Electromagnetic wave propagation with negative phase velocity in regular black holes
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Electromagnetic Wave Propagation with Negative Phase Velocity in Regular Black Holes1 M. Sharif* and R. Manzoor** Department of Mathematics, University of the Punjab 54590, Lahore, Pakistan *email: [email protected] **email: [email protected] Received April 24, 2012
Abstract—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 propaga tion, while its possibility in the region inside the event horizon is discussed. DOI: 10.1134/S1063776112100123 1
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 [1–7]. This occurs when the wave vector has a negative projection on the timeaverage Poynting 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 [1–5], NPV wave propagation was investi gated in different blackhole spacetimes and it was proved that a gravitationally affected vacuum resembling a medium can admit NPV wave propa gation 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 prop erties in a Veselago medium for particular black holes were discussed in [9]. In a recent paper [10], we have explored conditions of NPV for static charged black strings. 1 The article is published in the original.
Here, we extend the above work to investigate propagation 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 Section 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 [1–7] to explain the electrodynamics of a gravitationally affected vacuum serving as an equivalent instanta neously responding medium for wave propagation. The general line element of a charged regular black hole is of the form 2
–1
2
ds = ( 1 – h ( r ) )dt – ( 1 – h ( r ) ) dr 2
2
2
2
2
(1)
– r ( dθ + sin θdφ ). The metric can be expressed in Cartesian coordi nates as
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ELECTROMAGNETIC WAVE PROPAGATION WITH NEGATIVE PHASE VELOCITY
g ab
⎛ 1 – h(r) 0 0 0 ⎜ 2 ⎜ h ( r )x xyh ( r ) xzh ( r ) 0 – 1 – – – ⎜ 2 2 2 r (1 – h(r)) r (1 – h(r)) r (1 – h(r)) ⎜ 2 = ⎜ xyh ( r
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