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Lobachevsky geometry simulates a medium with special constitutive relations Di = ϵ0ϵikEk, Bi = μ0μikHk, where two matrices coincide: ϵik(x) = μik(x). The situation is specified in quasi-Cartesian coordinates (x, y, z) in Lobachevsky space, they are appropriate for modeling a medium nonuniform along the axis z. Exact solutions of the Maxwell equations in complex form of Majorana-Oppenheimer have been constructed. The problem reduces to a second-order differential equation for a certain primary function which can be associated with the one-dimensional Schrödinger problem for a particle in external potential field U(z) = U0e2z. In the frames of the quantum mechanics, Lobachevsky geometry acts as an effective potential barrier with reflection coefficient R = 1; in electrodynamic context, this geometry simulates a medium that effectively acts as an ideal mirror distributed in space. Penetration of the electromagnetic field into the effective medium along the axis z depends on the parameters of an electromagnetic waves ω, k2 1 + k2 2 and the curvature radius ρ of the used Lobachevsky model. The generalized quasi-plane wave solutions f(t, x, y, z) = E + iB and the relevant system of equations are transformed into the real form, which permit us to relate geometry characteristics with expressions for effective tensors of electric and magnetic permittivities.
Maxwell equations, Majorana-Oppenheimer formalism, Lobachevsky geometry, exact solutions, effective constitutive relations
Introduction
To treat Maxwell equations we make use of complex representation
of them according to the known approach by Majorana-
Oppenheimer [1–11], also see [12, 13] and references
therein for extending this approach to curved space-time
models.
The situation is specified in quasi-Cartesian coordinates
in Lobachevsky space, they are appropriate for modeling a
medium nonuniform along the axis z. Exact solutions of the
covariant Maxwell equations in complex E +iB form of Majorana-
Oppenheimer have been constructed. The problem reduces
to a second order differential equation for a certain primary
function which can be associated with the one-dimensional
Schrödinger problem for a particle in external potential
field U(z) = U0e2z. In quantum mechanics, curved geometry
acts as an effective potential barrier with reflection coefficient
R = 1; in electrodynamic context results are similar:
Lobachevsky geometry simulates a medium that effectively
acts as an ideal mirror. Penetration of the electromagnetic
field into the effective medium along the axis z depends on
the parameters of the electromagnetic waves ω, k2
1 +k2
2 and
the curvature radius ρ of the used Lobachevsky space. These
generalized quasi-plane solutions f(t, x, y, z) = E + iB
and the relevant system of equations are transformed into the
real form, which permit us to relate geometry characteristics
with expressions for effective tensors of electric and magnetic
permittivities.
1. Cartesian coordinates in Lobachevsky space
We will apply the coordinate system in Lobachevsky space
dS2 = dt2 − e−2z(dx2 + dy2) − dz2,
dV = e−2zdxdydz. (1)
It is helpful to have at hand some details of the parametrization
of the modelH3 by coordinates (x, y, z). It is known that
this model can be identified with a branch of hyperboloid in
4-dimension flat space
u20
− u21
− u22
− u23
= ρ2, u0 = +
√
ρ2 + u2.
Coordinates (x, y, z) are referred to ua by relations
u0 =
1
2
[
(ez + e−z) + (x2 + y2)e−z]
, u1 = xe−z,
u2 = ye−z, u3 =
1
2
[
(ez − e−z) + (x2 + y2)e−z]
.
We will employ the Poincare realization for Lobachevsky
space as the inside part of the 3-sphere
qi =
ui
u0
=
√ ui
ρ2 + u21
+ u22
+ u23
, qiqi < 1.
Quasi-Cartesian coordinates (x, y, z) are referred to qi as
follows
q1 =
2x
x2 + y2 + e2z + 1
,
q2 =
2y
x2 + y2 + e2z + 1
,
q3 =
x2 + y2 + e2z − 1
x2 + y2 + e2z + 1
. (2)
Inverses to (2) relations are
x =
q1
1 − q3
, y =
q2
1 − q3
, ez =
√
1 − q2
1 − q3
. (3)
In particular, note that on the axis q1 = 0, q2 = 0, q ∈
(−1, +1) relations (3) assume the following parametrization
of the axis z
x = 0, y = 0, ez =
√
1 + q3
1 − q3
,
so that
q3 → +1, ez → +∞, z → +∞;
q3 → −1, ez → +0, z → −∞.
Solutions of the Maxwell equations, constructed in the following
way, can be of interest for description of electromagnetic
waves in special media because Lobachevsky geometry
simulates effectively a special medium [12, 13], inhomogeneous
along the axis z. Effective electric permittivity tensor
ϵik(x) is given by
ϵik(x) = −
√
−gg00(x)gik(x) =
1 0 0
0 1 0
0 0 e−2z
,
whereas the effective magnetic permittivity tensor is
(μ−1)ik(x) =
√
−g
g22g33 0 0
0 g33g11 0
0 0 g11g22
=
=
1 0 0
0 1 0
0 0 e2z
.
The constitutive relations read
Di = ϵ0ϵikEk, Bi = μ0μikHk;
two tensors coincide ϵik(x) = (μ−1)ik(x).
2. Maxwell equations in complex form, separation
of the variables
In the coordinates (1), we will use the tetrad
eβ
(α) =
1 0 0 0
0 ez 0 0
0 0 ez 0
0 0 0 1
,
e(α)β =
1 0 0 0
0 −e−z 0 0
0 0 −e−z 0
0 0 0 −1
.
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59
In this tetrad, the matrix equation (see notations in [12, 13])
has the form
(
−i∂t + α1ez∂x + α2ez∂y + α3∂z−
−α1s2 + α2s1
)(
0
E + iB
)
= 0. (4)
Let us apply the substitution
(
0
E + iB
)
= e−iωteik1xeik2y
(
0
f(z)
)
,
ei(k1x+k2y−ωt) = eiφ.
Eq. (4) gives
(−ω + α1ezik1 + α2ezik2 + α3 d
dz
−
−α1s2 + α2s1)(0, f1(z), f2(z), f3(z))t = 0.
Here ()t stands for transposition. After calculation with
the use of explicit expressions for all involved matrices
(see [12, 13]), we derive the first-order system for functions
f1(z), f2(z), f3(z)
ik1ezf1 + ik2ezf2 +
(
d
dz
− 2
)
f3 = 0,
−ωf1 −
(
d
dz
− 1
)
f2 + ik2ezf3 = 0,
−ωf2 +
(
d
dz
− 1
)
f1 − ik1ezf3 = 0,
−ωf3 − ik2ezf1 + ik1ezf2 = 0.
Allowing for three last equations in the first one, we get
the identity 0 = 0. So, there exist only three independent
equations (we will simplify notations: k1 = a, k2 = b)
ωf3 = −ibezf1 + iaezf2,
ωf1 = −
(
d
dz
− 1
)
f2 + ibezf3,
ωf2 =
(
d
dz
− 1
)
f1 − iaezf3. (5)
With substitutions f1 = ezF1(z), f2 = ezF2(z) from Eqs.
(5) we get
ωf3 = −ibe2zF1 + iae2zF2, ωF1 = − d
dz
F2 + ibf3,
ωF2 =
d
dz
F1 − iaf3. (6)
There exists a particular case readily treatable, when
a = 0, b = 0, f3 = 0:
ωF1 = − d
dz
F2, ωF2 =
d
dz
F1,
that is
F1(z) = e±iωz, F2(z) = ±ie±iωz,
which leads to the following plane wave solutions
Φ± =
(
0
E + iB
)
= e−iωtez (
0, e±iωz,±ie±iωz, 0
)t
,
whence we get
E+
1 + iB+
1 = cos(ωt − ωz) − i sin(ωt − ωz),
E+
2 + iB+
2 = sin(ωt − ωz) + i cos(ωt − ωz),
and
E
−
1 + iB
−
1 = cos(ωt + ωz) − i sin(ωt + ωz),
E
−
2 + iB
−
2 = −sin(ωt + ωz) − i cos(ωt + ωz).
Let us present this solution in the real form
E+
1 = cos(ωt − ωz), E+
2 = sin(ωt − ωz), E+
3 = 0,
B+
1 = −sin(ωt − ωz), B+
2 = cos(ωt − ωz), B+
3 = 0
and
E
−
1 = cos(ωt + ωz), E
−
2 = −sin(ωt + ωz), E
−
3 = 0,
B
−
1 = −sin(ωt+ωz), B
−
2 = −cos(ωt+ωz), B
−
3 = 0.
In turn, from complex-valued identities (in this case, we have
φ = −ωt)
E + iB = eiφf(z) = eiφ(F(z) + iG(z)) =
= (cos φ + i sin φ)(F(z) + iG(z)),
F∗ = F, G∗ = G, φ = k1x + k2y − ωt
we derive expressions for real vectors E and B:
E = cos φF(z) − sinφG(z),
B = sin φF(z) + cosφG(z), φ = −ωt.
Let us turn back to the general system (6); with the help of
the first equation we eliminate the variable f3, so producing
the system of linked equations for F1 and F2 (
d
dz
+
abe2z
ω
)
F2 =
b2e2z − ω2
ω
F1,
(
d
dz
− abe2z
ω
)
F1 =
ω2 − a2e2z
ω
F2. (7)
In the new variable Z, ez =
√
ωZ two last equations are
written as
Z
(
d
dZ
+ abZ
)
F2 = (b2Z2 − ω)F1,
Z
(
d
dZ
− abZ
)
F1 = −(a2Z2 − ω)F2. (8)
This system can be solved straightforwardly in terms of
the Heun confluent functions. Indeed, from (8) it follows a
second order differential equation for F1
d2F1
dZ2
− a2Z2 + ω
Z(a2Z2 − ω)
dF1
dZ
+
+
(
ω2
Z2 +
2abω
a2Z2 − ω
− (a2 + b2)ω
)
F1 = 0,
60
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where we note the presence of an additional singular point
Z = ±
√
ωa−1. In the new variable y = a2Z2ω−1, we
arrive at the equation
d2F1
dy2 +
(
1
y
− 1
y − 1
)
dF1
dy
+
+
(
ω2
4y2
− 2abω + (a2 + b2)ω2
4a2y
+
bω
2a(y − 1)
)
F1 = 0.
With the use of the substitution F1 = ycg1(y), c =
±iω/2, further we derive
d2g1
dy2 +
(
2c + 1
y
− 1
y − 1
)
dg1
dy
+
+
(
2c − ω2/2 − bω/a − b2ω2/(2a2)
2y
+
+
−2c + bω/a
2(y − 1)
)
g1 = 0,
which can be identified with the confluent Heun equation. Below
we will develop a method that makes possible to construct
solutions of the system (7) in terms of more simple
Bessel functions.
3. Solutions in terms of the Bessel functions
Let us perform a linear transformation over the system
(7):
F1 = αG1 + βG2, F2 = mG1 + nG2;
G1 = nF1 − βF2, G2 = −mF1 + αF2; (9)
suppose the constraint αn−βm = 1. Combining equations
from (7), we get
nZ
(
d
dZ
− abZ
)
F1 − βZ
(
d
dZ
+ abZ
)
F2 =
= −n(a2Z2 − ω)F2 − β(b2Z2 − ω)F1,
−mZ
(
d
dZ
− abZ
)
F1 + αZ
(
d
dZ
+ abZ
)
F2 =
= m(a2Z2 − ω)F2 + α(b2Z2 − ω)F1,
whence it follows
Z
d
dZ
G1 − Z2ab(nF1 + βF2) =
= −Z2(na2F2 + βb2F1) + ω(nF2 + βF1),
Z
d
dZ
G2 + Z2ab(mF1 + αF2) =
= Z2(ma2F2 + αb2F1) − ω(mF2 + αF1). (10)
Taking into account (9), we reduce Eqs. (10) to other form
[
Z
d
dZ
− Z2ab(nα + mβ) + Z2(a2mn + b2αβ)−
−ω(nm+αβ)
]
G1 =
[
−Z2(an−bβ)2+ω(n2+β2
]
G2,
[
Z
d
dZ
+ Z2ab(nα + mβ) − Z2(a2mn + b2αβ)+
+ω(nm+αβ)
]
G2 =
[
Z2(am−bα)2−ω(m2+α2)
]
G1.
Let us impose additional restrictions:
the first one is
an − bβ = 0, that is β
n
=
a
b
,
[
Z
d
dZ
− Z2ab(nα + mβ) + Z2(a2mn + b2αβ)−
−ω(nm + αβ)
]
G1 = ω(n2 + β2)G2,
[
Z
d
dZ
+ Z2ab(nα + mβ) − Z2(a2mn + b2αβ)+
+ω(nm+αβ)
]
G2 =
[
Z2(am−bα)2−ω(m2+α2)
]
G1;
(11)
the second one is
am − bα = 0, that is α
m
=
a
b
,
[
Z
d
dZ
− Z2ab(nα + mβ) + Z2(a2mn + b2αβ)−
−ω(nm+αβ)
]
G1 =
[
−Z2(an−bβ)2+ω(n2+β2)
]
G2,
[
Z
d
dZ
+ Z2ab(nα + mβ) − Z2(a2mn + b2αβ)+
+ω(nm + αβ)
]
G2 = −ω(m2 + α2)G1.
These two possibilities are equivalent to each other, for
definiteness we will use the variant (11). It can be presented
in more symmetrical form
F1 = αG1 + βG2 =
√ b
a2 + b2
G1 +
√ a
a2 + b2
G2,
F2 = mG1 + nG2 = −√ a
a2 + b2
G1 +
√ b
a2 + b2
G2;
(12)
at this Eqs. (6) lead to
Z
d
dZ
G1 = ωG2,
Z
d
dZ
G2 =
[
Z2(a2 + b2) − ω
]
G1. (13)
From (13) we derive a second order equation for G1:
(
Z2 d2
dZ2 + Z
d
dZ
+ ω2 − ω(a2 + b2)Z2
)
G1 = 0.
(14)
It is convenient to transform this equation into the initial variable
z, then it reads
ez =
√
ωZ,
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61
(
Z2 d2
dZ2 + ω2 − (a2 + b2)e2z
)
G1 = 0. (15)
It can be associated with the Schrödinger equation
(
d2
dz2 + m − U(z)
)
φ(z) = 0 (16)
with the potential function U(z) = (a2 + b2)e2z, the corresponding
effective force acts on the left Fz = −2(a2 +
b2)e2z. The situation described by Eq. (15) can be illustrated
in Fig. 1.
z
U(z)
ǫ = ω2
Figure 1. Effective potential curve.
Рисунок 1. Эффективная потенциальная кривая.
Therefore, we should expect the properties of the electromagnetic
solutions similar to those existing in the relevant
quantum-mechanical problem. Note that when a = k1 =
0, b = k2 = 0, this force vanishes. In accordance with
(16), an equation below ω2 = U(z) = (a2 + b2)e2z determines
a critical point z0 in which behavior of the function
G1(x) must change dramatically. To such a point z0,
x0 = i
√
a2 + b2ez0 = iω. Expression for the turning point
z0 is given by the formula
z0 = ρ ln
ω
ρ
√
k2
1 + k2
2
.
The last relation is written in the usual units. The ρ is a curvature
radius of Lobachevsky space, it is a free parameter of
the model description.
The primary variable G1(x) determine all remaining
ones. Let us turn back to Eq. (14). In the variable x =
i
√
ω(a2 + b2)Z = i
√
a2 + b2ez it takes the Bessel form
(
d2
x2 +
1
x
d
x
+ 1 +
ω2
x2
)
G1 = 0. (17)
The first-order system (13), being transformed to the variable
x, reads
x
d
x
G1 = ωG2, x
d
x
G2 = −ω2 + x2
ω
G1.
The second function is determined by relation
G2 =
1
ω
x
d
dx
G2 =
1
ω
d
dz
G1.
In turn, taking into account the transformation (12), we get
(see (6))
f3 =
e2z
ω
(−ibF1 + iaF2) =
√
a2 + b2
iω
e2zG1(z).
Let us write down the final expressions for obtained solutions
E(z) + iB(z) = (cos φ + i sin φ)f(z),
φ = ax + by − iωt,
where
f1(z) = ezF1(z) =
= ez
(
√ b
a2 + b2
G1 +
√ a
a2 + b2
G2
)
,
f2(z) = ezF2(z) =
= ez
(
−√ a
a2 + b2
G1 +
√ a
a2 + b2
G2
)
,
f3(z) = −i
√
a2 + b2
ω
e2zG1(z),
where G1(z) is the solution to equation (17),
G2(z) =
1
ω
d
dz
G1(z), x = i
√
a2 + b2ez.
Conclusion
In the frames of the quantum mechanics, Lobachevsky
geometry acts as an effective potential barrier with reflection
coefficient R = 1. In electrodynamic context, results
are similar: this geometry simulates a medium that effectively
acts as an ideal mirror distributed in space. Penetration
of the electromagnetic field into the effective medium along
the axis z depends on the parameters of an electromagnetic
waves ω, k2
1 + k2
2 and the curvature radius ρ of the used
Lobachevsky model. The generalized quasi-plane wave solutions
f(t, x, y, z) = E + iB and the relevant system of
equations are transformed into the real form, which permit us
to relate geometry characteristics with expressions for effective
tensors of electric and magnetic permittivities.
The authors declare no conflict of interest.
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