Maxwell equations in Riemannian space-time, …npcs.j-npcs.org/Procc/v13p207.pdf · Maxwell equations in Riemannian space-time, geometrical modeling of medias which depends upon ten

Nonlinear Dynamics and Applications. Vol. 13 (2006) 207 - 228

Maxwell equations in Riemannian space-time,

geometrical modeling of medias

Red’kov V.M.∗ and Tokarevskaya N.G., E.M. Bychkouskaya

B.I.Stepanov Institute of Physics of the National Academy of Science of Belarus68 Nezavisimosti Ave., 220072, Minsk, Belarus

George J. Spix

BSEE Illinois Institute of Technology, USA

In the paper, the known possibility to consider the (vacuum) Maxwell equationsin a curved space-time as Maxwell equations in flat space-time (Mandel’stam L.I.,Tamm I.E. [1,2]) but taken in an effective media the properties of which are deter-mined by metrical structure of the initial curved model gαβ(x) is studied. Metricalstructure of the curved space-time generates the ”material equations” for electro-magnetic fields:

Di = εik(x) Ek + αik(x) Bk , H i = βik(x) Ek + µik(x) Bk ,

the form of four symmetrical ”matherial” tensors εik(x), αik(x), βik(x), µik(x) isfound explicitly for general case of an arbitrary Riemannian space-time geometrygαβ(x):

εik(x) = ε0 [ g00(x) gik(x)− g0i(x) g0k(x) ] , µik(x) =1µ0

12

εimn gml(x)gnj(x) εljk ,

αik(x) = +ε0c gij(x) g0l(x) εljk , βik(x) = −ε0c g0j(x) εjil glk(x) .

Several, the most simple examples are specified in detail: it is given geometricalmodeling of the anisotropic media (magnetic crystals)

εi = ε0ε ni , µi = µ0µ ni, n2 = 1 ,

gab(x) =1√ε

∣∣∣∣∣∣∣∣∣∣∣∣

1√εµ

1√n1n2n3

0 0 0

0 −√εµ√

n2n3n1

0 0

0 0 −√εµ√

n3n1n2

0

0 0 0 −√εµ√

n1n2n3

∣∣∣∣∣∣∣∣∣∣∣∣and the geometrical modeling of a uniform media in moving reference frame in thebackground of Minkowsky electrodynmamics – the latter is realized trough the useof a non-diagonal metrical tensor determined by 4-vector velocity of the movinguniform media gam = [ gam + (εµ− 1) uaum ]/

√µ .

The main peculiarity of the geometrical generating for effective elec-tromagnetic medias characteristics consists in the following: four tensorsεik(x), αik(x), βik(x), µik(x) are not independent and obey some additional con-straints between them.

PACS numbers: 04.62.+vKeywords: Maxwell equations, curved space-time

∗E-mail: [email protected]

207

Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix

1. Introduction: Riemannian geometry and Maxwell theory

Let us start with the Maxwell equations in Minkowski space: in vector notation they are[3-6]

(I) div B = 0 , rot E = −∂B

∂t,

(II) εε0 div E = ρ ,1

µµ0

rot B = J + εε0∂E

∂t. (1)

With the use of material equations

H = B/µµ0 , D = εε0 E (2)

eqs. (1) can be written in terms of four vectors as follows

(I) div cB = 0 , rot E = −∂cB

∂x0.

(II) div D = j0 , rotH

c= j + εε0

∂E

∂x0(3)

where x0 = ct , ja = (J0 = ρ,J/c) , In terms of two electromagnetic tensors:

(Fαβ) =

∣∣∣∣∣∣∣∣

0 −E1 −E2 −E3

+E1 0 −cB3 +cB2

+E2 +cB3 0 −cB1

+E3 −cB2 +cB1 0

∣∣∣∣∣∣∣∣, (Hαβ) =

∣∣∣∣∣∣∣∣

0 −D1 −D2 −D3

+D1 0 −H3/c +H2/c+D2 +H3/c 0 −H1/c+D3 −H2/c +H1/c 0

∣∣∣∣∣∣∣∣

eqs. (3) take the form

(I) ∂aFbc + ∂bFca + ∂cFab = 0 , (II) ∂bHba = ja . (4)

When extending Maxwell theory to the case of space-time with non-Euclidean geometry,which can describe gravity according to General Relativity [6], one must change previous equa-tions to a more general form [6]:

(I) ∇αFβγ +∇βFγα +∇γFαβ = 0 , (II) ∇βF βα = Jα . (5)

Here again two tensors are used. However, now instead of Lorentz symmetry, one requiresinvariance under any continuous coordinate transformations x

′α = fα(x0, x1, x2, x3). At this,electromagnetic tensors transform with respect to the coordinate change by the law (the ruleof summing over any repeated indices applies)

Fα′β′(x′) =

∂xα

∂xα′∂xβ

∂xβ′ Fαβ(x) , F α′β′(x′) =∂xα′

∂xα

∂xβ′

∂xβFαβ(x) ,

Hα′β′(x′) =

∂xα

∂xα′∂xβ

∂xβ′ Hαβ(x) , Hα′β′(x′) =∂xα′

∂xα

∂xβ′

∂xβHαβ(x) .

The role of the main relativistic invariant dS2 = c2(dx0)2 − (dx1)2 − dx2)2 − dx3)2 play ageneralized Riemannian interval

dS2 = gαβ(x)dxαdxβ = g00 (dx0)2 + 2g0i dx0dxi + gik dxidxk

208

Maxwell equations in Riemannian space-time, geometrical modeling of medias

which depends upon ten functions, components of 2-rank metrical tensor gαβ(x). The mainidea of general relativity is that metrical structure gαβ(x) of a physical space-time should besolution of the Einstein-Hilbert equation.

Connection between upper (contra-variant) and lower (covariant) indices in tensor quantitiesis realized through metrical tensor: Fαβ(x) = gαρ(x)gβσ(x) Fρσ(x) and so on .Also the identity holds gασ gσβ = δα

β . The main differential operator is the covariant derivative,with the help of which one can formulate physical differential equations, for instance Maxwellequations, explicitly. Its action on the most simple 1-rank tensor (vector) is given by theformula (more details see in [6])

∇αAβ = ∂αAβ − ΓσαβAσ , ∇αAβ = ∂αAβ + Γβ

ασAσ , ∇α = gαβ(x)∇β .

Symmetrical in lower indices quantities Γραβ(x), Γρ

αβ(x) = Γρβα(x) are called Christoffel symbols.

They determine the action of covariant derivative on tensors. The action of covariant derivativeon 2-rank tensor is determined as follows

∇αFβγ = ∂αFβγ − ΓσαβFσγ − Γσ

αγFβσ , ∇αF βγ = ∂αF βγ + ΓβασF

σγ + ΓγασF

βσ .

If one requires identity

∇ρgαβ(x) ≡ 0 ,

for Christoffel symbols one may derive representation in term of metrical tensor:

Γραβ(x) = gρσ(x) Γσ,αβ(x), Γσ,αβ(x) =

1

2[ −∂σgαβ(x) + ∂αgβσ(x) + ∂βgασ(x) ] .

When working with the covariant derivative symbol, one must be careful because two suchderivatives do not commute with each other. Just in this point the concept of Riemann curva-ture tensor arises:

(∇α∇β −∇β∇α) Aρ(x) = R ραβ γ(x) Aγ(x) .

Riemann tensor has some symmetry properties:

Rαβ ρσ(x) = −Rβα ρσ(x) = −Rαβ σρ(x) = Rρσ αβ(x) .

The tensor Rαβ ρσ(x) is a rather involved non-linear function of a metrical tensor gαβ(x).

2. Maxwell equations in curved space-time

In a curved space-time with arbitrary metrical tensor

dS2 = gαβ dxαdxβ

the Maxwell equations are [6]

(I) ∇αFβγ +∇βFγα +∇γFαβ = 0 , (II) ∇βF βα = Jα . (6)

Symmetry of Christoffel symbols, Γσαβ = Γσ

βα, enables us to substitute usual derivatives insteadof covariant ones. Indeed

∇αFβγ +∇βFγα +∇γFαβ =

= ∂αFβγ − ΓσαβFσγ − Γσ

αγFβσ + ∂βFγα − ΓσβγFσα − Γσ

βαFγσ + ∂γFαβ − ΓσγαFσβ − Γσ

γβFασ ,

209


all six terms without derivatives cancel out each others and equation I in (6) will take the form

(I) ∂αFβγ + ∂βFγα + ∂γFαβ = 0 . (7)

However, this equation though written in terms of usual derivatives proves its covariance undergeneral coordinate transformations. To prevent errors, one should note that equation of theform (7) with upper indices would not be correct; instead a correct equation of that type is

(I) ∇αF βγ +∇βF γα +∇γFαβ = 0 . (8)

In detailed form equation (8) looks as

gαρ ∇ρFβγ + gβρ ∇ρF

γα + gγρ ∇ρFαβ = gαρ [ ∂ρ F βγ + Γβ

ρσ F σγ + Γγρσ F βσ ]

+gβρ [ ∂ρFγα + Γγ

ρσ F σα + Γαρσ F γσ ] + gγρ [ ∂ρF

αβ + Γαρσ F σβ + Γβ

ρσ Fασ ] = 0

where the terms without derivatives do not compensate each other:

0 = (gαρ ∂ρ F βγ + gβρ ∂ρFγα + gγρ ∂ρF

αβ) + (gαρ Γβρσ − gβρ Γα

ρσ) F σγ +

+ (gαρ Γγρσ − gγρ Γα

ρσ) F βσ + (gβρ Γγρσ − gγρ Γβ

ρσ)F σα .

Now, in the Maxwell equations written in terms of covariant derivatives, let us make (3+1)-splitting. Equation (I) gives

∇1 F23 +∇2 F31 +∇3 F12 = 0 ; ∇0 F12 +∇1 F20 +∇2 F01 = 0 ,

∇0 F23 +∇2 F30 +∇3 F02 = 0 , ∇0 F31 +∇3 F10 +∇1 F03 = 0 ,

or with the use of index techniques (ε123 = +1 and so on)

εijk∇iFjk = 0 , ∇0 Fij +∇i Fj0 −∇j Fi0 = 0 . (9)

Eqs. (9) are generally covariant, so that equations with upper indices are correct as well:

εijk∇iF jk = 0 , ∇0 F ij +∇i F j0 −∇j F i0 = 0 . (10)

In turn, eq. (II) of eq (5) contains also four relations:

∇i F i0 = J0 , ∇0F0j +∇i F ij = J j . (11)

3. Maxwell equation, 3-dimensional form

The Maxwell equations (6) can be written in terms of ordinary derivatives as follows:

(I) ∂αFβγ + ∂βFγα + ∂γFαβ = 0 , (II)1√−g

∂β

√−g F βα = jα . (12)

where g(x) = det [gαβ(x)] < 0. Take special notice that in contrast to equation (I), equation(II) is written for contra-variant electromagnetic tensor. Tensor relation (I) is equivalent tofour equations:

∂1 F23 + ∂2 F31 + ∂3 F12 = 0 , ∂0 F12 + ∂1 F20 + ∂2 F01 = 0 ,

∂0 F23 + ∂2 F30 + ∂3 F02 = 0 , ∂0 F31 + ∂3 F10 + ∂1 F03 = 0 . (13)

210


In the following, let us use the 3-dimension notation:

E1 = F10 , E2 = F20 , E3 = F30, cB1 = −F23 , cB2 = −F31 , cB3 = −F12 .

Then eqs. (13) look as

∂1 B1 + ∂2 B2 + ∂3 B3 = 0 , ∂t B1 = ∂2 E3 − ∂3 E2 ,

∂t B2 = ∂3 E1 − ∂1 E3 , ∂t B3 = ∂1 E2 − ∂2 E1 . (14)

Equation (II) from (12) is equivalent to

1√−g∂i

√−gF i0 = j0 ,1√−g

∂

∂x0

√−gF 0j +1√−g

∂

∂xi

√−gF ij = jj , (15)

and further

1√−g∂i

√−g F i0 = j0 ,

1√−g

∂

∂x0

√−g F 01 − 1√−g

∂

∂x2

√−g F 12 +1√−g

∂

∂x3

√−g F 31 = j1 ,

1√−g

∂

∂x0

√−g F 02 +1√−g

∂

∂x1

√−g F 12 − 1√−g

∂

∂x3

√−g F 23 = j2 ,

1√−g

∂

∂x0

√−g F 03 − 1√−g

∂

∂x1

√−g F 31 +1√−g

∂

∂x2

√−g F 23 = j3 . (16)

In the 3-dimensional notation

Di(x) = F i0(x) ,H i(x)

c= −εijk F jk(x) ,

eqs. (16) read as

1√−g∂i

√−g Di = ρ ,

1√−g

∂

∂x2

√−g H3 − 1√−g

∂

∂x3

√−g H2 = J1 +1√−g

∂

∂t

√−g D1 ,

1√−g

∂

∂x3

√−g H1 − 1√−g

∂

∂x1

√−g H3 = J2 +1√−g

∂

∂t

√−g D2 ,

1√−g

∂

∂x1

√−g H2 − 1√−g

∂

∂x2

√−g H1 = J3 +1√−g

∂

∂t

√−g D3 . (17)

So, the full system of Maxwell equations in arbitrary curved space-time can be presented inthe form:

∂1B1 + ∂2B2 + ∂3B3 = 0 , ∂tB1 = ∂2E3 − ∂3E2 ,

∂tB2 = ∂3E1 − ∂1E3 , ∂tB3 = ∂1E2 − ∂2E1 , (18)1√−g

∂i

√−g Di = ρ ,

1√−g

∂

∂x2

√−gH3 − 1√−g

∂

∂x3

√−gH2 = J1 +1√−g

∂

∂t

√−gD1 ,

1√−g

∂

∂x3

√−gH1 − 1√−g

∂

∂x1

√−gH3 = J2 +1√−g

∂

∂t

√−gD2 ,

1√−g

∂

∂x1

√−g H2 − 1√−g

∂

∂x2

√−gH1 = J3 +1√−g

∂

∂t

√−gD3 . (19)

211


Let us take spacial notation to distinguish vacuum case from all others. For vacuum case inthe Minkowsky flat space-time, material equations [17]

D = ε0E = (Di) , H =1

µ0

B = (H i), (20)

being translated to tensor form will look

(F ab) =

∣∣∣∣∣∣∣∣

0 −E1 −E2 −E3

+E1 0 −cB3 +cB2

+E2 +cB3 0 −cB1

+E3 −cB2 +cB1 0

∣∣∣∣∣∣∣∣,

(Hab) =

∣∣∣∣∣∣∣∣

0 −D1 −D2 −D3


∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣

0 −ε0E1 −ε0E

2 −ε0E3

+ε0E1 0 −B3/cµ0 +B2/cµ0

+ε0E2 +B3/cµ0 0 −B1/cµ0

+ε0E3 −B2/cµ0 +B1/cµ0 0

∣∣∣∣∣∣∣∣,

from where, taking in mind identity ε0µ0 = 1/c2, we arrive at the rule

Hab(x) = ε0 F ab(x) . (21)

in other words, for vacuum case, electromagnetic tensors Hab(x) and F ab(x) are in essence thesame being different only in trivial measure factor. We face quite other situation when turningto the case of uniform media. Then material equations

D = ε0εE = (Di) , H =1

µ0µB = (H i), (22)

being translated to tensor form will look

(F ab) =

∣∣∣∣∣∣∣∣

0 −E1 −E2 −E3

+E1 0 −cB3 +cB2

+E2 +cB3 0 −cB1

+E3 −cB2 +cB1 0

∣∣∣∣∣∣∣∣, (Hab) =

=

∣∣∣∣∣∣∣∣

0 −D1 −D2 −D3


∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣

0 −ε0εE1 −ε0εE

2 −ε0εE3

+ε0εE1 0 −B3/cµ0µ +B2/cµ0µ

+ε0εE2 +B3/cµ0µ 0 −B1/cµ0µ

+ε0εE3 −B2/cµ0µ +B1/cµ0µ 0

∣∣∣∣∣∣∣∣

= ε0ε

∣∣∣∣∣∣∣∣

0 −E1 −E2 −E3

+E1 0 −cB3/εµ +cB2/εµ+E2 +cB3/εµ 0 −cB1/εµ+E3 −cB2/εµ +cB1/εµ 0

∣∣∣∣∣∣∣∣.

(23)

It is readily verified that with the help of a (4× 4) -matrix

ηam =

∣∣∣∣∣∣∣∣

1/k 0 0 00 −k 0 00 0 −k 00 0 0 −k

∣∣∣∣∣∣∣∣, k =

1√εµ

(24)

relation (23), can be written as follows:

Hab = ε0ε ηamηbn Fmn . (25)

So, the material equations in a uniform media, to be given in terms of two electromagnetictensors, requires the use of 4-rank tensor factorized in term of 2-rank tensor ηab.

212


4. Maxwell equations in orthogonal coordinates of a curved space

Let the metrical tensor gαβ of a curved space-time model have a diagonal structure

gαβ(x) =

∣∣∣∣∣∣∣∣

h20 0 0 0

0 −h21 0 0

0 0 −h22 0

0 0 0 −h23

∣∣∣∣∣∣∣∣, gαβ(x) =

∣∣∣∣∣∣∣∣

h−20 0 0 00 −h−2

1 0 00 0 −h−2

2 00 0 0 −h−2

3

∣∣∣∣∣∣∣∣,

√−g(x) =

√−det (gαβ) = h0h1h2h3 . (26)

We will consider vacuum Maxwell equations on the background of a non-Euclidean geometry.So Maxwell equations (I) of (12) looks in orthogonal coordinates the same as in arbitrary(non-orthogonal) coordinates:

∂1 B1 + ∂2 B2 + ∂3 B3 = 0 , (27)

∂t B1 = ∂2 E3 − ∂3 E2 , ∂t B2 = ∂3 E1 − ∂1 E3 , ∂t B3 = ∂1 E2 − ∂2 E1 .

Contra-variant components Di, H i are expressed through covariant Ei, Bi according to theformulas

D1 = −ε0E1

h20 h2

1

, D2 = −ε0E2

h20 h2

2

, D3 = −ε0E3

h20 h2

3

,

H1 =1

µ0

B1

h22h

23

, H2 =1

µ0

B2

h33h

21

, H3 =1

µ0

B3

h21h

22

. (28)

Maxwell equations (II)

1√−g∂i

√−g Di = ρ , (29)

1√−g

∂

∂x2

√−g H3 − 1√−g

∂

∂x3

√−g H2 = J1 +1√−g

∂

∂t

√−g D1 ,

1√−g

∂

∂x3

√−g H1 − 1√−g

∂

∂x1

√−g H3 = J2 +1√−g

∂

∂t

√−g D2 ,

1√−g

∂

∂x1

√−g H2 − 1√−g

∂

∂x2

√−g H1 = J3 +1√−g

∂

∂t

√−g D3 . (30)

will take form

− ε0

h0h1h2h3

(∂

∂x1

E1h2h3

h0 h1

+∂

∂x2

E2h3h1

h0 h2

+∂

∂x3

E3h1h2

h0 h3

) = ρ , (31)

1

µ0

1

h0h1h2h3

(∂

∂x2

B3h0h3

h1h2

− ∂

∂x3

B2h0h2

h3h1

) = J1 − ε01

h0h1h2h3

∂

∂t

E1h2h3

h0 h1

,

1

µ0

1

h0h1h2h3

(∂

∂x3

B1h0h1

h2h3

− ∂

∂x1

B3h0h3

h1h2

) = J2 − ε01

h0h1h2h3

∂

∂t

E2h3h1

h0 h2

,

1

µ0

1

h0h1h2h3

(∂

∂x1

B2h0h2

h3h1

− ∂

∂x2

B1h0h1

h2h3

) = J3 − ε01

h0h1h2h3

∂

∂t

E3h1h2

h0 h3

. (32)

Let g00 = 1 and spatial metric be static, gij = gij(x1, x2, x3), then Maxwell equations will be

simplified:

D1 = −ε0E1

h21

, D2 = −ε0E2

h22

, D3 = −ε0E3

h23

,

H1 =1

µ0

B1

h22h

23

, H2 =1

µ0

1

mu0

B2

h23h

21

, H3 =1

µ0

B3

h21h

22

, (33)

213


and

∂1 B1 + ∂2 B2 + ∂3 B3 = 0 , ∂t B1 = ∂2 E3 − ∂3 E2 ,

∂t B2 = ∂3 E1 − ∂1 E3 , ∂t B3 = ∂1 E2 − ∂2 E1 , (34)

−ε01

h1h2h3

(∂

∂x1

E1h2h3

h1

+∂

∂x2

E3h3h1E2

h2

+∂

∂x3

h1h2

h3

) = ρ ,

1

µ0

1

h1h2h3

(∂

∂x2

B3h3

h1h2

− ∂

∂x3

B2h2

h3h1

= J1 − ε0∂

∂t

E1

h21

,

1

µ0

1

h1h2h3

(∂

∂x3

B1h1

h2h3

− ∂

∂x1

B3h3

h1h2

) = J2 − ε0∂

∂t

E2

h22

,

1

µ0

1

h1h2h3

(∂

∂x1

B2h2

h3h1

− ∂

∂x2

B1h1

h2h3

) = J3 − ε0∂

∂t

E3

h23

. (35)

5. Maxwell equations in Riemannian space-time and a media

Let us discuss in detail the known possibility [1-2] to consider the (vacuum) Maxwellequations in a curved space-time as Maxwell equations in flat space-time but taken in aneffective media the properties of which are determined by metrical structure of the initialcurved model gαβ(x).

In the first place, let us restrict ourselves to the case of curved space-time models which areparameterized by the same quasi-Cartesian coordinate system xa. In flat space-time model,the Maxwell equations in a media are formulated with the help of two electromagnetic tensorsFab, Hab:

in a media

(I) ∂a Fbc + ∂b Fca + ∂c Fab = 0 , (II) ∂b Hba = Ja ; (36)

relationship between Hab and Fab is established by some (external) material equations, in thevacuum case they are Hab = ε0 Fab.

Let us write down the vacuum Maxwell equations but now in a Riemannian space-time,parameterized by the same quasi-Cartesian coordinates (to distinguish formulas referring to aflat and curved models let us use small letters to designates electromagnetic tensors in curvedmodel, fab and hab )

(I) ∂afbc + ∂bfca + ∂cfab = 0 , (II)1√−g

∂b

√−g hba = ja . (37)

hab(x) = ε0fab(x), hab(x) = ε0gam(x)gbn(x) fmn(x) .

One can immediately see that introducing new (formal) variables[18]

Fab = fab, Hba = ε0

√−g gam(x)gbn(x) Fmn(x), Ja =√−g ja (38)

equations (37) in the curved space can be re-written as Maxwell equations of the type (36) inflat space but in a media:

(I) ∂aFbc + ∂bFca + ∂cFab = 0 (II) ∂b Hba = Ja . (39)

At this, relations playing the role of material equations are determined by metrical structure:

Hβα(x) = ε0 [√−g(x) gαρ(x)gβσ(x) ] Fρσ(x) (40)

214


Such a re-interpretation can be extended to curvilinear coordinates as well. Indeed, let inthe flat (Minkowsky) space-time be given in a curved coordinate system (xσ) with metricaltensor Gαβ(x). The Maxwell equations in a media specified for these coordinates (xσ) are

in a media

Fαβ(x) , Hαβ(x) , Hαβ 6= ε0Fαβ

(I) ∂α Fβγ + ∂β Fγα + ∂γ Fαβ = 0 , (II)1√−G

∂β

√−G Hαβ = Jα ; (41)

connection between Hab and Fab is given by external material equations.Now, let a Riemannian space-time model be parameterized by formally the same curvilinear

coordinates (xσ) with metric gαβ(x). The Maxwell equations (in vacuum) in this curved space-time model are

(I) ∂αfβγ + ∂βfγα + ∂γfαβ = 0 , (II)1√−g

∂β

√−g hβα = jα . (42)

hαβ(x) = ε0fαβ(x), hαβ(x) = ε0gαρ(x)gβσ(x) fρσ(x) .

The second equation can be re-written as follows

(II)

√−G√−g

1√−G∂β

√−G

√−g√−Ghβα = jα . (43)

Now, introducing new formal variables

Fαβ(x) = fαβ(x), Hβα(x) =

√−g√−Ghβα(x) , Jα(x) =

√−g(x)√−G(x)

jα(x) , (44)

equations (42) are reduced to the Maxwell equations of the type (41) in the flat space-time (incurvilinear coordinates xσ and a media):

(I) ∂αFβγ + ∂βFγα + ∂γFαβ = 0 , (II)1√−G

∂β

√−G Hαβ = Jα . (45)

At this, relationships between electromagnetic tensors are established by the formulas

Hβα(x) = ε0

√−g(x)√−G(x)

gαρ(x)gβσ(x) Fρσ(x) .

6. Metrical tensor gαβ(x) and material equations

In this section let us consider the ”material equations” for electromagnetic fields which aregenerated by metrical structure of the curved space-time model (for simplicity let us omit thefactor

√−g(x) or

√−g/√−G – see (38) and (44))

Hρσ(x) = ε0 Fρσ , =⇒ Hρσ(x) = ε0 gρα(x) gσβ(x) Fαβ(x) . (46)

3-dimensional representation of the tensor is defined by the formulas:

E1 = F10 , E2 = F20 , E3 = F30 ,

cB1 = −F23 , cB2 = −F31 , cB3 = −F12 ,

D1 = ε0F10 , D2 = ε0F

20 , ε0D3 = F 30 ,

H1

c= −ε0F

23 ,H2

c= −ε0F

31 ,H3

c= −ε0F

12 . (47)

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Consider the case of arbitrary metrical tensor

gαβ(x) =

∣∣∣∣∣∣∣∣

g00 g01 g02 g03

g01 g11 g12 g13

g02 g12 g22 g23

g03 g13 g23 g33

∣∣∣∣∣∣∣∣, gαβ(x) =

∣∣∣∣∣∣∣∣

g00 g01 g02 g03

g01 g11 g12 g13

g02 g12 g22 g23

g03 g13 g23 g33

∣∣∣∣∣∣∣∣. (48)

We are to obtain a 3-dimensional form of relation (46)

Hρσ = (Di, H i/c), Fαβ = (Ei, cBi), (Di, H i) = f(Ei, Bi) ;

their general structure should be as follows[19]:

Di = εik(x) Ek + αik(x) Bk , H i = βik(x) Ek + µik(x) Bk . (49)

Four (3×3)-matrices εik(x), αik(x), βik(x), µik(x) should not be independent because they arebilinear functions of 10 independent components of the symmetrical tensor gαβ(x) = gβα(x).

First, let us turn to the first equation in (49). Because the relation holds

Di = ε0 giαg0β Fαβ = ε0 [−gi0g0j Fj0 + gijg00 Fj0 + gijg0l Fjl] ,

we have

Di = ε0 ( g00gik − gi0gk0 ) Ek + ε0c (− gijg0l εjlk) Bk . (50)

Now, let us turn to the second equation in (49); starting from

F ij = ε0 [giαgjβ Fαβ] = −ε0g0igjkFk0 + ε0g

ikgj0Fk0 + ε0gikgjlFkl ,

that is

F ij = ε0 (gnigj0 − gnjgi0) Fn0 + ε0 gikglj Fkl .

Multiplying it by (−12

εmij), we get

Hm = −ε0c1

2εmij (gnigj0 − gnjgi0) En + ε0 c2 1

2εmij gikglj εkln Bn . (51)

Thus, from (50)-(51) it follow expressions for four tensors:

εik(x) = ε0 [ g00(x) gik(x)− g0i(x) g0k(x) ] ,

µik(x) =1

µ0

1

2εimn gml(x)gnj(x) εljk ,

αik(x) = +ε0c gij(x) g0l(x) εljk ,

βik(x) = −ε0c g0j(x) εjil glk(x) . (52)

The tensor εik(x) is evidently symmetrical; it is easy to demonstrate the same property forµik(x). Indeed,

µki(x) =1

µ0

1

2εkmn gml (x)gnj(x) εlji ,

taking changes in mute indices, m ↔ j, n ↔ l, we get

µki(x) =1

µ0

1

2εkjl gjn(x) glm(x) εnmi =

1

µ0

εimnglm(x) gjn(x)εljk = µik(x) .

216


In the same manner, one can prove the identity βki(x) = +αik . Indeed,

βki = −ε0c g0j(x) εjkl gli(x) = ε0c gil(x) g0j(x)εjlk = +αik .

Thus, the above form the tensors obey special symmetry conditions:

εik(x) = +εki(x) , µik(x) = +µki(x) , βki(x) = −αik ; (53)

which mean that the (6× 6)-matrix defining material equations

∣∣∣∣Di(x)H i(x)

∣∣∣∣ =

∣∣∣∣∣εik(x) αik(x)

βik(x) µik(x)

∣∣∣∣∣∣∣∣∣Ek(x)Bk(x)

∣∣∣∣ (54)

is a symmetrical matrix. Introducing (3 + 1)-splitting for gαβ(x)

gαβ(x) =

∣∣∣∣g00(x) (g0i(x)) = g(x)

(gi0(x)) = g(x) (gik(x)) = g(x)

∣∣∣∣ ,

(g×)jk(x) ≡ g0l(x)εljk (55)

tensors (εik), (αik), (βik) can be written in the form

ε(x) = ε0 [ g00(x)g(x)− g(x) • g(x) ] ,

α(x) = ε0c g(x) g×(x) , β(x) = −ε0c g×(x) g(x) . (56)

Also, one can produce more simple representation for µik(x). Indeed,

µik(x) =1

µ0

1

2εimn gml gnj εljk = − 1

µ0

1

2(εimn gnj) (εkjl glm) ,

that is [20]

(µik)(x) = − 1

µ0

1

2Sp [ τig(x) τkg(x) ] , (57)

where

(τi)mn = εimn , (τk)jl = εkjl .

Thus, the material equations for electromagnetic vectors generated by Riemannian geom-etry are:

general case

Di = εik(x) Ek + αik(x) Bk ,

H i = βik(x) Ek + µik(x) Bk ,

ε(x) = ε0 [ g00(x)g(x)− g(x) • g(x) ] , ε(x) = +ε(x) ,

µ(x) = − 1

µ0

1

2Sp [ τig(x) τkg(x) ], µ(x) = +µ(x) , (58)

α(x) = ε0c (x) g×(x) , β(x) = −ε0c g×(x) g(x) , β(x) = +α(x) .

217


If the metrical tensor has quasi-diagonal structure, equations (58) will be much simplified:

gαβ(x) =

∣∣∣∣∣∣∣∣

g00 0 0 00 g11 g12 g13

0 g21 g22 g23

0 g31 g32 g33

∣∣∣∣∣∣∣∣, Di = εik(x) Ek , H i = µik(x) Bk ,

ε(x) = ε0 g00(x)g(x) , µ(x) = − 1

µ0

1

2Sp [ τig(x) τkg(x) ] . (59)

Explicit expression for tensor εik(x) and µik(x) given by (59) are:

(εik) = ε0 g00

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣, (µik) =

1

µ0

∣∣∣∣∣∣

G11 G12 G13

G21 G22 G23

G31 G32 G33

∣∣∣∣∣∣, (60)

where Gik(x) stands for (algebraic) cofactor to the element gik(x):

Gik(x) = (−1)j+kM ik(x) =

=

∣∣∣∣∣∣

(g22g33 − g23g32) (g31g23 − g21g33) (g21g32 − g22g31)(g32g13 − g33g12) (g33g11 − g31g13) (g31g12 − g32g11)(g12g23 − g13g22) (g13g21 − g11g23) (g11g22 − g12g21)

∣∣∣∣∣∣. (61)

For general case of a non-diagonal metric (gi0(x) = gi(x) 6= 0), the dielectric tensor εik(x)becomes more complex:

[ εik(x) ] = ε0 g00

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣− ε0

∣∣∣∣∣∣

g1 g1 g1 g2 g1 g3

g2 g1 g2 g2 g2 g3

g3 g1 g3 g2 g3 g3

∣∣∣∣∣∣; (62)

whereas the tensor µik(x) preserves its form. For additional tensors αik(x) and βik one readilyproduces

αik(x) = +ε0c gij(x) g0l(x) εljk , =⇒

α(x) = ε0c

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣

∣∣∣∣∣∣

0 +g3 −g2

−g3 0 +g1

+g2 −g1 0

∣∣∣∣∣∣=

= ε0c

∣∣∣∣∣∣

(−g12g3 + g13g2) (g11g3 − g13g1) (−g11g2 + g12g1)(−g22g3 + g23g2) (g21g3 − g23g1) (−g21g2 + g22g1)(−g32g3 + g33g2) (g31g3 − g33g1) (−g31g2 + g32g1)

∣∣∣∣∣∣(63)

βik(x) = −ε0c g0j(x) εjil glk(x) , =⇒

α(x) = ε0c

∣∣∣∣∣∣

0 −g3 +g2

+g3 0 −g1

−g2 +g1 0

∣∣∣∣∣∣

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣=

= ε0c

∣∣∣∣∣∣

(−g12g3 + g13g2) (−g22g3 + g23g2) (−g32g3 + g33g2)(g11g3 − g13g1) (g21g3 − g23g1) (g31g3 − g33g1)

(−g11g2 + g12g1) (−g21g2 + g22g1) (−g31g2 + g32g1)

∣∣∣∣∣∣. (64)

In conclusion, let us specify the formulas for the simplest case of a pure diagonal metric:

[gαβ(x)] =

∣∣∣∣∣∣∣∣

g00(x) 0 0 00 g11(x) 0 00 0 g22(x) 00 0 0 g33(x)

∣∣∣∣∣∣∣∣, (65)

218


then

Di = εik(x) Ek , H i = µik(x) Bk ,

(εik) = ε0 g00(x)

∣∣∣∣∣∣

g11(x) 0 00 g22(x) 00 0 g33(x)

∣∣∣∣∣∣,

(µik) =1

µ0

∣∣∣∣∣∣

G11(x) 0 00 G22(x) 00 0 G33(x)

∣∣∣∣∣∣,

Gik(x) =

∣∣∣∣∣∣

g22(x)g33(x) 0 00 g33(x)g11(x) 00 0 g11(x)g22(x)

∣∣∣∣∣∣. (66)

From (66) one can easily see constraints between diagonal elements of

[εik(x)] = diag[ε1(x), ε2(x), ε3(x)] , [µik(x)] = diag(µ1(x), µ2(x), µ3(x)] ,

they are

ε1(x) = ε0 g00(x)g11(x), ε2(x) = ε0 g00(x)g22(x), ε3(x) = ε0 g00(x)g33(x) ,

µ1(x) =1

µ0ε20

ε2(x)ε3(x)

[g00(x)]2, µ2(x) =

1

µ0ε20

ε3(x)ε1(x)

[g00(x)]2, µ3(x) =

1

µ0ε20

ε1(x)ε2(x)

[g00(x)]2.

(67)

7. 3+1 splitting in metrical tensor

Let us obtain several formulas needed below. Starting from

gαβ(x)gβρ(x) = δαρ ,

after splitting it into four equations we get

(g0i(x) = gi(x), g0i(x) = gi(x)) ,

δij = gigj + gikgkj , (68)

1 = g00g00 + gigi , (69)

0 = g00gj + glglj , (70)

0 = gjg00 + gjlgl . (71)

Taking gj from (70)

gj = −gjl gl

g00, (72)

and substituting it into (68) we get

δij = −gi glglj

g00+ gilglj , =⇒ δi

j = ( gil − gigl

g00) glj .

219


With the notation

γil(x) ≡ gil − gi gl/g00 , (73)

the previous relation will look

γil(x) glj(x) = δij . (74)

In other words, γil(x) turns to be an inverse matrix for the spatial part glj(x) of the metricaltensor gαβ(x). Now, in the same manner, taking gi

gi = −gil gl

g00

, (75)

and substituting it into (68) we get to

δij = −gil gl

g00

gj + gil glj =⇒ δij = gil (glj − gl gj

g00

) ,

which with notation:

γlj(x) = glj − gl gj/g00 (76)

will read

δij = gil(x) γlj(x) . (77)

So, the quantity γlj(x) turns to be an inverse matrix for the spacial part glj(x) of the contra-variant metrical tensor gαβ(x). One can produce two additional relations; to this it suffices gi

from (70) to substitute into (69), and gi from (75) substitute into (69):

1

g00gijg

igj = (g00g00 − 1) , (78)

1

g00

gijgigj = (g00g00 − 1) . (79)

8. Inverse material equations

Having (direct) material equations:

Hρσ(x) = ε0 gρα(x)gσβ(x) Fαβ(x) =⇒{

Di = εik(x) Ek + αik(x) Bk ,

H i = βik(x) Ek + µik(x) Bk ,(80)

let us revert the problem and derive inverse ones

Fρσ(x) =1

ε0

gρα(x)gσβ(x) Hαβ(x) =⇒{

Ei = Eik(x) Dk + Aik(x) Hk ,

Bi = Bik(x) Dk + Mik(x) Hk .(81)

By symmetry reason, one does not need to make any substantially new calculation in additionto these given in Section 6. Expressions for Eik(x), Aik(x), Bik(x),Mik(x) are:

Eik(x) =1

ε0

[ g00(x) gik(x)− gi(x) gk(x) ] ,

Mik(x) = µ01

2εimn gml(x) gnj(x) εljk ,

Aik(x) = +1

ε0cgij(x) gl(x) εljk ,

Bik(x) = − 1

ε0cgj(x) εjil glk(x) , (82)

220


with evident symmetries

Eik(x) = +Eki(x) , Mik(x) = +Mki(x) , Bki(x) = +Aik . (83)

So the (6× 6) - matrix determining direct material equations is symmetrical

∣∣∣∣Ek(x)Bk(x)

∣∣∣∣ =

∣∣∣∣∣Ekl(x) Akl(x)

Bkl(x) Mkl(x)

∣∣∣∣∣∣∣∣∣Dl(x)H l(x)

∣∣∣∣ . (84)

All formulas of section 5 have their counterparts (with lower indices). For instance, if themetrical tensor has a quasi-diagonal form

gαβ(x) =

∣∣∣∣∣∣∣∣

g00 0 0 00 g11 g12 g13

0 g21 g22 g23

0 g31 g32 g33

∣∣∣∣∣∣∣∣, (85)

the material equations are

Ei = Eik(x) Dk , Bi = Mik(x) Hk , (86)

and Eik(x) and Mik(x) look as

(Eik) =1

ε0

g00

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣, (Mik) = µ0

∣∣∣∣∣∣

G11 G12 G13

G21 G22 G23

G31 G32 G33

∣∣∣∣∣∣, (87)

and

Gik(x) = (−1)j+kMik(x) =

=

∣∣∣∣∣∣

(g22g33 − g23g32) (g31g23 − g21g33) (g21g32 − g22g31)(g32g13 − g33g12) (g33g11 − g31g13) (g31g12 − g32g11)(g12g23 − g13g22) (g13g21 − g11g23) (g11g22 − g12g21)

∣∣∣∣∣∣. (88)

For general case of an arbitrary metric, expression for Eik(x) is modified:

(Eik) =1

ε0

g00

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣− 1

ε0

∣∣∣∣∣∣

g1 g1 g1 g2 g1 g3

g2 g1 g2 g2 g2 g3

g3 g1 g3 g2 g3 g3

∣∣∣∣∣∣; (89)

221


whereas Mik preserves its form. For two additional tensors Aik, Bik we have explicit expressions

Aik(x) = +1

ε0cgij(x) gl(x) εljk , =⇒

A(x) =1

ε0c

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣

∣∣∣∣∣∣

0 +g3 −g2

−g3 0 +g1

+g2 −g1 0

∣∣∣∣∣∣=

=1

ε0c

∣∣∣∣∣∣

(−g12g3 + g13g2) (g11g3 − g13g1) (−g11g2 + g12g1)(−g22g3 + g23g2) (g21g3 − g23g1) (−g21g2 + g22g1)(−g32g3 + g33g2) (g31g3 − g33g1) (−g31g2 + g32g1)

∣∣∣∣∣∣(90)

Bik(x) = − 1

ε0cgj(x) εjil glk(x) , =⇒

B(x) =1

ε0c

∣∣∣∣∣∣

0 −g3 +g2

+g3 0 −g1

−g2 +g1 0

∣∣∣∣∣∣

∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣=

=1

ε0c

∣∣∣∣∣∣

(−g12g3 + g13g2) (−g22g3 + g23g2) (−g32g3 + g33g2)(g11g3 − g13g1) (g21g3 − g23g1) (g31g3 − g33g1)

(−g11g2 + g12g1) (−g21g2 + g22g1) (−g31g2 + g32g1)

∣∣∣∣∣∣. (91)

For the diagonal metric

[gαβ(x)]

∣∣∣∣∣∣∣∣

g00(x) 0 0 00 g11(x) 0 00 0 g22(x) 00 0 0 g33(x)

∣∣∣∣∣∣∣∣, (92)

the (inverse) material equation become much more simple

Ei = Eik(x) Dk , Bi = Mik(x) Dk , (93)

[Eik(x)] =1

ε0

g00(x)

∣∣∣∣∣∣

g11(x) 0 00 g22(x) 00 0 g33(x)

∣∣∣∣∣∣,

[Mik(x)] = µ0

∣∣∣∣∣∣

G11(x) 0 00 G22(x) 00 0 G33(x)

∣∣∣∣∣∣,

[Gik(x)] =

∣∣∣∣∣∣

g22(x)g33(x) 0 00 g33(x)g11(x) 00 0 g11(x)g22(x)

∣∣∣∣∣∣. (94)

Among diagonal elements of Eik, Mik, one can note specific connection

E1(x) =1

ε0

g00(x)g11(x), E2(x) =1

ε0

g00(x)g22(x), E3(x) =1

ε0

g00(x)g33(x),

M1(x) = µ0ε20

E2(x)E3(x)

[g00(x)]2, M2(x) = µ0ε

20

E3(x)E1(x)

[g00(x)]2, M3(x) = µ0ε

20

E1(x)E2(x)

[g00(x)]2.

(95)

Additionally, let us verify two identities

εik(x) Ekl(x) = δil , µik(x) Mkl(x) = δi

l

222


which being detailed are

∣∣∣∣∣∣

g00g11 0 00 g00g22 00 0 g00g33

∣∣∣∣∣∣

∣∣∣∣∣∣

g00g11 0 00 g00g22 00 0 g00g33

∣∣∣∣∣∣=

∣∣∣∣∣∣

1 0 00 1 00 0 1

∣∣∣∣∣∣,

∣∣∣∣∣∣

g22g33 0 00 g33g11 00 0 g11g22

∣∣∣∣∣∣

∣∣∣∣∣∣

g22g33 0 00 g33g11 00 0 g11g22

∣∣∣∣∣∣=

∣∣∣∣∣∣

1 0 00 1 00 0 1

∣∣∣∣∣∣.

In general case, symmetrical (6 × 6)-matrices determining (direct and inverse) materialequations:

∣∣∣∣Di(x)H i(x)

∣∣∣∣ =


βik(x) µik(x)

∣∣∣∣∣∣∣∣∣Ek(x)Bk(x)

∣∣∣∣∣∣∣∣Ek(x)Bk(x)

∣∣∣∣ =


Bkl(x) Mkl(x)

∣∣∣∣∣∣∣∣∣Dl(x)H l(x)

∣∣∣∣ =


Bkl(x) Mkl(x)

∣∣∣∣∣

must obey the following conditions


βik(x) µik(x)

∣∣∣∣∣


Bkl(x) Mkl(x)

∣∣∣∣∣ =

∣∣∣∣δkl 00 δk

l

∣∣∣∣ ,

that is four relations must hold

εik(x) Ekl(x) + αik(x) Bkl(x) = δkl ,

εik(x) Akl(x) + αik(x) Mkl(x) = 0 ,

βik(x) Ekl(x) + µik(x) Bkl(x) = 0

βik(x) Akl(x) + µik(x) Mkl(x) = δkl . (96)

Let us verify the simplest case of quasi-diagonal metrics (g0i(x) = 0), then we have identities

εik(x) Ekl(x) = δkl , µik(x) Mkl(x) = δk

l ; (97)

Let us verify that the first equation in (97) hold in fact. Remembering

εik(x) = ε0 g00(x)gik(x) , Ekl(x) =1

ε0

g00(x)gkl(x) ,

equation ε(x) E(x) = I become an identity

[ g00(x)g00(x) ] [ gik(x)gkl(x) ] = δik . (98)

Now, remembering

µik(x) =1

µ0

1

2εimn gma(x)gnb(x) εabk , Mkl(x) = µ0

1

2εkps gpc(x)gsd(x) εcdl ,

we get

µik(x)Mkl(x) =1

4εimn gma(x)gnb(x) [ εabkεkps] gpc(x)gsd(x) εcdl ;

223


ant further with the identity

εabkεkps = δapδbs − δasδbp

we have arrived at the identity:

µik(x)Mkl(x) =1

2εimn gmp(x)gns(x) gpc(x)gsd(x) εcdl =

=1

2εimn δm

c δnd εcdl =

1

2εimn εmnl = δi

l . (99)

9. Geometrical modeling of the uniform media

Let us consider one special form of the metrical tensor:

gαβ(x) =

∣∣∣∣∣∣∣∣

a2 0 0 00 −b2 0 00 0 −b2 00 0 0 −b2

∣∣∣∣∣∣∣∣, gαβ(x) =

∣∣∣∣∣∣∣∣

a−2 0 0 00 −b−2 0 00 0 −b−2 00 0 0 −b−2

∣∣∣∣∣∣∣∣, (100)

where a2 and b2 are arbitrary (positive) numerical parameters. The material equations gener-ated by that geometry are

Di = εik Ek , H i = µik Bk ,

(εik) = ε0g00

∣∣∣∣∣∣

g11 0 00 g22 00 0 g33

∣∣∣∣∣∣=

ε0

a2b2

∣∣∣∣∣∣

−1 0 00 −1 00 0 −1

∣∣∣∣∣∣,

(µik) =1

µ0

∣∣∣∣∣∣

G11 0 00 G22 00 0 G33

∣∣∣∣∣∣=

1

µ0b4

∣∣∣∣∣∣

1 0 00 1 00 0 1

∣∣∣∣∣∣, (101)

or differently

Di = − ε0

a2b2Ei , H i =

1

µ0b4Bi , (102)

and the Maxwell equations take the form

∂1 B1 + ∂2 B2 + ∂3 B3 = 0 , ∂t B1 = ∂2 E3 − ∂3 E2 ,

∂t B2 = ∂3 E1 − ∂1 E3 , ∂t B3 = ∂1 E2 − ∂2 E1 ,

∂iDi = ρ ,

∂

∂x2H3 − ∂

∂x3H2 = J1 +

∂

∂tD1 ,

∂

∂x3H1 − ∂

∂x1H3 = J2 +

∂

∂tD2 ,

∂

∂x1H2 − ∂

∂x2H1 = J3 +

∂

∂tD3 . (103)

In vector terms

B = (Bi) , E = (−Ei) , H = (H i) , D = (Di) , (104)

224


they can be rewritten as follows:

div B = 0 , rot E = −∂B

∂t,

div D = ρ , rot D = J +∂D

∂t(105)

At this eqs. (102) look

D =ε0

a2b2E , H =

1

µ0b4B , ?? (106)

they may be compared with

D = εε0 E , H =1

µ0µB ; (107)

from which it follows

b2 =√

µ , a2 =1

ε

1√µ

. (108)

Corresponding metrical tensor (100) is

gαβ(x) =

∣∣∣∣∣∣∣∣

a2 0 0 00 −b2 0 00 0 −b2 00 0 0 −b2

∣∣∣∣∣∣∣∣=

1√ε

∣∣∣∣∣∣∣∣

1/√

εµ 0 0 00 −√εµ 0 00 0 −√εµ 00 0 0 −√εµ

∣∣∣∣∣∣∣∣,

√−g = a b3 =µ

ε. (109)

The contra variant metrical tensor is

gαβ =√

ε

∣∣∣∣∣∣∣∣

√εµ 0 0 00 −1/

√εµ 0 0

0 0 −1/√

εµ 00 0 0 −1/

√εµ

∣∣∣∣∣∣∣∣, (110)

Thus, material equations look

Hab = ε0gamgbn Fmn, =⇒ ε0g

amgbn =

= εε0

∣∣∣∣∣∣∣∣

√εµ 0 0 00 −1/

√εµ 0 0

0 0 −1/√

εµ 00 0 0 −1/

√εµ

∣∣∣∣∣∣∣∣

am

⊗

∣∣∣∣∣∣∣∣

√εµ 0 0 00 −1/

√εµ 0 0

0 0 −1/√

εµ 00 0 0 −1/

√εµ

∣∣∣∣∣∣∣∣

am

.

(111)

Let us compare (111) with material equations in moving media [16]

Hab = ∆abmn Fmn ,

∆abmn = ε0εk2 [ gam + (εµ− 1) uaum ] [ gbn + (εµ− 1) ubun ] , (112)

225


which in motionless reference frame becomes

Hab = ∆abmnFmn, ∆abmn = εε0

1

εµ

∣∣∣∣∣∣∣∣

εµ 0 0 00 −1 0 00 0 −1 00 0 0 −1

∣∣∣∣∣∣∣∣am

⊗

∣∣∣∣∣∣∣∣

εµ 0 0 00 −1 0 00 0 −1 00 0 0 −1

∣∣∣∣∣∣∣∣bn

=

= εε0

∣∣∣∣∣∣∣∣

√εµ 0 0 00 −1/

√εµ 0 0

0 0 −1/√

εµ 00 0 0 −1/

√εµ

∣∣∣∣∣∣∣∣am

⊗

∣∣∣∣∣∣∣∣

√εµ 0 0 00 −1/

√εµ 0 0

0 0 −1/√

εµ 00 0 0 −1/

√εµ

∣∣∣∣∣∣∣∣am

.

(113)

So, two results coincide with each other.

9. Geometrical modeling of an anisotropic media

Let us extend the previous analysis and consider another metrical tensor:

gαβ(x) =

∣∣∣∣∣∣∣∣

a2 0 0 00 −b2

1 0 00 0 −b2

2 00 0 0 −b2

3

∣∣∣∣∣∣∣∣, gαβ(x) =

∣∣∣∣∣∣∣∣

a−2 0 0 00 −b−2

1 0 00 0 −b−2

1 00 0 0 −b−2

1

∣∣∣∣∣∣∣∣, (114)

where a2, b21, b

22, b

23, are arbitrary (positive) numerical parameters. The material equations gen-

erated by that geometry are

Di = εik Ek , H i = µik Bk , (115)

(εik) = ε0a−2

∣∣∣∣∣∣

−b−21 0 0

0 −b−22 0

0 0 −b−23

∣∣∣∣∣∣, (µik) =

1

µ0

∣∣∣∣∣∣

b−22 b−2

3 0 00 b−2

3 b−21 0

0 0 b−21 b−2

2

∣∣∣∣∣∣,

or differently

D1 = − ε0

a2b21

E1 , D2 = − ε0

a2b22

E2 , D3 = − ε0

a2b23

E3 ,

H1 =1

µ0 b22b

23

B1 , H2 =1

µ0 b23b

21

B2 , H3 =1

µ0 b21b

22

B3 , (116)

and the Maxwell equations in vector terms

B = (Bi) , E = (−Ei) , H = (H i) , D = (Di) ,

can be written as follows:

div B = 0 , rot E = −∂B

∂t,

div D = ρ , rot D = J +∂D

∂t(117)

226


At this material equations eqs. (117) may be compared with

D1 = −ε1 E1 , D2 = −ε2 E2 , D3 = −ε3 E3 ,

H1 =1

µ1

B1 , H2 =1

µ2

B2 , H3 =1

µ3

B3 ,

from which it follows

ε1 =ε0

a2b21

, ε2 =ε0

a2b22

, ε3 =ε0

a2b23

, (118)

µ1 = µ0 b22 b2

3 , µ2 = µ0 b23 b2

1 , µ3 = µ0 b21 b2

2 . (119)

From this it follows identities

µ1

ε1

=µ2

ε2

=µ3

ε3

=µ0

ε0

(a2 b21 b2

2 b23) = −g

µ0

ε0

,

−g =

√µ2

1 + µ22 + µ2

3

ε21 + ε2

2 + ε22

ε0

µ0

,µi√

µ21 + µ2

2 + µ23

=εi√

ε21 + ε2

2 + ε22

. (120)

The latter means that one may use four independent parameters, ε, µ, ni:

εi = ε0ε ni , µi = µ0µ ni, n2 = 1 (121)

From (110b) one can readily express b2i in terms of µi:

µ2µ3 = µ20 b4

1 (b22 b2

3) = µ0 b41 µ1 =⇒ b2

1 =

√µ2µ3

µ0µ1

=√

µ

√n2n3

n1

µ3µ1 = µ20 b4

2 (b23 b2

1) = µ0 b42 µ2 =⇒ b2

2 =

√µ3µ1

µ0µ2

=√

µ

√n3n1

n2

µ1µ2 = µ20 b4

3 (b21 b2

2) = µ0 b43 µ3 =⇒ b2

3 =

√µ1µ2

µ0µ3

=√

µ

√n1n2

n3

. (122)

In turn, from a2 b21 b2

2 b23 = µ/ε it follows

a2 =µ

ε

1

b21b

22b

23

=1

ε√

µ

1√n1n2n3

(123)

The formula (122)-(123) provide us with (anisotropic) extension

gab(x) =1√ε

∣∣∣∣∣∣∣∣∣∣∣

1√εµ

1√n1n2n3

0 0 0

0 −√εµ√

n2n3

n10 0

0 0 −√εµ√

n3n1

n20

0 0 0 −√εµ√

n1n2

n3

∣∣∣∣∣∣∣∣∣∣∣

(124)

of the previous (isotropic) metrical tensor

gαβ(x) =1√ε

∣∣∣∣∣∣∣∣

1√εµ

0 0 0

0 −√εµ 0 00 0 −√εµ 00 0 0 −√εµ

∣∣∣∣∣∣∣∣. (125)

227


One other, more involved, example of effective anisotropic media is provided by the materialequations for uniform media for a moving observer (more details see in [14-16]):

∆abmn = ε0εk2 [ gam + (εµ− 1) uaum ] [ gbn + (εµ− 1) ubun ] , (126)

Hρσ(x) = ∆abmn Fmn = ε0 gρα(x) gσβ(x) Fαβ(x) .

References

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[2] Mandel’stam L.I., Tamm I.E. Math. Annalen. 95, 154-160 (1925).[3] Abraham M., Becker R. Theories der Electrizitat, Leipzig,1933. Bd. II ; Rissian translation:

Liningrad- Moskva, GITTL, 1941.[4] Stratton J.A. Electromagnetic theory; Russina translation, Moskva-Leningrad, 1948.[5] Valer Novacu, Introducere in electrodinamica, 1955; Rissian translation: Novaky V. Introduction

to electrodynamics. Moskva, 1963.[6] Landay L.D., Lifshitz E.M. The field theory. Moscow, 1973. pages 326-329[7] Pham Man Quan. Compt. Rend/ 1957. Vol. 245. no 21. 1782-1785.[8] Tomil’chik L.M., Fedorov A.I. Magnetic anisotropy as metrical property of space. Kristalography.

1959. Vol. 4, no 4, pages 498-504.[9] Post E.J. Formal structure of electrodynamics. General covariance and electromagnetics/ Ams-

terdam, 1962.[10] De Lange O.L., Raab R.E. Post’s constrain for electromagnetic constitutive relations. J. Opt. A.

2001. Vol. 3. L23-L26.[11] Raab R.E., De Lange O.L. Symmetry constrains for electromagnetic constitutive relations. J.

Opt. A. 2001. Vol. 3. pages 446-451.[12] Barkovsky L.M., Furs A.N., Operator methods of description of optical fields in complex medias.

Minsk, 2003 (Chapter 2, Section 19. Clasiffication for medias )[13] Alexeeva T.A., Barkovsky L.M. To the history of electrodynamics constitutive equations. Proc.

of XII Ann. Sem. Nonlinear Phenomena in Complex Systems. Minsk. 2005. P. -[14] Bolotowskij B.M., Stoliarov C.N. Contamporain state of electrodynamics of moving medias (un-

limited medias) Eistein collection, 1974. Moscow, Nauka. 1976. pages 179-275.[15] Barykin V.N., Tolkachev E.A., Tomilchik L.M. On symmetry aspects of choice of material equa-

tions in micriscopic electrodynamics of moving medias. // Vesti AN BSSR, ser. fiz.-mat. 1982,no 2, P.96-98.

[16] V.M. Red’kov, George J. Spix. On the different forms of the Maxwell’s electromagnetic equationsin a uniform media. Proc. of XII Ann. Sem. Nonlinear Phenomena in Complex Systems. Minsk.2005. P. 118-133; hep-th/0604080.

[17] Take notice that Ei = −Ei, Di = −Di, Bi = +Bi, Hi = +Hi.[18] There exists one special case; namely, if g(x) does not depend on coordinates in fact then the factor

√−g canbe omitted from the formulas (40) and below.

[19] For discussion of different types of electromagnetic medias see in [7-12].[20] The symbol of trace-operation Sp that means the sum over diagonal elements of a matrix is used.

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