>K9m 2013, TOM 143, Bbin. 2, cTp. 257 262
© 2013
ELECTROMAGNETIC PLANE WAVES WITH NEGATIVE PHASE VELOCITY IN CHARGED BLACK STRINGS
M. Sharif* R. Manzoor**
Department of Mathematics, University of the Punjab 54590, Lahore, Pakistan
Received July 28, 2012
We investigate the propagation regions of electromagnetic plane waves with negative phase velocity in the ergosphere of static charged black strings. For such a propagation, some conditions for negative phase velocity are established that depend on the metric components and the choice of the octant. We conclude that these conditions remain unaffected by the negative values of the cosmological constant.
DOI: 10.7868/S0044451013020053
1. INTRODUCTION
The phenomenon of negative phase velocity (NPV) propagation is important duo to one of its consequences, negative refraction fl, 2]. This is the property of light propagation in a medium that occurs when the phase velocity of a plane wave has a negative projection onto the time-average Poynting vector. Alternatively, it is the plane wave propagation mode in which the wave vector and the time-average Poynting vector are oppositely aligned [3 5]. Negative refraction is an electromagnetic phenomenon in which light rays are refracted at the interface in a sense reverse to that normally expected. This property of negative refraction has generated considerable attention in the electromagnetic, optics, and material research communities [2, 3, 6].
Metamaterials are synthetic materials with unusual refractive index properties used to obtain the negative refraction effect. Negative values of the permittivity e and permeability //. are responsible for these unusual refractive index properties. This was originally proposed by Veselago [7]. The direct consequence of this property is the development of a wave propagation medium called the Veselago medium. The presence of the Veselago medium alters the propagation of plane waves such that the electric field, the magnetic field, and the wave vector follow a left-hand rule instead of
E-mail: msharif.math'&pu.edu.pk E-mail: rubabmanzoor9(öyahoo.<'om
the right-hand rule. This leads to the construction of left-handed metamaterials [8]. Lenses with extremely low distortion are one of the most useful applications of NPV supporting artificial metamaterial. These are widely used in the modern optics, for communication, entertainment, and data storage as well as for retrieval purposes [9 12].
The characteristics of NPV materials lead to the concept of anisotropic and bianisotropic materials that provide industrial benefits in modern technology [2, 4, 13]. The application of NPV propagation in astro-physical scenarios has been explored in the last few years. It was shown in [14,15] that the vacuum can support NPV propagation for particular spacctimes. The same authors [13] proved that the do Sitter spacetime supports NPV propagation, whereas the anti-de Sitter metric does not admit such a propagation. The propagation of electromagnetic plane waves with NPV in the Schwarzschild do Sitter spacetime was investigated in [16]. Some regions supporting NPV propagation within the ergosphere of an uncharged rotating and charged rotating black holes were explored in [17,18]. Plasma wave properties of the Schwarzschild and Schwarzschild do Sitter horizons in a Veselago medium were discussed in [19].
In this paper, we investigate propagation of electromagnetic plane waves of static charged black strings described by a cylindrical symmetric spacetime with a negative cosmological constant. The regions of NPV propagation are explored. The format of the paper is as follows. In the next section, we review the mathematical formalism. Section 3 describes the static charged
4 >K9T<E>, iibiii.2
257
black strings and piano wave propagation in R. In Sec. 4, wo investigato the conditions of NPV. Finally, we discuss and summarize the results in the last section.
2. REVIEW OF THE MATHEMATICAL FORMALISM
In this section, we review the mathematical formulation needed to discuss the propagation of electromagnetic waves in the vacuum in a curved spacetime. This is based on the formal analogy between electromagnetic waves in a flat spacetime in a fictitious instantaneously responding medium and in the curved spacetime in free space. Tamm [20] originally proposed this approach which was used by many authors [21 26].
The source-free covariant Maxwell equations for a curved spacetime are
F
Ctf3\V
¿w = o, Ff; = 0,
Ffj i,-a
a,/i = 0,1,2,3.
For a flat spacetime, these equations reduce to
¿W + W + ¿W = 0, (-.9)1/2F^ = 0. (1)
Here, FQ/ii and Fmi are the contravariant and covariant electromagnetic field tensors and
g = dot [.(/a/i] •
The semicolon (; ) and comma (,) respectively indicate covariant and ordinary derivatives. These equations can be rewritten as
(2)
Bi4 = 0, /?,.„ + eijkEjik = 0, /),., = 0, -A,0 + ZijkHj,k = o, i,j, k = 1,2,3,
where />',. Ej, I),. and II j are the components of the magnetic field vector B, electric field vector E, displacement field vector D, and magnetizing field vector H and Eiju is the three-dimensional Levi-Civita symbol.
The electromagnetic field vectors E, B, D, and H
are
Ei = Fm, Bi = (l/2)eijkFjk, Dt = (~g)1/2Fi0, Hi = (l/2)eijk(~g)1/2Fik.
(3)
These vectors satisfy the constitutive relations of an equivalent instantaneously responding medium that can describe the electromagnetic response of the vacuum in a curved spacetime. These constitutive relations are
D = eoO'E, B = /Í07H,
(4)
where e0 = 8.854 • 10 12 Fm 1 and //.o = 4?r x x 10-12 Hni-1 in SI units. The dyadic 7 can be expressed in the metric form
lab = -(-.9)
l/2f±
ab
.900
(5)
In the 3x3 dyadic form, Eqs. (2) and (4) can be written as [15, 17]
D(ei, r) = €ol(ct,r) ■ E(ct, r), B(ci,r) = fi.0l(ct,r) ■ H(ci,r).
(6)
(7)
3. CHARGED BLACK STRINGS AND WAVE PROPAGATION
Static charged black strings with a negative cos-mological constant have the line element of the form [27, 28]
d.s" =
2 2 a: r
b_
ar
dt2
2 2 a: r
b_
ar
dr"
■ r2d92 + a2r2dz2,
(8)
where
a2 = b = 4 G AI, c2 = 4 GQ2,
O
n \ 2Q h(r) = —
ar
const,
^oc < t < oc, 0 < r < oc,
^00 < ; < 00, 0 < 8 < 2n.
Here, M is the IIlclSS and Q is the linear charge density per unit length of the i line of the black strings, G is the gravitational constant, and A < 0 is the cosmo-logical constant. The black hole horizons are found by-setting g0o = 0,
r± =
b* ^J.s + v^.s2 - 4//2 - a) 2a
0)
where
s =
2
P = TT-
Here, r_ and r+ represent the inner and outer event horizons. In order to have a physical region, we take r+ only and neglect the inner event horizon.
For Q = 0, Eq. (8) yields the line element of static black strings
ds2 =
a2r2—— I dt2~ ar J
a2r2—— I dr2 + ar J
r2(102 + a2r2dz2, (10)
where mass is the only parameter and the respective event (outer) horizon is
r = r+ =
b3
a
(ID
Because 7 is a second-rank Cartesian tensor [13, 22, 23] we convert metric (8) into Cartesian coordinates as
9ab =
( -f 0 0
•<-2 + ,</2/ xy{l-f)
0 0
V 0
y>2 j- J-
*y{i-f) ,</2 + -<-2/
0
0
0 \
0
0
a2r2 )
(12)
where
2 4
g = —a r ,
j- 2 2 J = a: r
ar
The constitutive relations provide global description of the cylindrically symmetric spacotimo. To approximate a nonuniform metric 7a¡, by a uniform metric 7qím we consider the partition of the global spacotimo into sufficiently small and adjoining neighborhoods R at arbitrary locations (x.y.z). We usually solve differential equations with nonhomogeneous coefficients by this method. The uniform metric is defined as [16. 17]
[7 ab] =
/ afx2 + y2 —1 •
/
/
—1 + f).ïy nfy2 + X2
where
/ 0
dot 7 =
/ 0
0 1
«f )
(13)
a(fx2+y2)(x2+fy2
P
r 2 -2
J = a: r
To discuss the propagation of plane waves in the medium defined by constitutive relations (7), we consider the plane-wave solutions
E = Re E0 exp[¿(k • r — uit)], H = ReH0exp[¿(k • r — uit)],
(14)
where k is the wave vector, r is the position vector within the neighborhood R containing an arbitrary location (x, y, z), t denotes the time and ui is the angular frequency. Also, E0 and H0 represent complex-valued amplitudes. When Eq. (14) is used in Eq. (6), an eigenvector equation is obtained after some algebraic manipulations. The resulting equation is given by
[ (i-gdot [7] - k • 7 • k) £ + kk • 7] • E0 = 0, (15)
where
ko = uJyJeoHo-
The corresponding dispersion relation can be written
kldot [7] (i-gdot [7] - k • 7 • k)2 = 0. (16) Since dot 7 is nonzero, this equation leads to
k • 7 • k = kgdct [7] .
(17)
With this value used in Eq. (15), it follows that kk • 7 • E0 = 0, which shows that k • 7 and E0 are orthogonal.
Let the wave vector k be described as
k = kk =
k(ii3,AiyAi-_
(18)
259
4*
whoro u3., iiy, and 11 roprosont unit vectors along x, y, and z axes and k is the magnitude of the wave vector. Hence,
k • 7 =
k [(7ii + 721)11* + (712 + 722)% + 733Ut]
Furthermore,
, - , k'2 [(711 + 721) + (712 + 722) + 733] ,1n. k-pk =---, (19)
where 7n, 712, 721, 722. 733 are the components of the metric [7ah]. Substituting Eq. (19) in (17), we obtain
k2 =
9/,-gdet
(711 + 712) + (712 + 722) + 733
Inserting the values of 7n, 712, 721, 722. 733 in the above equation, we obtain
k2 = 9 k2a2(fx2+y2)(x2+y2f) P jVi2(/(.i + y)'2 + (J: — i/)2) + 1
which yields the wave numbers k =
k* = 'ikn
\
a2 (fx2 y2)(->'2 y2,f) P [n,2(/(.F: + y)2 + (x — y)'2) + 1
(20)
For propagation of waves, the values of wave numbers must be real, which lead to
/2 [n2(/(.F:+,(/)2 + (.F:^)2) + l] ^0, />0. (21)
We impose the condition / > 0 because / < 0 provides the nonphysical region r < r+. The general solution of Eq. (15) can be written as
EQ — C161 + C2G2,
(22)
where Ci and C2 are complex constants and ei and e2 are two linearly independent eigenvectors given by [15]:
ei = (711 + 722 )u3. - (711 + 712)%
(23)
e2 = 733(711 + 712 )u3. + 733(712 + 722)% -
- [(711 + 712 )2 + (712 + 722 )2] Uf (24)
The application of Fourier tr
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