Maxwell equations in Riemannian space-time, …npcs.j-npcs.org/Procc/v13p207.pdf · Maxwell...
Transcript of Maxwell equations in Riemannian space-time, …npcs.j-npcs.org/Procc/v13p207.pdf · Maxwell...
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 ]/
õ .
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]
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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
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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 ,
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Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix
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)
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Maxwell equations in Riemannian space-time, geometrical modeling of medias
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)
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Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix
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
+D1 0 −H3/c +H2/c+D2 +H3/c 0 −H1/c+D3 −H2/c +H1/c 0
∣∣∣∣∣∣∣∣=
∣∣∣∣∣∣∣∣
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
+D1 0 −H3/c +H2/c+D2 +H3/c 0 −H1/c+D3 −H2/c +H1/c 0
∣∣∣∣∣∣∣∣=
∣∣∣∣∣∣∣∣
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.
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Maxwell equations in Riemannian space-time, geometrical modeling of medias
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)
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Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix
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)
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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) .
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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) .
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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)
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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 .
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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)
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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)
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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
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Maxwell equations in Riemannian space-time, geometrical modeling of medias
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)
∣∣∣∣ =
∣∣∣∣∣εik(x) αik(x)
βik(x) µik(x)
∣∣∣∣∣∣∣∣∣Ek(x)Bk(x)
∣∣∣∣∣∣∣∣Ek(x)Bk(x)
∣∣∣∣ =
∣∣∣∣∣Ekl(x) Akl(x)
Bkl(x) Mkl(x)
∣∣∣∣∣∣∣∣∣Dl(x)H l(x)
∣∣∣∣ =
∣∣∣∣∣Ekl(x) Akl(x)
Bkl(x) Mkl(x)
∣∣∣∣∣
must obey the following conditions
∣∣∣∣∣εik(x) αik(x)
βik(x) µik(x)
∣∣∣∣∣
∣∣∣∣∣Ekl(x) Akl(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 ;
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Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix
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)
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Maxwell equations in Riemannian space-time, geometrical modeling of medias
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
1õ
. (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)
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Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix
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)
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Maxwell equations in Riemannian space-time, geometrical modeling of medias
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 =
õ2
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)
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Red’kov V.M., Tokarevskaya N.G., Bychkouskaya E.M., George J. Spix
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) .
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[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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