Aligned Electromagnetic Excitations of the Kerr-Schild Solutions

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by  Alexander Burinskii

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ALIGNED ELECTROMAGNETIC EXCITATIONS OF THE
KERR-SCHILD SOLUTIONS∗
ALEXANDERBURINSKII
Gravity Research Group, NSI Russian Academy of Sciences,
B. Tulskaya 52, Moscow 115191, Russia, [email protected]
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Aligned to the Kerr-Schild geometry electromagnetic excitations are investigated, and
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asymptoticallyexactsolutionsareobtainedforthelow-frequencylimit.
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1.In this paper we consider the alignedelectromagnetic excitations of the Kerr-
2
Schildgeometry,takingintoaccountthe backreactionofthe excitationsonmetric.
n
a To our knowledge,it is the first attempt to get in the Kerr-Schildformalism a self-
J consistentsolutionforthecaseγ 6=0.ElectromagneticfieldoftheexactKerr-Schild

9 solutions1 has to be aligned to the Kerr null congruence2,3 which is generated by

tangent vector kµ(x). Aligned e.m. excitations on the Kerr backgroundwere inves-
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v tigated in.2–4 Contrary to the usual ‘quasi-normal’ modes, the aligned excitations
6 are compatible with the Kerr congruence and type D of the metric. On the other
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hand they have very specific exhibition in the form of semi-infinite ‘axial’ singular
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linesproducingnarrowbeamswhichcanleadto somenewastrophysicaleffects like
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1 the holes inthe horizonsand jet formation.4 ‘Axial’singularities appearalso in the
6 particle aspect of the Kerr-Schild solutions.5
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2.ThevectorfieldkµisdeterminedbytheKerrTheoremviaacomplexfunction
/
c Y(x), k dxµ = P−1(du+Y¯dζ +Ydζ¯−YY¯dv), where P is a normalizing factor,
q µ
- providing k0 = 1. For the geodesic and shear-free congruences, satisfying to Y,2=
r Y, = 0, the Einstein-Maxwell field equations were integrated out in1 in a general
g 4
: form and reduced to the system of equations for electromagnetic field
v
i A,2−2Z−1Z¯Y,3A=0, A,4=0,
X DA+Z¯−1γ, −Z−1Y, γ =0, γ, =0, where D=∂ −Z−1Y, ∂ −Z¯−1Y¯, ∂
2 3 4 3 3 1 3 2
r
and for gravitationalfield, which will be discussed bellow.
a
Electromagnetic Sector, was discussedin.2–4 The firstequationhas the gen-
eral solution A=ψ/P2, where ψ, =ψ, =0. Therefore ψ has to be a holomorphic
2 4
function of variable Y, since Y, = Y, = 0. Function Y is a projective (complex)
2 4
angular coordinate Y ∈CP1 =S2, Y =eiφtanθ. A holomorphic function may be
representedasaninfiniteLaurentseriesψ(Y)=P2 ∞ Yn.IfthefunctionY ∈S2
n=−∞
is not constant,it has to containatleast one pole whichmay also be atY =∞ (or
θ = π). So, for exclusion of the Kerr-Newman solution having Y = e = const., we
has to consider solutions ψ(Y)=P qi which are singular at angular directions
i Y−Yi
Y = eiφitanθi, and represent a narrow beams in there angular directions. Note,
i 2
that for q = const. these solutions are exact self-consistent solutions of the full
i
system of Kerr-Schild equations.
A waveexcitationpropagatingin the directionY will be describedby the func-
i
∗TalkattheGT3sessionoftheMG11Meeting,thisworkisperformedintheframeofcollaboration
withE.Elizalde,S.R.HildebrandtandG.Magli.
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tion ψ (Y,τ) = q(τ)exp{iωτ} 1 . where τ is a retarded time. For the rotating
i Y−Yi
Kerrsourcethe retardedtime is complexa Inthe nonstationarycase,solutionforA
hastheonlydifferencethatthefunctionψ acquiresextradependencefromthe‘left’
retardedtimeτ .IntherestframethefunctionP hastheformP =2−1/2(1+YY¯).
L
The real operator D acts on the real slice as follows DY = DY¯ = 0, and DP = 0.
The explicit form of the retarded time is τ =t−r+iacosθ. Since cosθ = 1−YY¯,
L 1+YY¯
we have Dcosθ =0, , and Dτ =Dρ= 1.
P
The second e.m. equation takes the form A˙ =−(γP),Y¯ . Integration yields
21/2ψ˙
γ = +φ(Y,τ)/P, (1)
P2Y
where we neglected recoil and φ is an arbitrary analytic function of Y and τ.
3. Gravitationalsector is:
M, −3Z−1Z¯Y, M =Aγ¯Z¯, (2)
2 3
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DM = γγ¯, M, =0. (3)
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Solutions of this system were given in1 only for stationary case, corresponding to
γ = 0. We assume that the energy of electromagnetic wave excitation is much
lower then the mass of rotating object m, and does not affect on the motion of the
center of mass of the solution. However, influence of the electromagnetic field on
the metric occurs also via the function H = mr−ψψ¯/2 , in the K-S metric form
r2+a2cos2θ
g = η +2Hk k , where η is the metric of auxiliary Minkowski space-time.
µν µν µ ν µν
This is a more thin effect, leading to a deformation of the metric tensor around
rotating black hole by electromagnetic excitations. The poles in function ψ which
cause the ‘axial’ singular electromagnetic beams deform strongly the function H.
Theequation(2)acquiresthe form(MP3), =Aγ¯Z¯.Theequation(3)takesthe
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form m˙ = 1P4γγ¯. It is known,6,7 that it determines the loss of mass by radiation.
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The right sides of (2) and (3) will be small for the small (low-frequency) aligned
waveexcitations,sincethefunctionsψ˙ andγwillbeoforder∼iωψ.Inthissensethe
alignedexcitationswillbeasymptoticallyexactsolutionsinthelow-frequencylimit.
However, since ψ contains the singular poles in Y, the limit γ → 0 is not uniform
one,andanextratrickis necessary-a regularization. Sucharegularizationmaybe
performedbythefreefunctionφ(Y,τ)in(1).Thefunctionγ isrepresentedasasum
of simple poles P ai(Yi,Y¯i,τ)−bi(Yi,τ)Pi, where the coefficients a are determined by
i Pi2(Y−Yi) i
function ψ˙, and coefficients b are chosenfrom free function φ to provide cancelling
i
of the poles. It allows us to perform regularization of the most of poles in γ. If
all the poles in the function γ will be cancelled, the result of integration will be a
stochastic radiationwhich will reduce to zero for weakexcitations,andsolutions of
aThe Kerr solution is described by a complex ‘point-like’ source propagating along a complex
worldline.6 Therearedifferent‘left’and‘right’complexconjugateworldlinesandcorresponding
‘left’and‘right’retardedtimesτL andτR.
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(2) and (3) will be asymptotically exact. However, the pole at Y = ∞ can not be
regularized by this method and demands especial treatment.
4.Structureofthesolutionsnearthebeams(pp-waves)isdiscussedin.2,4 Itwas
shown that such beams pierce the horizons forming the tube-like holes connecting
internal and external regions. So the classical structure of black hole turns out to
be destroyed.
Our solution turns out to be exact in the asymptotic limit γ → 0, which cor-
responds to the weak and slowly changed electromagnetic field. In particular, it
shall tend to exact one for a black hole immersed into the zero point field of
virtual photons. In this case we have a sum of excitations in diverse directions
ψ(Y,τ) = P qi(τ) which leads to a flow and migration of many singular beams
i Y−Yi
leading to an instantaneous appearance and disappearance of the holes in hori-
zon, as it is shown on fig.1. One can assume that it may be a mechanism of BH
evaporation.
Fig.1. Thevacuum flowofvirtualphotons piercestheblackholehorizon.
Note, that this picture is reminiscent of the haired black hole which was sug-
gested by the approach from the loop quantum gravity, where singular hairs were
formed from the horizon contrary to the appearance of the holes in horizon.9
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