Hot Plasma Waves Surrounding the Schwarzschild Event Horizon in a Veselago Medium

0.68 MB·English

by  M. Sharif

Hot Plasma Waves Surrounding the
Schwarzschild Event Horizon in a
1
Veselago Medium
1
0
2

n M. Sharif and Noureen Mukhtar
a ∗ †
J Department of Mathematics, University of the Punjab,

5
Quaid-e-Azam Campus, Lahore-54590, Pakistan.

]
c
q
-
r
g
Abstract
[

1 Thispaperinvestigates wavepropertiesofhotplasmainaVeselago
v medium. For the Schwarzschild black hole, the 3+1 GRMHD equa-
4
tions are re-formulated which are linearly perturbed and then Fourier
8
8 analyzed forrotating (non-magnetized andmagnetized) plasmas. The
0 graphs of wave vector, refractive index and change in refractive are
.
1 used to discuss the wave properties. The results obtained confirm the
0
presence of Veselago medium for both rotating (non-magnetized and
1
1 magnetized) plasmas. This work generalized the isothermal plasma
: waves in the Veselago medium to hot plasma case.
v
i
X
Keywords: Veselago medium; 3+1 formalism; GRMHD equations; Isother-
r
a mal plasma; Dispersion relations.
PACS: 95.30.Sf; 95.30.Qd; 04.30.Nk
1 Introduction
Plasmas are found nearly everywhere in nature. These are electrically con-
ductive and give a strong respond to electromagnetic fields. Plasma fills the
interplanetary and interstellar medium and are the components of the stars.
∗[email protected]
†[email protected]
1
To explore the dynamics of magnetized plasma and characteristics of black
hole gravity when it acts in plasma’s magnetic field, the theory of general rel-
ativistic magnetohydrodynamics (GRMHD) is the most accurate academic
discipline. Schwarzschild black hole has zero angular momentum so plasma
present in the magnetosphere moves only in the radial direction.
The gravity of black hole perturbs the magnetospheric plasma. Rela-
tivists are always curious to study the effect of these perturbations in the
black hole regime. Regge and Wheeler [1] concluded that the Schwarzschild
singularity remains stable when a small non-spherical odd-parity perturba-
tion is introduced. Zerilli [2] explored the same stability problem by consid-
ering an even parity perturbation. The behavior of electric field generated
by a charged particle at rest near the Schwarzschild black hole was discussed
by Hanni and Ruffini [3]. Sakai and Kawata [4] examined electron-positron
plasma waves in the frame of two fluid equations for the Schwarzschild black
hole. Hirotoni and Tomimatsu [5] found that a small perturbation of polo-
dial magnetic field could highly disturb the plasma accretion. They assumed
non-stationary and axisymmetric perturbations of MHD accretion onto the
Schwarzschild magnetosphere. Zenginoglu et al. [6] solved a hyperboloidal
initial value problem for the Bardeen-Press equation to analyze the effect of
gravitational perturbation on the Schwarzschild spacetime.
The 3 + 1 formalism (also called Arnowitt, Deser and Misner (ADM)
[7]) is much helpful to study gravitational radiations from black hole as well
as analyzing the gravitational waves. To explore the magnificent aspects of
general relativity (GR), many authors [8]-[10] adopted this technique. The
electromagnetic theory in the black hole regime was developed by Thorne
and Macdonald [11, 12]. Durrer and Straumann [13] deduced some basic
results of GR by using this formalism. Holcomb and Tajima [14], Holcomb
[15] and Dettmann et al. [16] studied the wave properties for the Friedmann
universe. Buzzi et al. [17] investigated the plasma wave propagation close to
the Schwarzschild magnetosphere. Zhang [18] composed the laws of perfect
GRMHDin3+1formalismforageneralspacetime. Thesameauthor[19]also
describedtheroleofcoldplasmaperturbationinthevicinityoftheKerrblack
hole. Using this formulation, Sharif and his collaborators [20]-[23] discussed
plasma (cold, isothermal and hot) wave properties with non-rotating as well
as rotating backgrounds.
Metamaterials are artificial materials that have unusual electromagnetic
properties and Veselago medium or Double negative medium (DNG) is its
most significant class. This medium has both electric permittivity as well
2
as magnetic permeability less than zero. It is also known as negative refrac-
tive index medium (NIM) and negative phase velocity medium (NPV). Many
people [24]-[28] considered this medium to explore the unusual behavior of
physical laws and their plausible applications. Ziolkowski [29] studied wave
propagation in DNG medium both analytically and numerically. Valanju et
al. [30] found positive and very inhomogeneous wave refraction in this un-
usual medium. Ramakrishna [31] examined the problem of designing such
materials which have negative material parameters. He also discussed the
concept of perfect lens consisting of a slab of negative refractive materi-
als (NRM). Veselago [32] verified the concepts related to the energy, linear
momentum and mass transferred by an electromagnetic wave in a negative
refraction medium. In a recent paper [33], we have discussed the isothermal
plasma wave properties for the Schwarzschild magnetosphere in a Veselago
medium. The results verified the presence of this medium for only rotating
non-magnetized plasma.
In this paper, we investigate wave properties of hot plasma in the vicinity
of the Schwarzschild event horizon in a Veselago medium. The format of
the paper is as follows. In Section 2, the general line element in ADM
3+1 formalism and its modification for the Schwarzschild planar analogue
is given. Linear perturbation and Fourier analysis of the 3 + 1 GRMHD
equations for hot plasma is provided in section 3. Sections 4 and 5 give
the reduced form of the GRMHD equations for rotating (non-magnetized
and magnetized respectively) plasmas. In the last section, summary of the
results is given.
2 3+1 Foliation and Planar Analogue of the
Schwarzschild Spacetime
The 3 + 1 split of spacetime is an access to the field equations in which
four-dimensional spacetime is sliced into three-dimensional spacelike hyper-
surfaces. In ADM 3+1 formalism, the general line element is [19]
ds2 = α2dt2 +γ (dxi +βidt)(dxj +βjdt), (2.1)
ij
−
wheretheratioofthefiducialpropertimetouniversal time, i.e., dτ isdenoted
dt
by α (lapse function). When FIDO (fiducial observer) changes his position
from one hypersurface to another, the shift vector βi calculates the change
3
of spatial coordinates. The components of three-dimensional hypersurfaces
are denoted by γ (i,j = 1,2,3). A natural observer associated with the
ij
above spacetime is known as FIDO. The mathematical description of the
Schwarzschild planar analogue is given by
ds2 = α2(z)dt2 +dx2 +dy2 +dz2, (2.2)
−
where the directions z, x and y are analogous to the Schwarzschild coordi-
nates r, φ and θ respectively. The comparison of Eqs.(2.1) and (2.2) yields
α = α(z), β = 0, γ = 1 (i = j). (2.3)
ij
3 3+1 GRMHD Equations for the Schwarzschild
Planar Analogue in a Veselago Medium
Appendix A contains the 3+1 GRMHD equations in a Veselagho medium
for the plasma existing in the general line element and the Schwarzschild
planar analogue (Eqs.(2.1) and (2.2)). In the vicinity of the Schwarzschild
magnetosphere, the specific enthalpy for hot plasma is [19]
ρ+p
µ = , (3.1)
ρ
0
where the rest mass density, moving mass density, pressure and specific en-
thalpy are represented by ρ , ρ, p and µ respectively. For cold plasma, we
0
have p = 0 while for isothermal plasma, p = 0 but specific enthalpy is con-
6
stant. However, specific enthalpy is variable for the hot plasma. This is
the most general plasma which reduces to cold and isothermal plasmas with
some restrictions. This equation shows the exchange of energy between the
plasma and fluid’s magnetic field. It is obvious from the above equation that
for µ to be variable, p must be variable. In this paper, we have used the hot
plasma along with the Veselago medium in the vicinity of the Schwarzschild
magnetosphere. The 3 + 1 GRMHD equations (Eqs.(A10)-(A14)) for hot
plasma surrounding the Schwarzschild event horizon become
∂B
= (αV B), (3.2)
∂t −∇× ×
.B = 0, (3.3)
∇
∂(ρ+p) ∂V
+(ρ+p)γ2V. +(ρ+p)γ2V.(αV. )V
∂t ∂t ∇
+(ρ+p) .(αV) = 0, (3.4)
∇
4
B2 1 1 ∂
(ρ+p)γ2 + δ +(ρ+p)γ4V V B B
ij i j i j
(cid:26)(cid:18) 4π(cid:19) − 4π (cid:27)(cid:18)α∂t
B2 1
+V. )Vj +γ2V (V. )(ρ+p) δ B B VjVk
∇ i ∇ −(cid:18)4π ij − 4π i j(cid:19) ,k
1 1
= (ρ+p)γ2a p + (V B) .(V B) (αB)2
− i − ,i 4π × i∇ × − 8πα2 ,i
1 1
+ (αB ) Bj [B V ( (αV B)) ] , (3.5)
i ,j i
4πα − 4πα ×{ × ∇× × }
1 ∂ 1 ∂p
( +V. )(ρ+p)γ2 +2(ρ+p)γ2(V.a)+(ρ+p)
α∂t ∇ − α ∂t
1 ∂B 1 ∂B
γ2( .V) (V B).(V ) (V B).(B )
∇ − 4πα × × ∂t − 4πα × × ∂t
1
+ (V B).( αB) = 0. (3.6)
4πα × ∇×
In rotating background, plasma is assumed to flow in two dimensions, i.e., in
xz-plane. Thus FIDO’s measured magnetic field B and velocity V become
V = V(z)e +u(z)e , B = B[λ(z)e +e ], (3.7)
x z x z
where B is an arbitrary constant. The quantities λ, u and V are related by
[20]
VF
V = +λu, (3.8)
α
where VF is an integration constant. Thus the Lorentz factor γ = 1
√1−V2
takes the form
1
γ = . (3.9)
√1 u2 V2
− −
When the plasma flow is perturbed, the flow variables (mass density ρ,
pressure p, velocity V and magnetic field B) turn out to be
ρ = ρ0 +δρ = ρ0 +ρρ, p = p0 +δp = p0 +pp,
V = V0 +δV = V0 +v, B = B0 +δB = B0 +Bb, (3.10)
e e
whereunperturbedquantitiesaredenotedbyρ0, p, V0, B0 whileδρ, δp, δV,
δB represent linearly perturbed quantities. The following dimensionless
5
quantities ρ, p, v , v , b and b are introduced for the perturbed quan-
x z x z
tities
e e
ρ˜= ρ˜(t,z), p˜= p˜(t,z), v = δV = v (t,z)e +v (t,z)e ,
x x z z
δB
b = = b (t,z)e +b (t,z)e . (3.11)
x x z z
B
When we insert these linear perturbations in the perfect GRMHD equations
(Eqs.(3.2)-(3.6)), it follows that
∂(δB)
= (αv B) (αV δB), (3.12)
∂t −∇× × −∇× ×
.(δB) = 0, (3.13)
∇
1 ∂(δρ+δp) 1 ∂
+(ρ+p)γ2V.( +V. )v+(ρ+p)( .v)
α ∂t α∂t ∇ ∇
= 2(ρ+p)γ2(V.v)(V. )lnγ (ρ+p)γ2(V. V).v
− ∇ − ∇
+(ρ+p)(v. lnu), (3.14)
∇
B2 1 1 ∂vj
(ρ+p)γ2 + δ +(ρ+p)γ4V V B B
ij i j i j
(cid:26)(cid:18) 4π(cid:19) − 4π (cid:27) α ∂t
1 1 ∂(δB)
+ [B V ] +(ρ+p)γ2v Vj +(ρ+p)
i i,j
4π ×{ × α ∂t }
γ4V v VjVk +γ2V (V. )(δρ+δp)+γ2V (v. )(ρ+p)
i j,k i i
× ∇ ∇
1
+γ2v (V. )(ρ+p)+γ4(2V.v)V (V. )(ρ+p) (αδB )
i i i ,j
∇ ∇ − 4πα{
(αδB ) Bj = (δp) γ2 (δρ+δp)+2(ρ+p)γ2(V.v) a
j ,i i i
− } − − { }
1
+ (αB ) (αB ) δBj (ρ+p)γ4(v Vj +vjV )V Vk
i ,j j ,i i i k,j
4πα{ − } −
γ2 (δρ+δp)Vj +2(ρ+p)γ2(V.v)Vj +(ρ+p)vj V
i,j
− { }
γ4V (δρ+δp)Vj +4(ρ+p)γ2(V.v)Vj +(ρ+p)vj V Vk, (3.15)
i j,k
− { }
1 ∂(δρ+δp) 1 ∂(δp)
γ2 +v. (ρ+p)γ2 +(V. )(δρ+δp)γ2
α ∂t ∇ − α ∂t ∇
+2(ρ+p)γ4(V. )(V.v)+2(ρ+p)γ2(v.a)+4(ρ+p)γ4(V.v)
∇
(V.a)+2(δρ+δp)γ2(V.a)+(ρ+p)γ2( .v)+2(ρ+p)γ4
∇
1 ∂B ∂B
(V.v)( .V)+(δρ+δp)γ2( .V) = [v.(B. )V+V.(B. )v
∇ ∇ 4πα ∂t ∂t
6
∂B ∂B ∂δB
+V.(B.δB)V+V.(δB )V v.(B.V) V.(B.V)
∂t − ∂t − ∂t
∂δB ∂B 1 ∂v ∂δv
V.(B.v) V.(δB.V) ] [V.(B.B) V.(B. )B]
− ∂t − ∂t − 4πα ∂t − ∂t
1
+ [(v B+V δB).( B)+(V B).( δB)] (3.16)
4π × × ∇× × ∇×
The component form of these equations, using Eq.(3.11), become
1 ∂b
x
ub = (ub Vb v +λv ) lnα
x,z x z x z
α ∂t − − − ∇
(v λv λ′v +V′b +Vb u′b ), (3.17)
x,z z,z z z z,z x
− − − −
1 ∂b
z
= 0, (3.18)
α ∂t
b = 0, (3.19)
z,z
1 ∂ρ˜ 1 ∂p˜ 1 ∂v
ρ +p +(ρ+p)γ2V( x +uv )+(ρ+p)γ2u
x,z
α ∂t α ∂t α ∂t
1 ∂v
z +(ρ+p)(1+γ2u2)v = γ2u(ρ+p)[(1+2γ2V2)V′
z,z
×α ∂t −
u′
+2γ2uVu′]v +(ρ+p)[(1 2γ2u2)(1+γ2u2)
x
− u
2γ4u2VV′]v , (3.20)
z
−
B2 1 ∂v λB2
(ρ+p)γ2(1+γ2V2)+ x + (ρ+p)γ4uV
(cid:26) 4π(cid:27) α ∂t (cid:26) − 4π (cid:27)
1 ∂v B2
z + (ρ+p)γ2(1+γ2V2)+ uv + (ρ+p)γ4uV
x,z
×α ∂t (cid:26) 4π(cid:27)
(cid:8)
λB2 B2 B2
uv (1+u2)b α′(1+u2)+αuu′ b
z,z x,z x
− 4π (cid:27) − 4π − 4πα
(cid:8) (cid:9)
+γ2u(ρρ˜+pp˜) (1+γ2V2)V′ +γ2uVu′ +γ2uV(ρ′ρ˜+ρρ˜′
+p′p˜+pp˜′)+[((cid:8)ρ+p)γ4u (1+4γ2V2)u(cid:9)u′+4VV′(1+γ2V2)
B2uα′ (cid:8) (cid:9)
+ +γ2u(1+2γ2V2)(ρ′ +p′)]v +[(ρ+p)γ2 (1+2γ2u2)
x
4πα
(cid:8)
B2u
(1+2γ2V2)V′ γ2V2V′ +2γ2(1+2γ2u2)uVu′ (λα)′
− − 4πα
(cid:9)
+γ2V(1+2γ2u2)(ρ′ +p′)]v = 0, (3.21)
z
7
λ2B2 1 ∂v λB2
(ρ+p)γ2(1+γ2u2)+ z + (ρ+p)γ4uV
(cid:26) 4π (cid:27) α ∂t (cid:26) − 4π (cid:27)
1 ∂v λ2B2
x + (ρ+p)γ2(1+γ2u2)+ uv + (ρ+p)γ4uV
z,z
×α ∂t (cid:26) 4π (cid:27)
(cid:8)
λB2 λB2 B2
uv + (1+u2)b + (αλ)′ α′λ+uλ(uα′
x,z x,z
− 4π (cid:27) 4π 4πα { −
+u′α) b +(ρρ˜+pp˜)γ2 a +uu′(1+γ2u2)+γ2u2VV′
x z
}
+(1+γ2u2)(p′p˜+pp˜′)+(cid:8)γ2u2(ρ′ρ˜+ρρ˜′)+[(ρ+p)γ4 (cid:9)
λB2uα′
u2V′(1+4γ2V2)+2V(a +uu′(1+2γ2u2))
z
×{ }− 4πα
+2γ4u2V(ρ′ +p′)]v +[(ρ+p)γ2 u′(1+γ2u2)(1+4γ2u2)
x
(cid:8)λB2u
+2uγ2(a +(1+2γ2u2)VV′) + (αλ)′ +2γ2u(1
z
4πα
(cid:9)
+γ2u2)(ρ′ +p′)]v = 0, (3.22)
z
1 ∂ρ˜ 1 ∂p˜
γ2ρ + γ2p +γ2(ρ′ +p′)v +uγ2(ρρ˜ +pp˜ +ρ′ρ˜
z ,z ,z
α ∂t α ∂t
1 ∂p˜
+p′p˜) p +2γ2u(ρρ˜+pp˜)a +γ2u′(ρρ˜+pp˜)+2(ρ
z
− α ∂t
+p)γ4(uV′ +2uVa +u′V)v +2(ρ+p)γ2(2γ2uu′ +a γ4
z x z
+2γ2u2a )v +2(ρ+p)γ4uVv +(ρ+p)γ2(1+2γ2u2)
z z x,z
B2
v [(V2 +u2)λb +(V2 +u2)b λV(λV
z,z x z
× − 4πα −
∂b ∂b B2
x z
+u) u(λV +u) ] [(V λu)v +λ(uλ
x,t
∂t − ∂t − 4πα −
B2
V)v ]+ (λλ′v λ′v λ′Vb
z,t z x z
− 4π − −
+λ′ub Vb +uλb ) = 0. (3.23)
x x,z x,z
−
For the purpose of Fourier analysis, the following harmonic spacetime depen-
dence of perturbation is assumed
ρ(t,z) = c e−ι(ωt−kz), p(t,z) = c e−ι(ωt−kz),
1 2
v (t,z) = c e−ι(ωt−kz), v (t,z) = c e−ι(ωt−kz),
z 3 x 4
e e
b (t,z) = c e−ι(ωt−kz), b (t,z) = c e−ι(ωt−kz). (3.24)
z 5 x 6
Here k and ω are the z-component of the wave vector (0,0,k) and angular
frequency respectively. Plasma wave properties near the event horizon can
8
be explored by the wave vector which is also used to obtain refractive index.
We define wave vector and refractive index as follows:
Wave Vector: The direction in which a plane wave propagates is
•
represented by a wave vector. Its magnitude gives the wave number.
Refractive Index: When light travels from one medium to another
•
(usually from vacuum) then its ratio between the two mediums is given
by the refractive index. The change in the refractive index with respect
to angular frequency decides whether the dispersion will be normal or
anomalous.
Using Eq.(3.24) in Eqs.(3.17)-(3.23), we get their Fourier analyzed form
c (α′ +ιkα) c (αλ)′ +ιkαλ c (αV)′ c (αu)′+ιω
4 3 5 6
− { }− − {
+ιkuα = 0, (3.25)
}
ιω
c (− ) = 0, (3.26)
5
α
c ιk = 0, (3.27)
5
ιω ιω ιω
c (− ρ)+c (− p)+c (ρ+p)[− γ2u+(1+γ2u2)ιk
1 2 3
α α α
u′ ιω
(1 2γ2u2)(1+γ2u2) +2γ4u2VV′]+c (ρ+p)γ2[(−
4
− − u α
+ιku)V +u(1+2γ2V2)V′ +2γ2u2Vu′] = 0, (3.28)
c [ργ2u (1+γ2V2)V′ +γ2Vuu′ +γ2Vu(ρ′ +ιkρ)]
1
{ }
+c [pγ2u (1+γ2V2)V′ +γ2Vuu′ +γ2Vu(p′+ιkp)]
2
{ }
ιω
+c [(ρ+p)γ2 (1+2γ2u2)(1+2γ2V2)V′ +(− +ιku)γ2Vu
3
{ α
γ2V2V′ +2γ2(1+2γ2u2)uVu′ +γ2V(1+2γ2u2)(ρ′ +p′)
− }
B2u λB2 ιω
(λα)′ + ( ιku)]+c [(ρ+p)γ4u (1+4γ2V2)
4
−4πα 4π α − {
ιω
uu′ +4VV′(1+γ2V2) +(ρ+p)γ2(1+γ2V2)(− +ιku)
× } α
B2uα′ B2 ιω
+γ2u(1+2γ2V2)(ρ′ +p′)+ ( ιku)]
4πα − 4π α −
B2
c [αuu′+α′(1+u2)+(1+u2)ιkα] = 0, (3.29)
6
− 4πα
9
c [ργ2 a +(1+γ2u2)uu′+γ2u2VV′ +γ2u2(ρ′ +ιkρ)]
1 z
{ }
+c [pγ2 a +(1+γ2u2)uu′ +γ2u2VV′ +(1+γ2u2)
2 z
{ }
ιω
(p′ +ιkp)]+c [(ρ+p)γ2 (1+γ2u2)(− +ιku)
3
× { α
+u′(1+γ2u2)(1+4γ2u2)+2uγ2(a +(1+2γ2u2)VV′)
z
}
λB2u λ2B2 ιω
+2γ2u(1+γ2u2)(ρ′ +p′)+ (λα)′ ( ιku)]
4πα − 4π α −
ιω
+c [(ρ+p)γ4 (− +ιku)uV +u2V′(1+4γ2V2)+2V(a
4 z
{ α
λB2 ιω
+(1+2γ2u2)uu′) +2γ4u2V(ρ′ +p′)+ ( ιku)
} 4π α −
λB2uα′ B2
]+c [ (λα)′ +α′λ uλ(uα′+u′α)
6
− 4πα 4πα{− − }
λB2
+ (1+u2)ιk] = 0, (3.30)
4π
ιω ιω
c (− γ2 +ιkuγ2 +2uγ2a +γ2u′)ρ+uρ′γ2 +c ( (1 γ2)
1 z 2
{ α } { α −
+ιkuγ2 +2γ2ua +γ2u′)p+uγ2p′ +c γ2 (ρ′ +p′)+2
z 3
} {
λB2
(2γ4uu′+a +2γ2u2a )(ρ+p)+(1+2γ2u2)(ρ+p)ιk +
z z
× 4πα
(λu V)ιω +αλ′ +c [2(ρ+p)γ2 (uV′ +2uVa +u′V)+uVιk
4 z
× − } { }
B2 B2
+ (V uλ)ιω αλ′]+c [− (V2 +u2)λ+λV(λV +u)ιω
6
4πα − − 4πα{ }
αλ′u+ιkα(V uλ)] = 0. (3.31)
− −
Dispersion relations will be obtained by using these equations.
4 Rotating Non-Magnetized Flow with Hot
Plasma
In this section, rotating non-magnetized background of plasma flow is as-
sumed, i.e., B = 0. Thus, the evolution equations (3.2) and (3.3) of magnetic
field are satisfied. Substituting B = 0 = λ and c = 0 = c in the Fourier
5 6
10