ELECTROMAGNETIC WAVE PROPAGATION WITH NEGATIVE PHASE VELOCITY IN REGULAR BLACK HOLES

M. Sharif* R. Manzoor**

Department of Mathematics, University of the Punjab 5459O, Lahore, Pakistan

Received April 24, 2012

We discuss the propagation of electromagnetic plane waves with negative phase velocity in regular black holes. For this purpose, we consider the Bardeen model as a nonlinear magnetic monopole and the Bardeen model coupled to nonlinear electrodynamics with a cosmological constant. It turns out that the region outside the event horizon of each regular black hole does not support negative phase velocity propagation, while its possibility in the region inside the event horizon is discussed.

1. INTRODUCTION

The phenomenon of electromagnetic plane wave propagation with negative phase velocity (NPV) in a curved spacetime has gained much attention during the last few years fl 7]. This occurs when the wave vector has a negative projection on the time-average Poynt-ing vector. A wide range of useful phenomena such as the negative Doppler effect, inverse Cerenkov radiation, and negative refraction have been predicted for NPV materials [8].

In fl 5], NPV wave propagation was investigated in different black-hole spacetimes and it was proved that a gravitationally affected vacuum resembling a medium can admit NPV wave propagation in certain curved spacetimes. The same authors [6, 7] showed that the erogosphere of an uncharged rotating and charged rotating black holes admits NPV wave propagation. Plasma wave properties in a Veselago medium for particular black holes were discussed in [9]. In a recent paper [10], we have explored conditions of NPV foist at ic charged black strings.

Here, we extend the above work to investigate prop-

E-mail: msharif.math'&pu.edu.pk E-mail: rubabmanzoor9(öyahoo.<'om

agation of electromagnetic plane waves with NPV in regular black holes. In the next section, we briefly review the relevant formulation. Section 3 explicitly deals with wave propagation and NPV conditions for two types of regular black holes. In Sec. 4, we discuss the results and the possibility of NPV wave propagation in the region inside the event horizon.

2. GENERAL FORMULATION: AN OVERVIEW

We briefly review the formulation in fl 7] to explain the electrodynamics of a gravitationally affected vacuum serving as an equivalent instantaneously responding medium for wave propagation. The general line element of a charged regular black hole is of the form

ds2 = (1 — h(r))dt2 — (1 — h(r))~1dr2 —

— V2 (dß2 + sin"2 0 d(j)2). (1) The metric can be expressed in Cartesian coordinates

( 1 - h(r) 0

9ab =

0

h(r)x2

0

xyh(r)

0

xzh(r)

r2(l-h(r)) r2(l-h(r))

xyh(r)

h(r)y2

r2(1 - h(r)) zyh(r)

r2(l - h(r))

-///H» >(1 -h(r))

r2( 1 -h(r)) r2 ( 1 — /7. (r))

-///M>)

whose inverso is

/ 1

if =

1 - /î(r) 0

0

xyh(r) -///M>)

V

The electromagnetic response of the vacuum in a curved spacetime is described by the constitutive relations

D(r,f) = e07 • E(r,f), B(r,í) = m-H(r,i). (2)

The dyadic 7 is a second-rank Cartesian tensor that can be expressed in the metric [%(,] form with

"fab =

,9oo '

(3)

We note that only a global observer based on a curved spacetime can observe electromagnetic wave propagation in a gravitationally affected vacuum, and we therefore observe this phenomenon globally [2]. The constitutive relations describe the regular black hole spacetime globally.

To approximate a nonuniform metric 7a¡, by a uniform metric 7qím we partition the global spacetime into adjoining and sufficiently small neighborhoods R at an arbitrary location (x.y. z) and formulate a global solution by stitching together the solutions evaluated from the neighborhoods. Also, it is assumed that the wavelength is shorter than the linear dimension of the neighborhood R, which is in turn shorter than the curvature radius of the spacetime. Differential equations with nonhomogonoous coefficients are solved by this method.

7*2 (1 — /7 (7* ) )

-1

h(r)

0 0

h(r)x2 xyh(r)

7-2(l — h(r)) )

0 \

xzh(r)

-1

h(r)y2 zyh(r)

zyh(r)

-1

h(r)

7

~ /

The uniform metric is defined as 1

7 = [lab] = (

1 - h(f)

x

h(f)x2 h(f)xy jt 2 h ( }x : \ jt2

h(f)xy h(f)y2 jt2 h(f)yz jt2

h ( }x : ~2 h(f)yz jt 2 h(f)z2 1 f2 /

where

and

-2 -2 /• = x

det [7] = (1 - h(f)y

Wo now find the dispersion relation of the medium and the complex time-average Poynting vector in a time-harmonic electromagnetic field [11, 12]. We consider plane wave solutions

E(r,t) = ReEo exp[i(k • r -urt)]. H(r,f) = ReH0 oxp[¿(k • r — uit)],

(4)

where r is the position vector within the neighborhood R containing (x.y.z), k is the wave vector, and ui is the angular frequency. The complex-valued amplitudes are represented by E0 and H0.

The source-free Maxwell curl postulates in R are given by

VxEM + ®=0. VXH(r,f)-®M = 0.

v at

(5)

We seek a plane-wave solution of Eq. (5) and obtain an eigenvector equation, after some algebraic manipulations. as

(6)

¿•¿dot 7 — k • 7 • k I / + kk • 7 • E0 = 0

where

k0 = oJyJeoHo

and kk represents the tensor product of vectors. The corresponding dispersion relation is obtained by setting

det y dot gj - k • 7 • kj J_ + kk • 7J =0, which is [5]

¡4dot [7] (frgdct [7] - k • 7 • k)2 = 0. (7) Since 7 is nonsingular, the above equation leads to

k-7-k = i-gdct [7] . (8)

Inserting this value in Eq. (6) yields

kk • 7 • E0 = 0, (9)

which shows that k • 7 and E0 are orthogonal.

Because spacetime (1) is spherically symmetric, we can take

k = kk = kii-

without any loss of generality. Here, 11 is a unit vector along the z axis and k is the magnitude of the wave vector. Then

k-g=k [71311a- + 723% + 733Ut] •

The terms 713, 723. and 733 are given by

h(f)xz _ h(f)yz

7i3 =

(1 - h(f))f2

733 =

where

(h(r))-_ = 1

. 723 = -

(ll(r)): (l -h(f))'

h(f):

(1 - h(f))r2

and U3. and Uj, are unit vectors along the x and y axes. Equation (8) yields

k

(h(r)):

kf,

1 - h(f) whose roots are

(1 - /1(f))2

k = ±k0 [(1 - h(r))(h(r)) For to be real, we must have

(i -h(f))(h(f)), >0

= 0,

-1/2

(10)

(ii)

which is possible if both terms (1 — h(f)) and (h(f))~ have the same sign. Equation (9) is satisfied if two linearly independent eigenvectors are

el = 723- 713%.

e2 = 7337i3Ua- + 733723% - [(713 )2 + (723 )2] Uf The general solution of Eq. (6) can be written as

E0 = fiei + c2e2, (12)

where c\ and c2 are complex constants. Combining Eqs. (2), (4), (5), and (12), wo obtain

k U(1 - h(f))e2 - (-■2(h(r))-:ei

Ho = —-i. (13)

u>/io

The NPV wave propagation is defined by k- (P), < 0, where

<P), = iRo{E0xHS}

is the time-average complex Poynting vector and Hg is the complex conjugate of H0. Using Eqs. (12) and (13) in the above equation, we have

(14)

k- (P), =

1

2u;//.o

k(h(f))~ f2( 1 - h(f))

(i2 +f) x

x R

K'2

(Hr)), 1 - h(f)

(h(f)),.

Using this value in Eq. (14) along with (11), we see that NPV exists if

(/'(''")): < 0.

(15)

Because this condition is derived for the neighborhood R on an arbitrary location within the spacetime, it holds generally.

3. NPV AND REGULAR BLACK HOLES

The concept of a regular black hole has played an important role in understanding the hidden interior of black holes. These are more general black-hole solutions in which true singularities of the black hole such as the Schwarzschild, Roissnor Nordstrom, Kerr and Kerr Nowmann black holes would be replaced by a special matter core. The work on NPV propagation has been done on black-hole solutions experiencing true singularities and therefore the conditions of NPV wave propagation are restricted to the outside of the event horizon. We are interested in extending this study to the interior of the black hole.

In this section, we apply the above formalism to two types of regular black holes. In Refs. [13, 14], the Bardeen model was described as a nonlinear magnetic monopolo. This is obtained for

h(r) =

2 mr

(r2 +1,2)3/2

in Eq. (1), where m is the rilciSS and q is the monopole charge of a self-gravitating magnetic field of a nonlinear oloctrodynamic source. We note that when

2 16 2 q < 7p m <

there is an event horizon. For q = 0, this reduces to the Schwarzschild black hole.

It follows from Eq. (15) that NPV wave propagation is possible if the condition

(Mr)): =

1

h(f)

< 0

holds, which implies that

2 mr'2

(f2 + q2)3/2

>

(16)

The event horizon for this regular black hole lies at r+ that is obtained from

1

2mr

(r2+q2)3/2

= 0.

It is difficult to find an explicit expression for r+ from here, and hence the regions lying outside or inside the event horizon could not be explored. However, we use the following inequality to investigate NPV wave propagation in these regions. The regions r < r+ and r > r+ can be determined by respectively taking

1

2 mr1'

(r2 +i/2)3/2

< 0, 1

2 mr1'

(r2 +i/2)3/2

> 0.

It follows from (16) that

2 mr'2

Because

1 ('F2 + q2)3/2 < 1 z'2 '

r = .iT + tj-'

(17)

it follows that f2 > z2 for nonzero x'2 and y'2. For f > r+, Eq. (17) does not hold because the term (1 — r2/z2) is negative and cannot be greater than any-positive term. On the other hand, for f < r+, Eq. (17) holds.

The line element of a regular black hole coupled to nonlinear electrodynamics with a cosmological constant [15. 1G1 is obtained for

h(r) =

2 mr

2 2

(r2+i/2)3/2 (/*2 + q2

Ar2 3

(18)

where in. q, and A are the rilctSS, the electric charge, and the cosmological constant. For A > 0 and A < 0, Eq. (18) respectively represents de Sitter and anti-do Sitter type spacetimes. Furthermore, if m = q = 0, this becomes the do Sitter spacetime, and if q = A = 0, it reduces to the Schwarzschild spacetime. The event horizon is obtained from

1

2mrl

2 2 q-rl

A r2

(r2 +i/2)3/2 (rl+q2

= 0.

As in the preceding case, it is difficult to evaluate the explicit function of r+ and the regions r < r+ and r > > r+. But we can obtain the regions r < r+ and r > r+

1

1

2 mr

2 2 fr

(r2 +q2)3/'2 2mr2

(r2 + q'2

2 2 fr

Ar2 3

Ar2

< 0,

> 0.

(r2+i/2)3/2

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