Hot plasma waves in Veselago medium around Reissner-Nordström black hole

Hot plasma waves in Veselago medium aroundReissner-Nordstrom black hole

M. Hossain Ali1 and M. Ashrafuzzaman Khan2

Department of Applied Mathematics,Rajshahi University,

Rajshahi - 6205, Bangladesh

ABSTRACT

We investigate the general relativistic magnetohydronadynamic (GRMHD) equationsfor hot plasmas in a Veselago medium around the Reissner-Nordstrom (RN) black hole.Using the 3+1 formalisms of spacetime, we write the GRMHD equations and perturbthem linearly. These are then Fourier analyzed for the magnetized and nonmagnetizedplasmas in rotating and nonrotating backgrounds. We derive dispersion relations andanalyze the wave properties by the graphs of wave vector, refractive index and changein refractive. The results confirm the presence of Veselago medium for rotating magne-tized/nonmagnetized and nonrotating nonmagnetized plasmas.

Subject headings: 3+1 formalism, Dispersion relations, GRMHD equations, Hot plasma,Veselago medium

1E-mail: m−hossain−ali−[email protected]: [email protected]

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1. Introduction

Black holes are one of the most intriguing predic-tions of general relativity (GR). In fact, they belongto the most fascinating objects predicted by the Ein-stein’s field equations. They depict a region of spacewith such a strong gravitational field that nothing,not even light, can escape from their grips. Blackholes are made when matter is compressed to a point.They are commonly created from deaths of massivestars or by the collapse of a supergiant star. Theyevoke the mysterious aspect of gravity and are a re-ality today. These objects are usually found in X-raybinaries and in the centers of galaxies and quasars.Their rotational behaviors are predicted by the exis-tence of rotating stars. By extracting rotation energy,a rotating black hole gradually reduces to a nonrotat-ing (Schwarzschild) black hole. The black hole createsenergy flux that generates a relatively large magneticfield (Raine and Thomas 2005).

The most common form of matter in the universe isplasma. Indeed, more than 99% of all known matteris in the plasma state. Stars are made of plasma andthe inner-stellar space is filled with plasma, normallyoccurring in a hot state. Plasma gathers around theblack hole in the form of an accretion disk. Sincethe moving plasma creates a magnetic field, there ex-ists a region of magnetic field surrounding the blackhole, called a magnetosphere. The study of stationaryconfigurations and dynamical evolution of conductingfluid in a magnetosphere of massive black holes de-mands the theory of general relativistic magnetohy-drodynamics (GRMHD). The equations of GRMHDtheory describe the aspects of interaction of relativis-tic gravity with plasma’s magnetic field. An isolatedblack hole can be endowed with a net electric charge(Rahman and Ali 2010b) and so it can have an elec-tromagnetic field. Since a collapsed object can havea very strong effect on an electromagnetic field, itis of interest to determine this effect using GRMHDequations when a black hole is placed in an externalelectromagnetic field.

Gravitational fields are stronger near the surfaceof a nonrotating black hole (Petterson 1974). Themagnetospheric plasma moves along the radial direc-tion of a nonrotating black hole and is perturbed bythe black hole gravity. The effect of these perturba-tions in the black hole regime has always been of in-terest to investigate. Regge and Wheeler (Regge andWheeler 1957) discussed the stability of Schwarzschild

singularity by considering a small non-spherical odd-parity perturbation, while Zerilli (Zerilli 1970a, Zer-illi 1970b, Zerilli 1970c) discussed it in terms of asmall even-parity perturbation. The exact solutionsto the Regge-Wheeler equations (Regge and Wheeler1957), which describe the axial perturbations of theSchwarzschild metric in linear approximation, hadbeen derived (Fiziev 2007). Price (Price 1972a, Price1972b) investigated the dynamics of non-sphericalperturbations during the collapse of stars with ascalar-field analogue. By applying second-order per-turbations, Gleiser et al. (Gleiser et al. 1996) dis-cussed the stability of black holes. Hanni and Ruffini(Hanni and Ruffini 1973) described the quasi-staticproblem of the electric field and Sakai and Kawata(Sakai and Kawata 1980) developed a special rela-tivistic approach for a linearized treatment of plasmawaves around the Schwarzschild black hole.

The more appropriate formalism in investigatingthe fluid equations in the vicinity of a black hole isbased on the 3+1 split of spacetime, called the ADM(Arnowitt, Deser, and Misner) formalism (Arnowittet al. 1962). This formalism had been used in numer-ical relativity (Evans et al. 1986, Smarr and York1978) and in exploring the magnificent aspects ofGR (Macdonald and Suen 1985, Thorne and Hartle1985, Wheeler 1968). Ellis (Ellis 1973) decomposedthe Maxwell equations in the black hole environmentin the 3+1 formalism. Thorne, Price, and Macdon-ald (TPM) (Macdonald and Thorne 1982, Thorneand Macdonald 1982, Thorne et al. 1986) devel-oped a full program of the electromagnetic theory inthe black hole regime using ADM formalism. TheTPM formalism had been used to deduce some ba-sic results of GR (Durrer and Straumann 1988) andanalyze the wave properties for the Friedmann uni-verse (Dettmann et al. 1993, Holcomb and Tajima1989, Holcomb 1990. Buzzi et al. (Buzzi et al.1995a, Buzzi et al. 1995b) analyzed the proper-ties of a relativistic two-fluid plasma wave near theSchwarzschild black hole. Adopting the technique ofBuzzi et al. (Buzzi et al. 1995a), Ali and Rahman in-vestigated the transverse electromagnetic waves prop-agating in a plasma in the vicinity of nonrotatingblack holes in the asymptotically flat and dS spaces(Ali and Rahman 2008, Ali and Rahman 2009a, Rah-man and Ali 2010a, Rahman and Ali 2010b). Zhang(Zhang 1989a) described the laws of perfect GRMHDin 3+1 formalism for a general spacetime and ana-lyzed cold plasma perturbation around the Kerr black

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hole (Zhang 1989b). Applying Zhang’s formulation,a lot of works had been carried out in investigatingwave properties of plasma (cold, isothermal and hot)around nonrotating as well as rotating black holes(Ali and Hasan 2009b, Hasan and Ali 2010, Hasanand Ali 2009a, Hasan and Ali 2009b, Hasan andAli 2009c, Hasan and Ali 2009d, Hasan and Ali2009e, Sharif and Sheikh 2007a, , Sharif and Sheikh2007c, Sharif and Mustafa 2008, Sharif and Sheikh2008a, Sharif and Sheikh 2008b, Sharif and Sheikh2008c, Sharif and Sheikh 2009a, Sharif and Sheikh2009b, Sharif and Rafique 2010).

The most significant class of metamaterials (i.e.artificial materials) is Veselago medium, which isnamed after Russian physicist Veselago (Veselago1968). This medium has unusual electromagneticproperties, such as, simultaneous negative electricpermittivity (ϵ < 0) and magnetic permeability (µ <0). The Veselago medium is also known as dou-ble negative medium (DNM), negative refractive in-dex medium (NIM), negative phase velocity medium(NPV), left-handed medium (LHM), or backward-wave medium (BWM). Wave refraction in this un-usual medium is found always positive and very in-homogeneous (Valanju et al. 2002). Ziolkowskiand Heyman (Ziolkowski and Heyman 2001) inves-tigated properties of electromagnetic wave in DNMmedium. The unusual behavior of physical laws forthis medium and their plausible applications havebeen investigated by many authors (Smith and Kroll2000, Lakhtakia et al. 2002, Bliokh and Bliokh2004, Mackay and Lakhtakia 2006). The importanceof NIM for perfect lensing was discussed in (Ramakr-ishna 2005). Presence of charge raises the tendencyof a rotating black hole to support the NPV propaga-tion in its ergosphere (Mackay and Lakhtakia 2006),and the concept of NPV medium is generalized toacoustic waves (Li 2007). Veselago (Veselago 2009)has analyzed the process of transferring energy, linearmomentum and mass by an electromagnetic wave ina negative refraction medium. Recently, Sharif andMukhtar analyzed isothermal plasma waves (Sharifand Mukhtar 2011a) and hot plasma waves (Sharifand Mukhtar 2011b) in the Veselago medium aroundthe Schwarzschild black hole.

The most general plasma, the basic constituent ofnature, is hot plasma and it reduces to a cold orisothermal plasma with some restrictions. Wave prop-erties of cold (Ali and Hasan 2009b, Hasan and Ali2010) and isothermal (Hasan and Ali 2009c) plasma

around the RN black hole have been investigated for-merly. In the present paper we have considered hotplasma in the vicinity of the RN black hole event hori-zon in a Veselago medium and analyzed the possibil-ity of receiving information. We apply perturbationand Fourier analysis techniques to the GRMHD equa-tions of the theory. The dispersion relations are cal-culated by using the software Mathematica to obtainthe wave vector. From this wave vector we evaluatethe refractive index, phase and group velocities, andrate of change in refractive index with respect to theangular frequency. The refractive index in the usualmedium is always greater than unity, but for a Vese-lago medium its value must be less than one. Thewave properties are ascertained through these quanti-ties for the hot plasma in a Veselago medium aroundthe RN black hole. The dispersion is normal if thephase velocity is greater than the group velocity andis anomalous otherwise (Achenbach 1973). Equiva-lently, dispersion is normal or anomalous accordingas the change in the refractive index with respect toangular frequency is positive or negative. When thedispersion is normal, information of energy exchangewithin the magnetosphere is extractible.

The paper is organized as follows. In Sect. 2,we summarize the 3+1 formulation of general rela-tivity and describe the GRMHD equations in a Vese-lago medium for the plasma existing in the planaranalogue of the RN spacetime. In Sect. 3, the3+1 GRMHD equations for hot plasma in Veselagomedium are linearized by perturbation method andFourier analyzed. In Sect. 4, we present our study ofmagnetized plasma in rotating environment. In Sect.5, we investigate nonmagnetized plasma in rotatingenvironment and in Sect. 6, we study nonmagnetizedplasma in nonrotating environment. Finally, in Sect.7 we present our concluding remarks. We use unitsin which G = c = kB = 1.

2. 3+1 spacetime split and GRMHD equa-tions for RN planar analogue in Veselagomedium

The RN spacetime is described by the metric

ds2 ≡ gµνdxµdxν = −∆2dt2 +∆−2dr2

+r2(dθ2 + sin2 θdφ2),

∆2 = 1− 2M

r+

Q2

r2, Q2 = Q2

e +Q2m. (1)

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The vector potential associated with the metric (1)has nonvanishing components: At = Qe/r, Aφ =−Qm cos θ, with corresponding field strength: F01 =Qe/r

2, F23 = Qm/r2, describing electric charge Qe

and magnetic charge Qm. The M denotes the massof the gravitating body. The matric (1) describes theSchwarzschild black hole for Q = 0 and the extremalRN black hole forQ = M . The extremely charged RNblack hole describes an extreme limit in the contextof the cosmic censorship hypothesis. The null hyper-surface equation, gtt = 0, yields an event horizon atr = r+ and an inner (Cauchy) horizon at r = r−,where

r± = M ±√M2 −Q2. (2)

When Q < M , the curvature singularity at r = 0is hidden behind these horizons. As no real valueis found in (2) for Q > M , the matric (1) does notdescribe a black hole at all but rather a physically for-bidden naked singularity in this case. The mass pa-rameter M is inseparably connected with the chargeparameter Q, i.e., M = 0 only when Q = 0. WhenQ = 0, r+ = 2M is the usual Schwarzschild hori-zon. In the extremal case of Q = M , the two valuesr± coincide and the horizon becomes degenerate. AnM > Q black hole will tend to Hawking radiate downto its extremal M = Q state. For Q = 0, the extremalspacetime is just the flat space vacuum.

The hypersurfaces of constant universal time t de-fine an absolute three-dimensional space describedby the metric ds2 = gijdx

idxj , where the indicesi, j = 1, 2, 3 refer to coordinates in absolute space.The observers remaining at rest with respect to thisabsolute space, called the fiducial observers (FIDOs),measure their proper time τ by using clocks that theycarry with them and make local measurements ofphysical quantities. The lapse function (or redshiftfactor), which is the ratio of the rate of FIDO propertime to that of universal time, is defined by

α(r) ≡ dτ/dt = ∆ = r−1(r − r+)12 (r − r−)

12 . (3)

The planar analogue of the RN metric near theevent horizon is obtained by expressing it in theRindler coordinates,

ds2 = −α2(z)dt2 + dx2 + dy2 + dz2, (4)

where

x = r+(θ − π/2), y = r+φ, z = 2r+α. (5)

The standard lapse function (3) simplifies in Rindlercoordinates to α = z/2r+. This function vanishesat the horizon which we can place at z = 0 and itincreases monotonically as z increases from 0 to ∞.

In Veselago medium, the Maxwell equations aregiven by

∇ ·B = 0, ∇×B = −µj+∂E

∂t= 0, (6)

∇ ·E = −ρeϵ, ∇×E+

∂B

∂t= 0. (7)

The perfect GRMHD equations for RN planar ana-logue are as follows (Sharif and Mukhtar 2011b,Zhang 1989a):

∂B

∂t= −∇× (αV ×B), (8)

∇ ·B = 0, (9)

∂(ρ0µ)

∂t+ (αV · ∇)ρ0µ

+ρ0µγ2V · ∂V

∂t+ ρ0µ∇ · (αV)

+γ2V · (αV · ∇)V = 0, (10)(ρ0µγ

2 +B2

4π

)δij + ρ0µγ

4ViVj −1

4πBiBj

×(1

α

∂

∂t+V · ∇

)V j − 1

4π

(B2δij −BiBj

)×V j

,kVk + ρ0γ

2Vi

(1

α

∂µ

∂t+V · ∇µ

)= −ρ0µγ

2ai − p,i +1

4π(V ×B)i∇ · (V ×B)

− 1

8πα2(αB)2,i +

1

4πα(αBi),jB

j

− 1

4πα[B× V × (∇× (αV ×B))]i , (11)(

1

α

∂

∂t+V · ∇

)(ρ0µγ

2)− 1

α

∂p

∂t

+ρ0µγ2 (2V · a) +∇ ·V

− 1

4π(V ×B) ·

(V × 1

α

∂B

∂t

)− 1

4π(V ×B) ·

(B× 1

α

∂V

∂t

)+

1

4πα(V ×B) · (∇× αB) = 0. (12)

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3. 3+1 GRMHD equations for hot plasma inVeselago medium around RN black hole

In the vicinity of the RN black hole, the equationof state for hot plasma is given by (Zhang 1989b)

µ =ρ+ p

ρ0, (13)

where µ is the specific enthalpy, ρ is the total densityof mass-energy, p is the pressure as seen in the fluid’srest frame, and ρ0 is the fluid’s rest-mass density. Thepressure vanishes (p = 0) for the cold plasma, while µis constant and p = 0 for the isothermal plasma. Butfor the hot plasma, the specific enthalpy µ is variable.Equation (13) shows the exchange of energy betweenthe plasma and fluid’s magnetic field. Obviously, thisequation states that p must be variable, if µ is vari-able.

The 3+1 GRMHD equations (8)–(12) with (13) de-scribe the hot plasma along with the Veselago mediumin the vicinity of the RN magnetosphere. When back-ground is rotating, plasma is assumed to flow in twodimensions, i.e., in xz-plane. Hence, FIDO-measuredmagnetic field B and velocity V can be described by

V = V (z)ex+u(z)ez, B = B[λ(z)ex+ez]. (14)

The variables λ, u and V are related by (Sharif andMukhtar 2011b)

V =V F

α+ λu, (15)

with V F an integration constant. Then, the Lorentzfactor γ = 1/

√1− v2 becomes

γ =1√

1− u2 − V 2. (16)

If the plasma flow is perturbed, the flow parame-ters take the the following form:

ρ = ρ0 + δρ = ρ0 + ρρ,

p = p0 + δp = p0 + pp,

B = B0 + δB = B0 +Bb,

V = V0 + δV = V0 + v, (17)

where B0, V0, p0 and ρ0 are unperturbed quantitiesand δρ, δp, δV and δB represent linearly perturbedquantities. The waves propagate in z-direction dueto gravitation with respect to time t and thus per-turbed quantities must depend on z and t. Using

the perturbed quantities in terms of the dimension-less quantities ρ, p, bx, bz, vx and vz:

ρ = ρ(t, z), p = p(t, z),

b = δB/B = bx(t, z)ex + bz(t, z)ez,

v = δV = vx(t, z)ex + vz(t, z)ez, (18)

in the perfect GRMHD equations (8)–(12), we obtain

∂(δB)

∂t= −∇× (αv ×B) + (αV × δB), (19)

∇ · (δB) = 0, (20)

1

α

∂(δp+ δρ)

∂t− (p+ ρ)(v · ∇ lnu−∇ · v)

+(p+ ρ)γ2

V ·

(1

α

∂

∂t+V · ∇

)v

+(V · ∇V) · v + 2(V · v)(V · ∇) ln γ

= 0,

(21)((p+ ρ)γ2 +

B2

4π

)δij + (p+ ρ)γ4ViVj

− 1

4πBiBj

1

α

∂vj

∂t+ γ2Vi(V · ∇)(δp+ δρ)

+1

4π

[B×

V × 1

α

∂(δB)

∂t

]i

+ (p+ ρ)γ2

×(vi,jV

j + γ2Vivj,kVjV k

)+ γ2[Vi(v · ∇)

+vi(V · ∇) + γ2(2V · v)Vi(V · ∇)](p+ ρ)

=1

4πα[(αδBi),j − (αδBj),iBj + (αBi),j

−(αBj),iδBj ]− (δp)i − γ2(δp+ δρ)

+2(p+ ρ)γ2(V · v)ai − (p+ ρ)γ4(viVj

+vjVi)Vk,jVk − γ2(δp+ δρ)V j

+2(p+ ρ)γ2(V · v)V j + (p+ ρ)vjVi,j

−γ4Vi(δp+ δρ)V j + 4(p+ ρ)γ2(V · v)V j

+(p+ ρ)vjVj,kVk, (22)

γ2 1

α

∂(δp+ δρ)

∂t+ v · ∇(p+ ρ)γ2 − 1

α

∂(δp)

∂t

+(V · ∇)(δp+ δρ)γ2 + 2(p+ ρ)γ2

×γ2(V · ∇)(V · v) + 2γ2(V · v)(V · a)+(v · a)+ (δp+ δρ)γ22(V · a) + (∇ ·V)+(p+ ρ)γ2(∇ · v) + 2γ2(V · v)(∇ ·V)

=1

4πα

[v ·

(B · ∂B

∂t

)V +V ·

(B · ∂B

∂t

)v

+V · (B · δB)V +V ·(δB · ∂B

∂t

)V

5

−v · (B ·V)∂B

∂t−V · (B ·V)

∂δB

∂t

−V · (B · v)∂δB∂t

−V · (δB ·V)∂B

∂t

−V · (B ·B)∂v

∂t+V · (B · ∂δv

∂t)B

]+

1

4π[(v ×B+V × δB) · (∇×B)

+(V ×B) · (∇× δB)]. (23)

Using (18), the component form of (19)–(23) can bewritten as follows:

1

α

∂bx∂t

− ubx,z

= (ubx − V bz − vx + λvz)∇ lnα− (vx,z

−λvz,z − λ′vz + V ′bz + V bz,z − u′bx), (24)

1

α

∂bz∂t

= 0, (25)

bz,z = 0, (26)

ρ1

α

∂ρ

∂t+ p

1

α

∂p

∂t+ (ρ+ p)γ2V

(1

α

∂vx∂t

+ uvx,z

)+(ρ+ p)γ2u

1

α

∂vz∂t

+ (ρ+ p)(1 + γ2u2)vz,z

= −(ρ+ p)γ2u(1 + 2γ2V 2)V ′ + 2γ2uV u′ vx

+(ρ+ p)(1− 2γ2u2)(1 + γ2u2)u′

uvz

−2(ρ+ p)γ4u2V V ′vz, (27)(p+ ρ)γ2(1 + γ2V 2) +

B2

4π

(1

α

∂vx∂t

+ uvx,z

)+

(p+ ρ)γ4uV − λB2

4π

(1

α

∂vz∂t

+ uvz,z

)−B2

4π(1 + u2)bx,z −

B2

4παα′(1 + u2)

+αuu′bx + γ2u(pp+ ρρ)(1 + γ2V 2)V ′

+γ2uV u′+ γ2uV (p′p+ pp′ + ρ′ρ+ ρρ′)

+[(p+ ρ)γ4u(1 + 4γ2V 2)uu′ + 4V V ′

×(1 + γ2V 2)+ B2uα′

4πα+ γ2u(p′ + ρ′)

×(1 + 2γ2V 2)]vx +

[(p+ ρ)γ2(1 + 2γ2u2)

×(1 + 2γ2V 2)V ′ − γ2V 2V ′

+2γ2(1 + 2γ2u2)uV u′ − B2u

4πα(λα)′

+γ2V (1 + 2γ2u2)(p′ + ρ′)]vz = 0, (28)

(p+ ρ)γ2(1 + γ2u2) +

λ2B2

4π

×(1

α

∂vz∂t

+ uvz,z

)+

(1

α

∂vx∂t

+ uvx,z

)×(p+ ρ)γ4uV − λB2

4π

+

λB2

4π

×[(1 + u2)bx,z +

1

λα

(αλ)′ − α′λ

+uλ(uα)′bx

]+ (pp+ ρρ)γ2

×az + uu′(1 + γ2u2) + γ2u2V V ′+(1 + γ2u2)(p′p+ pp′) + γ2u2(ρ′ρ+ ρρ′)

+

[(p+ ρ)γ4u2V ′(1 + 4γ2V 2)

+2V (az + uu′(1 + 2γ2u2))

−λB2uα′

4πα+ 2γ4u2V (p′ + ρ′)

]vx

+

[(p+ ρ)γ2

u′(1 + 4γ2u2)(1 + γ2u2)

+2uγ2(az + (1 + 2γ2u2)V V ′)

+λB2u

4πα(αλ)′ + 2γ2u(1 + γ2u2)

×(p′ + ρ′)

]vz = 0, (29)

1

αγ2

(ρ∂ρ

∂t+ p

∂p

∂t

)+ γ2(p′ + ρ′)vz + u(ρρ,z

+pp,z + ρ′ρ+ p′p) − 1

αp∂p

∂t+ γ2(pp+ ρρ)

×(2uaz + u′)+ 2(p+ ρ)γ2

γ2(uV ′

+2uV az + u′V )vx + (2γ2uu′ + azγ4

+2γ2u2az)vz + γ2uV vx,z +1

2(1

+2γ2u2)vz,z

− B2

4πα

[(V 2 + u2)λbx

−λV (λV + u)∂bx∂t

+ (V 2 + u2)bz

−u(λV + u)∂bz∂t

]− B2

4πα

[(V − λu)vx,t

+λ(uλ− V )vz,t]+

B2

4π(λλ′vz − λ′vx

−λ′V bz + λ′ubx − V bx,z + uλbx,z) = 0. (30)

In order to perform Fourier analysis, we assumethe following harmonic spacetime dependence of per-

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turbation:

ρ(t, z) = c1e−i(ωt−kz), p(t, z) = c2e

−i(ωt−kz),

vz(t, z) = c3e−i(ωt−kz), vx(t, z) = c4e

−i(ωt−kz),

bz(t, z) = c5e−i(ωt−kz), bx(t, z) = c6e

−i(ωt−kz),

(31)

where cs, s = 1, · · · , 6 are arbitrary constants, ωrepresents angular frequency and k denotes the z-component of the wave vector (0, 0, k). The wavevector is used to determine refractive index and theproperties of plasma near the event horizon. The wavevector gives the direction in which a plane wave prop-agates and its magnitude is the wave number. The re-fractive index is the ratio of light traveling from onemedium (usually from vacuum) to another and itschange with respect to angular frequency determineswhether the dispersion is normal or anomalous.

Inserting the linear perturbations (31) in (24)–(30),we obtain the Fourier analyzed form as follows:

−c3(αλ)′ + ikαλ+ c4(α′ + ikα)− c5(αV )′

−c6(αu)′ + iω + ikuα = 0, (32)

c5

(−iω

α

)= 0, (33)

c5ik = 0, (34)

c1

(−iω

αρ

)+ c2

(−iω

αp

)+c3(p+ ρ)

−iω

αγ2u+ (1 + γ2u2)ik

−(1− 2γ2u2)(1 + γ2u2)u′

u+ 2γ4u2V V ′

+c4(p+ ρ)γ2

(−iω

α+ iku

)V

+u(1 + 2γ2V 2)V ′ + 2γ2u2V u′

= 0, (35)

c1[ργ2u(1 + γ2V 2)V ′ + γ2V uu′+γ2V u(ρ′ + ikρ)] + c2[pγ

2u

×(1 + γ2V 2)V ′ + γ2V uu′

+γ2V u(p′ + ikp)] + c3

[(p+ ρ)γ2

×(1 + 2γ2u2)(1 + 2γ2V 2)V ′

+

(−iω

α+ iku

)γ2V u− γ2V 2V ′

+2γ2(1 + 2γ2u2)uV u′+ γ2V (p′ + ρ′)

×(1 + 2γ2u2)− B2u

4πα(λα)′

+λB2

4π

(iω

α− iku

)]+ c4

[(p+ ρ)γ4u

×(1 + 4γ2V 2)uu′ + 4V (1 + γ2V 2)V ′

+(p+ ρ)γ2(1 + γ2V 2)

(−iω

α+ iku

)+γ2u(1 + 2γ2V 2)(p′ + ρ′) +

B2uα′

4πα

−B2

4π

(iω

α− iku

)]− c6

B2

4πα

×αuu′ + (α′ + ikα)(1 + u2) = 0, (36)

c1[ργ2az + (1 + γ2u2)uu′ + γ2u2V V ′+γ2u2(ρ′ + ikρ)] + c2[pγ

2az + uu′

×(1 + γ2u2) + γ2u2V V ′+(1 + γ2u2)(p′ + ikp)]

+c3

[(p+ ρ)γ2

(1 + γ2u2)

(−iω

α+ iku

)+u′(1 + γ2u2)(1 + 4γ2u2)

+2γ2u(az + (1 + 2γ2u2)V V ′)

+2γ2u(1 + γ2u2)(p′ + ρ′) +

λB2u

4πα(λα)′

+λ2B2

4π

(−iω

α+ iku

)]+ c4

[(p+ ρ)γ4

×(

−iω

α+ iku

)uV + u2V ′(1 + 4γ2V 2)

+2V (az + (1 + 2γ2u2)uu′)

− λB2uα′

4πα

+2γ4u2V (p′ + ρ′)− λB2

4π

(−iω

α+ iku

)]+c6

[B2

4πα−(λα)′ + α′λ− uλ(uα)′

+λB2

4π(1 + u2)ik

]= 0, (37)

c1γ2

(−iω

α+ iku+ 2uaz + u′

)ρ+ uρ′

+c2

[iω

α(1− γ2)p+ γ2(iku+ 2uaz

+u′)p+ up′]+ c3γ

2

(p′ + ρ′)

+2(2γ4uu′ + az + 2γ2u2az)(p+ ρ)

7

+(1 + 2γ2u2)(p+ ρ)ik

+λB2

4πα(λu− V )iω + αλ′

+c4

[2γ2(p+ ρ)(uV ′ + 2uV az + u′V )

+uV ik+ B2

4πα(V − uλ)iω − αλ′

]+c6

[−B2

4πα(V 2 + u2)λ+ λV (λV + u)iω

−αλ′u+ ikα(V − uλ)

]= 0. (38)

The determinant of the coefficients ci, i = 1, · · · , 6 in(32)–(38) equated to zero gives a complex relation ink, called the dispersion relation.

4. Hot plasma in rotating magnetized back-ground

We assume rotating magnetized plasma with fluid’svelocity and magnetic field lying in xz-plane. In thisgeneral case, Fourier analyzed perturbed GRMHDequations are (32)–(38), given in the preceding sec-tion.

4.1. Numerical solution modes

For rotating magnetized plasma, we use the lapsefunction α = z/(2(M +

√M2 −Q2)), 0 ≤ Q2/M2 ≤

1, B =√8π. We consider the black hole mass

M ∼ 1M⊙, Q2/M2 = 0.49, and stiff fluid: p =ρ = µ/2. Specific enthalpy is µ =

√(1 + α2)/2 and

fluid velocity components u = V = −(2√1 + α2)−1

gives the Lorentz factor γ =√1 + 1/(1 + 2α2), λ =

1+2√1 + 1/α2 for V F = 1 and λ = 1 for V F = 0. So,

the magnetic field diverges close to the horizon. Un-der these restrictions, the perfect GRMHD equations(8)–(12) are satisfied for the region 0.2 ≤ z ≤ 10,0 ≤ ω ≤ 10. With these assumptions, we obtainfrom (32)–(38) two dispersion relations. The imagi-nary part is the dispersion relation

B1(z)k5 +B2(z, ω)k

4 +B3(z, ω)k3 +B4(z, ω)k

2

+B5(z, ω)k +B6(z, ω) = 0, (39)

which gives five values for k, one of which is real butnot interesting and other four values are imaginary.The real part gives

A1(z, ω)k4 +A2(z, ω)k

3 +A3(z, ω)k2

+A4(z, ω)k +A5(z, ω) = 0, (40)

which yields four real values of k and one of them doesnot provide any interesting result. From the remain-ing three real values of k, we have obtained graphi-cally the wave vector, phase velocity vp ≡ ω/k andgroup velocity vg ≡ (n + ωdn/dω)−1. The refractiveindex, computed as the ratio of the speed of light ina vacuum to the speed of light through the material,is n(= 1/vp) and its change with respect to angularfrequency, dn/dω, determines whether the dispersionis normal or not. The results are displayed in Figs.1, 2 and 3. They show that no wave exists near thehorizon at 0 ≤ z < 0.2, but waves are found in theregion 0.2 ≤ z ≤ 10, 0 ≤ ω ≤ 10. The wave vector,refractive index, phase velocity and group velocity areall negative in Fig. 1, however they are all positive inFigs. 2 and 3.

Surface plot of wave vector

02

46

8

10

z

0

2

4

6

8

10

Ω

-3

-2

-1

0

02

46

8z

Surface plot of phase velocity

02

46

8

10

z

0

2

4

6

8

10

Ω

-15

-10

-5

0

02

46

8z

Surface plot of group velocity

02

46

8

10

z

0

2

4

6

8

10

Ω

-15

-10

-5

0

02

46

8z

Surface plot of refractive index

02

46

8

10

z

0

2

4

6

8

10

Ω

-0.6

-0.4

-0.2

0

02

46

8z

Surface plot of D@n, ΩD

02

46

8

10

z

0

2

4

6

8

10

Ω

-1·10-16

0

1·10-16

02

46

8z

Fig. 1.— The waves move away from the event hori-zon. Normal dispersion exists but most of the disper-sion is anomalous.

Figure 1 indicates that waves move away from theevent horizon, but Figs. 2 and 3 display that wavesare directed towards the event horizon. The refrac-tive index in Fig. 1 is less than zero and it decreasestowards the horizon but increases away from the hori-zon. The phase and group velocities increase towards

8

the event horizon. The phase velocity has greatervalues than the group velocity for the low frequencywaves and they are equal in most of the region. Thechange in refractive index with respect to angular fre-quency has positive as well as negative values. Hence,there exist normal dispersion but most of the wavesare dispersed anomalously. In Figs. 2 and 3, thephase and group velocities decrease towards the eventhorizon. The refractive index increases towards thehorizon and decreases away from the horizon withn < 1. The phase velocity is greater than the groupvelocity for low frequency mode and they are equal inmost of the region. The dn/dω shows positive values,and hence, normal and anomalous dispersions occurin Figs. 2 and 3, but most of the waves are dispersedanomalously.


02

46

8

10

z

0

2

4

6

8

10

Ω

0

1

2

3

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

20

40

60

80

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

20

40

60

80

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

0.2

0.4

0.6

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-4·10-16-2·10-16

02·10-164·10-16

02

46

8z

Fig. 2.— The waves damp as they move away fromthe event horizon. Most of the region has anomalousdispersion but normal dispersion exists as well.

5. Hot plasma in rotating nonmagnetized back-ground

In rotating nonmagnetized background of plasmaflow the magnetic field is assumed to be zero, i.e.,


02

46

8

10

z

0

2

4

6

8

10

Ω

0

1

2

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

10

20

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

10

20

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

00.1

0.2

0.3

0.4

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-2·10-17-1·10-17

01·10-172·10-17

02

46

8z

Fig. 3.— The waves move towards the event horizon.The dispersion is shown normal.

B = 0. Hence, with B = 0 = λ and c5 = 0 = c6,the Fourier analyzed perturbed GRMHD equations(35)–(38) reduce to the following form:

c1

(−iω

αρ

)+ c2

(−iω

αp

)+ c3(p+ ρ)

×−iω

αγ2u+ (1 + γ2u2)ik

−(1− 2γ2u2)(1 + γ2u2)u′

u+ 2γ4u2V V ′

+c4(p+ ρ)γ2

(−iω

α+ iku

)V

+u(1 + 2γ2V 2)V ′ + 2γ2u2V u′

= 0, (41)

c1[ργ2u

(1 + γ2V 2)V ′ + γ2V uu′

+γ2V u(ρ′ + ikρ)] + c2[pγ2u

×(1 + γ2V 2)V ′ + γ2V uu′

+γ2V u(p′ + ikp)] + c3

[(p+ ρ)γ2

9

×(1 + 2γ2u2)(1 + 2γ2V 2)V ′

+

(−iω

α+ iku

)γ2V u− γ2V 2V ′

+2γ2(1 + 2γ2u2)uV u′+ γ2V

×(1 + 2γ2u2)(p′ + ρ′)

]+ c4

[(p+ ρ)γ4u

×(1 + 4γ2V 2)uu′ + 4V (1 + γ2V 2)V ′

+(p+ ρ)γ2(1 + γ2V 2)

(−iω

α+ iku

)+γ2u(1 + 2γ2V 2)(p′ + ρ′)

]= 0, (42)

c1[ργ2

az + (1 + γ2u2)uu′

+γ2u2V V ′+ γ2u2(ρ′ + ikρ)]

+c2[pγ2

az + (1 + γ2u2)uu′ + γ2u2V V ′

+(1 + γ2u2)(p′ + ikp)]+ c3

[(p+ ρ)γ2

×(1 + γ2u2)

(−iω

α+ iku

)+ u′(1 + γ2u2)

×(1 + 4γ2u2) + 2γ2u(az + V V ′

×(1 + 2γ2u2))

+ 2γ2u(1 + γ2u2)(p′ + ρ′)

]+c4

[(p+ ρ)γ4

(−iω

α+ iku

)uV

+u2V ′(1 + 4γ2V 2) + 2V(az + uu′

×(1 + 2γ2u2))

+ 2γ4u2V (p′ + ρ′)

]= 0,

(43)

c1γ2

(−iω


)ρ+ uρ′

+c2

[iω

α(1− γ2)p+ γ2

(iku+ 2uaz + u′)p

+up′]

+ c3γ2

(p′ + ρ′) + 2(2γ4uu′ + az

+2γ2u2az)(p+ ρ) + (1 + 2γ2u2)(p+ ρ)ik

+c4

[2γ4(p+ ρ)

(uV ′ + 2uV az + u′V )

+ikuV]

= 0. (44)


In this section we consider the same assumptionsas applied in Sect. 4 for the values of lapse function,velocity, pressure, density and specific enthalpy withthe restriction on the magnetic field that B = 0 =λ. Under these assumptions the GRMHD equations(12)–(16) are satisfied for the region 0.2 ≤ z ≤ 10, 0 ≤ω ≤ 10. Using these restrictions in (41)–(44), weobtain two dispersion relations. The real part gives

A1(z)k4 +A2(z, ω)k

3 +A3(z, ω)k2

+A4(z, ω)k +A5(z, ω) = 0. (45)

This gives four values of k two of which are real andinteresting and the remaining two are complex conju-gate. The imaginary part is

B1(z, ω)k3 +B2(z, ω)k

2 +B3(z, ω)k +B4(z, ω) = 0,(46)

which yields one real value of k and the remaining twovalues are complex conjugate to each other. The realvalues of k from (45) and (46) have been used to showgraphically wave vector k, phase velocity vp, group ve-locity vg, refractive index n and rate of change of re-fractive index with respect to frequency dn/dω. Theresults, displayed in Figs. 4–6, depict that k is notfound near the horizon at 0 ≤ z < 0.2, but at theregion 0.2 ≤ z ≤ 10, 0 ≤ ω ≤ 10, k is negative inFigs. 4 and 6 and positive in Fig. 5. Hence, there isno wave in the region 0 ≤ z < 0.2. Waves move awayfrom the horizon in Figs. 4 and 6, but move towardsthe horizon in Fig. 5. In Fig. 4, the phase and groupvelocities decrease as z increases. The phase veloc-ity has values greater than the group velocity for lowfrequency waves and they are equal for most of theregion. Also, refraction index is less than zero andthe change in refractive index with respect to angu-lar frequency is negative. So, the dispersion is mostlyanomalous but normal in a small region. In Fig. 6,the phase and group velocities increase towards theevent horizon. The refraction index is less than zeroand the change in refractive index with respect to an-gular frequency is positive at random points, whichdepicts that the waves disperse normally at thosepoints. The dispersion is anomalous at rest of thepoints on account of the negative values of dn/dω.As shown in Fig. 5 the phase and group velocitiesincrease as z increases. The phase velocity has valueslarger than the group velocity for low frequency mode,but they are equal in most of the region. The refrac-

10

tive index is less than one and dn/dω shows positivevalues. So, the dispersion is normal in this case.


02

46

8

10

z

0

2

4

6

8

10

Ω

-2

-1

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-20-15-10

-5

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-20-15-10

-5

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-0.4

-0.2

0

02

46

8z


02

46

810

z

0

2

4

6

8

10

Ω

-1·10-14-7.5·10-15-5·10-15

-2.5·10-150

02

46

8z

Fig. 4.— The waves move away from the event hori-zon. The dispersion is normal as well as anomalous.

6. Hot plasma in nonrotating nonmagnetizedbackground

In nonrotating nonmagnetized background of plasmaflow the magnetic field is assumed to be zero, i.e.,B = 0. Fluid velocity V = u(z)ez, i.e., flow is onlyalong z-axis. With choosing B = 0 = λ, V = 0 andc4 = c5 = c6 = 0, the Fourier analyzed perturbedGRMHD equations (35)–(38) reduce to the followingform:

c1

(−iω

αρ

)+ c2

(−iω

αp

)+ c3(p+ ρ)

×−iω

αγ2u+ (1 + γ2u2)ik

−(1− 2γ2u2)(1 + γ2u2)u′

u

= 0, (47)

c1[ργ2az + (1 + γ2u2)uu′+ γ2u2

×(ρ′ + ikρ)] + c2[pγ2az + (1 + γ2u2)


02

46

8

10

z

0

2

4

6

8

10

Ω

01

2

3

4

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

5

10

15

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

5

10

15

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

0.2

0.4

0.6

02

46

8z


02

46

810

z

0

2

4

6

8

10

Ω

02.5·10-145·10-14

7.5·10-141·10-13

02

46

8z

Fig. 5.— The waves move towards the event horizon.The dispersion is normal.

×uu′+ (1 + γ2u2)(p′ + ikp)]

+c3

[(p+ ρ)γ2

(1 + γ2u2)

×(−iω

α+ iku

)+ u′(1 + γ2u2)

×(1 + 4γ2u2) + 2γ2uaz

+2γ2u(1 + γ2u2)(p′ + ρ′)

]= 0, (48)

c1γ2

(−iω


)ρ+ uρ′

+c2

[iω

α(1− γ2)p+ γ2(iku+ 2uaz

+u′)p+ up′]+ c3γ

2(p′ + ρ′)

+2(2γ4uu′ + az + 2γ2u2az)(p+ ρ)

+(1 + 2γ2u2)(p+ ρ)ik= 0. (49)

11


02

46

8

10

z

0

2

4

6

8

10

Ω

-4

-3

-2

-1

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-15

-10

-5

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-15

-10

-5

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-0.6

-0.4

-0.2

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-1·10-170

1·10-172·10-17

02

46

8z

Fig. 6.— The waves move away from the event hori-zon. The dispersion is normal as well as anomalousat random points


We consider the same assumptions as applied inSect. 4 for the values of lapse function, velocity, pres-sure, density and specific enthalpy with the restric-tion that B = 0 = λ, V = 0. Under these conditionsthe GRMHD equations (12)–(16) are satisfied for theregion 0.2 ≤ z ≤ 10, 0 ≤ ω ≤ 10. Applying theseassumptions in (47)–(49), we obtain two dispersionrelations. The real part gives a quartic equation in k

A1(z, ω)k2 +A2(z, ω)k +A3(z, ω) = 0, (50)

which yields four values of k, out of which two arereal. The imaginary part gives a cubic equation in k

B1(z)k3+B2(z, ω)k

2+B3(z, ω)k+B4(z, ω) = 0, (51)

from which one real value of k is obtained and theremaining two are complex conjugate to each other.The real values for k from (50) and (51) have beenemployed to show graphically wave vector k, phasevelocity vp, group velocity vg, refractive index n and


02

46

8

10

z

0

2

4

6

8

10

Ω

0

1

2

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

05

10

15

20

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

05

10

15

20

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

0.2

0.4

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-1

-0.5

0

0.5

1

02

46

8z

Fig. 7.— The waves move towards the event horizon.The dispersion is normal as well as anomalous.

rate of change of refractive index with respect to fre-quency dn/dω. The results are shown in Figs. 7–9.Waves move away from the horizon in Fig. 8, but theymove towards the horizon in Figs. 7 and 9. The wavenumber k takes on values in the region 0.2 ≤ z ≤ 10,0.2 < ω ≤ 10, which are positive in Figs. 7 and 9and negative in Fig. 8. No wave exists in the region0 ≤ z ≤ 0.2. The phase and group velocities bothincrease as z increases in Figs. 7 and 9. The phasevelocity has greater values than the group velocity inthe low frequency region, but they are equal in mostof the region. The refractive index is less than unityand the change in refractive index with respect to an-gular frequency is positive at random points in Fig. 9and negative in Fig. 7. Hence, the dispersion is nor-mal as well as anomalous in Figs. 7 and 9. In Fig. 8the phase velocity has greater values than the groupvelocity for low frequency region, but they are equalin most of the region and both increase towards thehorizon. The refractive index is less than zero andthe dispersion is normal as well as anomalous.

12


02

46

8

10

z

0

2

4

6

8

10

Ω

-3

-2

-1

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-20

-15

-10

-5

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-20

-15

-10

-5

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-0.4

-0.2

0

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-1

-0.5

0

0.5

1

02

46

8z

Fig. 8.— The waves move away from the event hori-zon. The dispersion is normal as well as anomalous.

7. Concluding remarks

Our main concern has been the investigation ofthe wave properties of hot plasmas near the Reissner-Norstrom black hole event horizon in a Veselagomedium. For this unusual medium, we reformu-late the 3+1 GRMHD equations, consider linear per-turbation and obtain their component form. Theseequations are then Fourier analyzed to obtain disper-sion relations for the rotating magnetized, rotatingnonmagnetized and nonrotating nonmagnetized back-grounds of plasmas. The nonrotating background ex-presses the existence of a perfect Schwarzschild orRN-type geometry outside the event horizon. Butthe rotating background describes the restricted Kerror Kerr-Newman-type geometry (Zhang 1989b) nearthe event horizon which has a variable lapse functionwith negligible rotation. In our study we have consid-ered RN black hole with rotating background. Thisis reasonable due to the fact that the RN black holeis the minimum configuration of the Kerr-Newmanblack hole. The properties of plasma waves are an-alyzed on the basis of the quantities: wave number,


02

46

8

10

z

0

2

4

6

8

10

Ω

012

3

4

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

5

10

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

5

10

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

0

0.2

0.4

0.6

02

46

8z


02

46

8

10

z

0

2

4

6

8

10

Ω

-2·10-17-1·10-17

01·10-172·10-17

02

46

8z

Fig. 9.— The waves move towards the event hori-zon. The dispersion is normal as well as anomalousat random points.

phase and group velocities, and refractive index, de-rived from the dispersion relations. We have solvedthe dispersion relation numerically and presented theresults graphically. There is no wave near the eventhorizon at 0 ≤ z < 0.2 due to immense gravitationalfield and waves in growing/damping mode are foundin the region 0.2 ≤ z ≤ 10, 0 ≤ ω ≤ 10, dependingon both the frequency and radial distance from theblack hole’s event horizon.

For a rotating magnetized plasma, Fig. 1 indicatesthat waves are directed away from the event horizon,while Figs. 2 and 3 display that waves move towardsthe event horizon. The value of refractive index nsatisfies n < 1. The phase and group velocities areseen to have antiparallel values in all the figures forwaves with less angular frequencies. The change inrefractive index with respect to angular frequency ispositive in Figs. 2 and 3, but positive as well as neg-ative in Fig. 1. So, the dispersion is normal as wellas anomalous in Fig. 1, and it is normal in Figs. 2and 3.

13

In a rotating nonmagnetized plasma, waves arefound to move away from the event horizon in Figs.4 and 6, and they are towards the event horizon inFig. 5. The refractive index is less than one and thephase and group velocities have antiparallel values inFigs. 4 and 5 for waves with less angular frequencies.The change in refractive index with respect to angu-lar frequency is negative in Fig. 4, positive in Fig. 5,and positive at random points in Fig. 6. Thus, thewave disperses normally in Fig. 5, but it dispersesnormally as well as anomalously in Figs. 4 and 6.

Figures 7 and 9 display that waves move towardsthe event horizon, while in Fig. 8 waves go awayfrom the event horizon when the plasma backgroundis nonrotating nonmagnetized. The refractive indexis less than one and the phase and group velocitieshave antiparallel values in all these figures for waveswith less angular frequencies. The change in refrac-tive index with respect to angular frequency is neg-ative in Figs. 7, 8, but positive at random points inFig. 9. Hence, waves disperse normally as well asanomalously in Figs. 7–9.

As Mackay (Mackay et al. 2005) studied, rotationof a black hole is required for the existence of negativephase velocity propagation and the waves of less an-gular velocity are evanescent. It is interesting that ouranalysis shows negative phase velocity propagates inthe rotating as well as nonrotating backgrounds whenthe black hole is nonrotating. The results displayedin Figs. 1–9 show that the value of refractive index isless than one and it increases in a small region nearthe event horizon. The phase velocity has greatervalue than that of the group velocity for both mag-netized and nonmagnetized backgrounds. These arethe prominent features of the Veselago medium, whichdemonstrate the validity of this unusual medium forrotating (magnetized and nonmagnetized) and nonro-tating nonmagnetized backgrounds around Reissner-Norstrom black hole event horizon. The presenceof Veselago medium is thus confirmed for the rotat-ing (magnetized and nonmagnetized) and nonrotatingnonmagnetized plasmas.

The result of this paper reduces to the case of theSchwarzschild black for Q = 0. In this study we havetaken Q2 = Q2

e + Q2m, Q2/M2 = 0.49. Using the

formalism of this paper with Q = M , one can ob-tain the result for the extremal Reissner-Norstromblack hole. Black holes will greatly affect the sur-rounding plasma medium (which is highly magne-tized) with their enormous gravitational fields. A

successful study of waves and emissions from plas-mas falling into a black hole will be of great value inaiding the observational identification of black holecandidates. For this reason, plasma physics in theblack hole environment has become a subject of ob-vious interest in astrophysics. Nevertheless, there arerelatively few relativistic cosmological investigationsthat take into account of plasma effects and the be-havior of matter in the presence of electromagneticfields. The general relativistic treatment of plasmas,both in astrophysics as well as in cosmology, could bea field open to investigation.

The basic constituent of nature is the hot plasma,which reduces to cold and isothermal plasmas withsome restrictions. There have been renewed interestin artificial media with negative real parts of the ma-terial parameters, i.e., Veselago media. Indeed, suchmedia are experimentally realized and have potentialapplications in sub-wavelength imaging. The work ofplasma fluid in Veselago medium has attracted a lotof interest in recent years.

In view of the above considerations, our study ofthe dispersion relation of the hot plasma for Veselagomedium in the vicinity of the RN black hole is wellmotivated. The work of this paper can be extendedto the RN black hole generalized with a cosmologicalparameter. This type of extension may be interestingfrom the point of view of an inflationary scenario ofearly universe.

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Hot plasma waves in Veselago medium around Reissner-Nordström black hole

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