

(1)

,ξ

(2)

,ξ

(3)

}, form the Abelian sub-

group G

, and the ﬁrst of them, ξ

(1)

, is the null covariantly

constant four-vector. Mathematically, this means that, ﬁrst,

the Lie derivative of the metric is equal to zero, £

(a)

= 0

(a = 1, 2, 3, 4, 5); second, g

(1)

= 0, and third,

∇

(1)

= 0.

123

剩余13页未读，继续阅读

评论收藏

内容反馈

weixin_38645266

粉丝: 4
资源: 948

pp波模式在电磁有源动态醚中引起的双折射

电磁场电磁波实验报告电磁波的反射和折射

电磁波穿透界面时折射与反射的Matlab仿真

非扰动QED真空双折射

模拟电磁波传播

电磁场与电磁波 郭辉萍

不同色散分布的双折射光纤中电磁波的调制不稳定性分析.pdf

电磁波在导电媒质中传播时的matlab仿真.pdf

电磁场与电磁波 ppt课件

电磁波穿透界面时折射与反射的Matlab仿真 下载

电磁场与电磁波公式总结.pdf

电磁波在均匀等离子体中的传输.m

电磁波在导体表面的反射与折射.pdf

基于MATLAB的电磁波斜射到介质表面的反射折射动态仿真

均匀平面电磁波传播的动态显示

通信与网络中的基于Matlab的均匀平面电磁波的仿真

电磁场与电磁波答案电磁场与电磁波答案

【电磁波斜射】基于matlab电磁波反射折射动态仿真【含Matlab源码 2787期】.zip

电磁场与电磁波 电磁场与电磁波

Qt 5实现串口调试助手 （源工程文件、0积分下载）

【SystemVerilog】路科验证V2学习笔记（全600页）.pdf

AutoSAR标准协议4.2.2

光伏-储能并网系统仿真.rar

GD32替换STM32注意事项.pdf

XCP协议的规范文档

NPPJSONViewer.zip

CANoe通过CAPL脚本实现自动测试

蓝牙BLE协议中文版.pdf

VS2015安装证书，JavaScript_ProjectSystem.msi，JavaScript_LanguageService.msi

最新资源

电磁场与电磁波郭辉萍

电磁波穿透界面时折射与反射的Matlab仿真下载

电磁场与电磁波电磁场与电磁波

Qt 5实现串口调试助手（源工程文件、0积分下载）