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Holographic structure of the Kerr-Schild black-holes, Twistor-b eam excitations and fluctuating horizon Alexander Burinskii
NSI, Russian Academy of Sciences, Moscow, Russia

MSU, 22-th August, 2009
based on A.B. First Award of Gravity Research Foundation 2009, Gen.Rel.Grav.July 2009 A.B., Elizalde E., Hildebrandt S.R., Magli G., Phys. Rev. D 74(2006) 021502 A.B., Elizalde E., Hildebrandt S.R., Magli G., Phys. Lett. B 671(2009) 486 A.B., arXiv: 0903.2365 [hep-th]

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Recent ideas and metho ds in the black hole physics are based on complex analyticity and conformal field theory, which unifies the black hole physics with (sup er)string theory and physics of elementary particles [G.`t Ho oft, NPB(1990)] Kerr-Newman solution: as a Rotating Black-Hole and as a Kerr Spinning Particle: Carter (1968), (g = 2 as that of the Dirac electron), Israel (1970),
AB (1974-2009), Lop ez (1984) ...

Ab out 40 years of the Kerr-Schild Geometry and Twistors. Based on twistors Kerr Theorem and Kerr-Schild geometry (1969). Twistor Algebra, R. Penrose, (1967). З Real and Complex Twistor Structures of the Kerr-Schild Geometry З Twistor-b eams - New results on Black-Holes (AB, First Award of GRF 2009, arXive: 0903.3162).

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Kerr-Schild form of the rotating black hole solutions: g
Е

=

Е

+ 2 H k Е k ,

mr - e2/2 H= 2 . r + a2 cos2

(1)

Vector field kЕ(x) is tangent to Principal Null Congruence (Kerr congruence). ? ? ? kЕdxЕ = P -1(du + Y d + Y d - Y Y dv ), (2) 1 1 1 ? where Y (x) = ei tan 2 , and = (x + iy )2- 2 , = (x - iy )2- 2 , u = (z - t)2- 2 , v = 1 (z + t)2- 2 are the nul l Cartesian coordinates. The Kerr Theorem: The geo desic and shear-free null congruences (typ e D metrics) are determined by holomorphic function Y (x) which is analytic solution of the equation F (T a) = 0 , where F is an arbitrary analytic function ? of the projective twistor coordinates T a = {Y , - Y v , u + Y }.

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Z
10

5

0

-5

-10 10 5 0 0 -5 -10 -10 -5 5 10

Figure 1: The Kerr singular ring and the Kerr congruence.

The Kerr singular ring r = cos = 0 is a branch line of space on two sheets: "negative (-)" and "p ositive (+)" where the fields change their directions. Twosheetedness! Mystery of the Kerr source! The Kerr ring as a "mirror gate" to "Alice world". Stringy source: E.Newman 1964, A.B. 1974-1999, W.Israel 1975, ... Rotating disk. W.Israel (1969), Hamity (1973), L`op ez (1983)9; A.B. (1989) Superconducting bag ( U (1) з U (1) mo del), A.B. (2002-2004). New Lo ok: Holographic BH interpretation. A.B.(2009) based on the ideas C.R.Stephens, G. t' Ho oft and B.F. Whiting (1994), `t Ho oft (2000).
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8

6

out- photons

out- photons

4

2

I+

I+

0

-2

I- in-photons conformal diagram of Minkowski space-time
0 2 4 6 8 10

I

-

-4

-6

unfolded conformal diagram of the Kerr space-time
12 14 16 18 20

-8

Figure 2: Penrose conformal diagrams.

Unfolded Kerr-Schild spacetime corresp onds to holographic BH spacetime. Kerr congruence p erforms holographic pro jection of 3+1 dim bulk to 2+1 dim b oundary.Desirable structure of a quantum BH spacetime ( Stephens, t' Ho oft and Whiting (1994)).

Exact solutions demand Alignment of the electromagnetic field to Kerr congruence, AЕk Е = 0 ! (+) (-) Twosheetedness k Е(+) = k Е(-) gЕ = gЕ . It is ignored in p erturbative approach. Exact solutions have twistor-b eams!
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Twistor-Beams. The exact time-dep endent KS solutions.
Debney, Kerr and Schild (1969). The black-hole at rest: g ? P = 2-1/2(1 + Y Y ). Tetrad comp onents of electromagnetic field Fab = eЕe FЕ , ab F12 = AZ 2, F31 = Z - (AZ ),1 ,
Е

= Е +2H kЕk ,

(3)

here Z = -P /(r + ia cos ) is a complex expansion of the congruence. Stationarity = 0. Kerr-Newman solution is exclusive: (Y ) = const. In general case (Y ) is an arbitrary holomorphic function of Y (x) = ei tan 2 , which is a pro jective co ordinate on celestial sphere S 2, A = (Y )/P 2, and there is infinite set of the exact solutions, in which (Y ) is singular at the set of p oints {Yi}, (Y ) = i Y (xq)i-Yi , corresp onding to angular directions i, i. Twistor-b eams. Poles at Yi pro duce semi-infinite singular lightlike beams, supported by twistor rays of the Kerr congruence. The twistor-b eams have very strong backreaction to KS metric g
Е

=

Е

- 2H k k ,

Е

mr - | |2/2 H= 2 . r + a2 cos2

(4)

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How act such b eams on the BH horizon? Black holes with holes in the horizon, A.B., E.Elizalde, S.R.Hildebrandt and G.Magli, Phys. Rev. D74 (2006) 021502(R) Singular b eams lead to formation of the holes in the black hole horizon, which op ens up the interior of the "black hole" to external space.
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event horizon r+ r -
g g

singular ring

z
0

axial singularity g00 =0
-30 -30 0 30

Figure 3:

Near extremal black hole with a hole in the horizon. The event horizon is a closed surface surrounded by surface g00 = 0.

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Twistor-beams are exact stationary and time-dep endent Kerr-Schild solutions (of type D) which show that `elementary' electromagnetic excitations have generally singular beams supported by twistor nul l lines. Interaction of a blackhole with external, even very weak, electromagnetic field results in app earance of the b eams, which have very strong back reaction to metric and horizon and form a fine-grained structure of the horizon pierced by fluctuating microholes. [A.B., E. Elizalde, S.R. Hildebrandt and G. Magli, Phys.Lett. B 671 486 (2009), arXiv:0705.3551[hep-th]; A.B., arXiv:gr-qc/0612186.]

Figure 4:

Horizon covered by fluctuating micro-holes.

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Time-dep endent solutions of DKS equations for electromagnetic excitations, = 0, A.B. (2004-2008)
a) Exact solutions for electromagnetic field on the Kerr-Schild background, (2004), b) Asymptotically exact wave solutions, consistent with Kerr-Schild gravity in the low frequency limit, (2006-2008) c) Self-regularized solutions, consistent with gravity for averaged stressenergy tensor, (A.B. 2009) Electromagnetic field is determined by functions A and , ? A,2 -2Z -1Z Y ,3 A = 0, A,4 = 0, (5) (6)

? D A + Z - 1 , 2 - Z - 1 Y ,3 = 0 ,

and Gravitational sector: has two equations for function M , which take into account the action of electromagnetic field ? M ,2 -3Z -1Z Y ,3 M = A Z , ?? 1 DM = . ? 2 ?? where cD = 3 - Z -1Y ,3 1 - Z -1Y ,3 2 .
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(7) (8)


Similar to the exact stationary solutions, typical time-dependent (type D) solutions contain outgoing singular beam pulses which have very strong back reaction to metric and perforate horizon. Eqs. of the electromagnetic sector were solved (2004). GSF condition Y ,2 = Y ,4 = 0, k ЕЕY = 0. Stationary Kerr-Schild solutions A = /P 2, where ,2 = ,4 = 0 (Y ) alignment condition k ЕЕ = 0. Time-dep endent solutions need a complex retarded time parameter , ob eying ,2 = ,4 = 0, and = (Y , ). There app ears a dep endence b etween A and ? ?? (3 - Z -1Y ,3 1 - Z -1Y ,3 2)A + Z -1 ,2 -Z -1Y ,3 = 0. Integration yields 21/2 = 2 + 0(Y , )/P, (9) PY which shows that nonstationarity, = i ci( )/(Y - Yi) = 0, creates generally the p oles in i qi/(Y - Yi), leading to twistor-b eams in directions Yi = ei tan 2 .

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Self-regularization.
Structure of KS solutions inspire the regularization which acts immediately on the function . Free function 0(Y , ) of the homogenous solution may b e tuned, to cancel the p oles of function = i ci( )/(Y - Yi). i-th term 21/2ci( ) (tun) i(reg) = + i (Y , )/P. (10) Y (Y - Yi)P 2 Condition to comp ensate i-th p ole is i(reg)(Y , )|Y ? We obtain
(tun) i ? =Yi

= 0.

(11)

21/2ci( ) (Y , ) = - , Y (Y - Yi)Pi
-1/2

(12) (13)

where

? Pi = P (Y , Yi) = 2

? (1 + Y Yi)

is analytic in Y , which provides analyticity of tun(Y , ). i

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As a result we obtain ? ? ci( )(Yi - Y ) i(reg) = 2 . P Pi(Y - Yi) First gravitational DKS equation gives m = m0(Y ) +
i,k

(14)

cick (Yk - Y ) ? (Y - Yi)

? Yk

? dY ?? ?. P Pk (Y - Yk )

(15)

Using the Cauchy integral formula, we obtain m = m0(Y ) + 2 i
i

(Yk - Y ) (Y - Yi)

k

cick ? . |Pk |2

(16)

Functions ci and ck for different b eams are not correlated, ? < cick >= 0. Time averaging retains only the terms with i = k , ? < m >t= m0 - 2 i
k

ck ck ? . |Pk |2

(17)

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Representing ci( ) = qi( )e-ii via amplitudes qi( ) and carrier frequencies i of the b eams. The mass term retains the low-frequency fluctuations and angular non-homogeneity caused by amplitudes and casual angular distribution of the b eams, qk qk ? >. (18) < m >t= m0 + 2 k < |Pkk |2
k k

Second gravitational DKS equations is definition radiation, 1 1 i(reg)k(reg) = - ? m = - P2 2 2 P
i,k i,k

of the loss of mass in cick ? 2P P ? ik (19)

Time averaging removes the terms with i = k and yields 1 ck ck ? < m >t = - . 2 P 2|Pk |2
k

(20)

In terms of the amplitudes of b eams we obtain 1 qk qk ? 2 < m >t = - k < >, 2 |Pkk |4
k

(21)

which shows contribution of a single b eam pulse to the total loss of mass. The obtained solutions are consistent with the Einstein-Maxwell system of equations for the time-averaged stress-energy tensor.
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Obtained results.
З Exact time-dep endent solutions for Maxwell eqs. on the Kerr-Schild background singular twistor-beam pulses. З Exact back reaction of the b eams to metric and horizon fluctuating metric and horizon p erforated by twistor-b eam pulses. З Exact time-dep endent solutions for Maxwell eqs. on the Kerr-Schild background leading to regular, but fluctuating radiation regular < T Е >, but metric and horizon are covered by fluctuating twistor-b eams! З Consistency with averaged Einstein equations R
Е

1 - Rg 2

Е

=< T

Е

>.

(22)

We arrive at a semiclassical geometry of fluctuating twistorb eams which takes an intermediate p osition b etween the Classical and Quantum gravity. THE END. THANK YOU FOR YOUR ATTENTION!
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