Consequences of Analyticity of

the Kerr-Schild Geometry from

Black Holes to Spinning

Particles.

Alexander Burinskii

Orleans, July, 2008

Kerr-Schild Geometry, and Conformal Field

Theory

Black Hole Evaporation ⇔ Conformal Field Theory⇔ Superstrings:

[1] A. Asthekar, V. Taveras and M. Varadarajan, arXiv:0801.1811.[2] A. Strominger, JHEP 02 009 (1998). [3] S. Carlip,Phys. Rev. Lett. 82, 2828 (1999). [4] O. Coussaertand M. Henneaux, Phys. Rev. Lett. 72, 183 (1994).[5] M. Banados, Phys. Rev. Lett. 82, 2030 (1999). [6]J.A. Cardy, Nucl. Phys B270, 186 (1986).

Superstring Theory ⇔ Conformal Field Theory:

[1] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys.B247, 83 (1984). [2] A.A. Belavin, A.M. Polyakov andA.B. Zamolodchikov, Nucl. Phys. B241, 333 (1984).

Kerr-Schild Geometry ⇔ Conformal Field Theory⇔ Superstrings:

[1]G.C. Debney, R.P. Kerr and A.Schild, J. Math. Phys.10, 1842 (1969). [2] A.B., Twistorial Analyticity and

Three Stringy Structures of the Kerr Spinning Particle.

Phys. Rev. D 70, 086006 (2004). [3] A.B., Grav.& Cosmol. 11, 301 (2005); [4] A.B., Int. J. Geom.Methods Mod. Phys. 4, 437 (2007); [5] A.B., Phys.Rev. D 70, 086006 (2004); [6] A.B., String-like Struc-

tures in Complex Kerr Geometry. ( gr-qc/9303003,) In:“Relativity Today”, Ed. by R.P.Kerr and Z.Perjes,1994,p.149.

1

Kerr-Schild Formalism. Debney, Kerr and Schild,J. Math. Phys. 10 (1969) 1842

Complex Projective Angular Coordinate:

Y = eiφ tanθ

2∈ CP 1

The Kerr-Schild ansatz for metric,

gµν = ηµν + 2he3µe3ν ,

√−g = 1.

ηµν - auxiliary Minkowski space-time.

Principal Null Direction

e3 = du+ Y dζ + Y dζ − Y Y dv

in the null Cartesian coordinates√

2ζ = x+ iy,√

2ζ =x− iy,

√2u = z − t,

√2v = z + t.

Null tetrad ea, a = 1,2,3,4. Real directions e3 and e4 =dv+ he3, and two complex conjugate directions

e1 = dζ − Y dv, e2 = dζ − Y dv.

The Kerr Theorem. The geodesic and shear-free(GSF) null congruences (Y,2 = Y,4 = 0.) are deter-mined by function Y (x) which is a solution of theequation F (Y, l1, l2) = 0, where F is an arbitrary an-alytic function of the projective twistor coordinates

Y, l1 = ζ − Y v, l2 = u+ Y ζ

2

Integration of the Einstein-Maxwell field

equations for the geodesic and shear-free con-

gruences was fulfilled in DKS, leading to the

following form of the function

h =1

2M(Z + Z) − 1

2AAZZ , (1)

of the Kerr-Schild ansatz:

gµν = ηµν + 2he3µe3ν . (2)

Necessary functions Z = P/r, Y (x) and param-

eters are determined by the generating func-

tion F.

PZ−1 = r = − dF/dY (3)

is complex radial distance, factor P is con-

nected with Killing vector or the boost of the

source.

There was obtained a system of differential

equations for functions A, and M.

3

Electromagnetic sector:

A,2−2Z−1ZY,3A = 0, A,4 = 0, (4)

DA+ Z−1γ,2−Z−1Y,3 γ = 0, γ,4 = 0, (5)

where D = ∂3 − Z−1Y,3 ∂1 − Z−1Y ,3 ∂2.

The strength tensor of self-dual electromag-

netic field is given by the tetrad components

F12 = AZ2, F31 = γZ − (AZ),1 . (6)

Gravitational sector:

Real function M.

M,2−3Z−1ZY,3M = AγZ, (7)

DM =1

2γγ, M,4 = 0. (8)

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Integration: For any holomorphic F (Y ) ⇒GSF congruence ⇒ a family of algebraically

special solutions.

The Kerr-Newman solution is a very impor-

tant particular case:

Metric gµν = ηµν+2hkµkν, where h =mr−e2/2

r2+a2 cos2 θ.

Electromagnetic field Aµ = err2+a2 cos2 θ

kµ.

Kerr congruence kµ(x) = e3µ(Y )√

2/(1 + Y Y )

which is determined by function Y (x) (solution

of the eq. F (Y, x) = 0.

Singular ring at r+ ia cos θ ≡ ∂Y F = 0.

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The Kerr congruence is vortex of null lines

(twistors)

[] −10

−5

0

5

10

−10

−5

0

5

10−10

−5

0

5

10

Z

The Kerr singular ring and the Kerr congru-

ence.

The Kerr singular ring is a branch line of

space on two sheets: ”negative” and ”posi-

tive” where the fields change their signs and

directions. Congruence covers the space-time

twice. Problem of the Kerr source ⇒ twosheet-

edness !!!

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−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−3

−2

−1

0

1

2

3

Z

r>0

θ=const.

r=const

Twosheeted oblate coordinate system.

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

8

in−photons

out− photons

out− photons

conformal diagram ofMinkowski space−time

unfolded conformal diagram of the Kerr space−time

I−

I+

I−

I+

Twosheetedness of the Kerr space-time. The‘in’ and ‘out’ electromagnetic fields are posi-tioned on different sheets, they are aligned toKerr congruence and don’t interact with eachother.

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Electromagnetic field is to be aligned with the Kerrcongruence:

F µνkµ = 0.

Kerr -Schild form of metric gµν = ηµν − 2hkµkν, where

h = 2mr − |ψ|2

r2 + a2 cos2 θ

General stationary solutions, ψ(Y ) is arbitrary holo-morphic function of Y .

There is an infinite set of the exact solutions !, in whichfunction ψ(Y ) is holomorphic, but singular at the set ofpoints {Yi, i = 1,2, ...}, ψ(Y ) =

∑i

qiY (x)−Yi . These solu-

tions have a set of lightlike beams in the correspondingangular directions φi, θi, (singular pp-strings) along cor-responding rays of the Kerr congruence.

How act such beams on the BH horizon? Singularbeams lead to formation of the holes in the black holehorizon, which opens up the interior of the “black hole”to external space.

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Black holes with holes in the horizon A.B.,E.Elizalde, S.R.Hildebrandt and G.Magli, Phys. Rev.D74 (2006) 021502(R)

−25 −20 −15 −10 −5 0 5 10 15 20 25−25

−20

−15

−10

−5

0

5

10

15

20

25

M=10, a=9.5, q=0.03

g00

=0

event horizonsr+

r−

Singular ring

axial singularity

Singular beam forms a small hole in the horizon.

The boundaries of ergosphere (punctured surfaces) aredetermined by g00 = 0. The event horizons,r+ and r−,are two null surfaces. These surfaces turn out to bejoined by a tunnel, forming a simply connected surfaces.

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Wave excitations propagating in the direction Yi are de-scribed by means of the function

ψi(Y, τ) = qi(τ) exp{iωτ} 1

Y − Yi, (9)

where the extra dependence is on the retarded time τ.The right hand sides of the gravitational KS equationsare proportional to γ and γγ and are quite small for low-frequency aligned wave excitations, since the function γwill be of the order: γ ∼ ψ ∼ iωψ.

Electromagnetic field deforms horizon! Beams per-forate horizon!

H = {mr − ψ(Y )2/2}/(r2 + a2 cos2 θ).

Solutions tend in the low-frequency limit to exact sta-tionary Kerr-Schild solutions with singular beams.

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Beam-like basis functions.

Plane waves don’t exist on the Kerr’s twosheeted back-ground. Twistor-string conjecture of scattering by Nairand Witten (2003) leads to beam-like basis wave func-tions for ingoing vacuum photons which are supportedon twistor lines. Basis |in > - state is a wave functionon twistor space ZA ∈ Tw = CP 3. Amplitude of scatter-ing and stress-energy tensor < T µν >vac=

∑i T

µν(φi) aredetermined by the system of Kerr-Schild equations viathe vacuum in-fields γin related with the basis primaryholomorphic field φ(Y ) =

∑i φi(Y ) =

∑i

qiY−Yi . The both

fields contain beams performing horizon.

Algebraically special solutions ⇒ beam-like basisfunctions ⇒ Horizon is not irresistible with respectto outgoing beams!

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Twistorial structure of interaction.

The beams propagate along the null twistor lines of theKerr principal null congruence.

The case of quadratic Kerr’s generating functionsF(Y) is well studied. A.B. and G. Magli, Phys.Rev.D61044017 (2000).

F = AY 2 +BY + C,⇒two solutions of the equation F = 0 for the functionY (x) ⇒ twosheetedness of the Kerr geometry.

The functions F (Y ) of higher degrees are formed as aproduct of quadratic blocks corresponding to n differentparticles

F (Y ) =

n∏

i

Fi(Y ).

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The particles i and j are positioned on different Rie-mannian sheets of the function F (Y ) and interact witheach other only via a common twistor line of the i-thand j-th congruences, by forming a singular null stringconnecting these particles.

−15−10

−50

510

15

−15−10

−50

510

15

0

2

4

6

8

10

P1

P2

P3

P4

P5

Interaction of multiparticle solutions.

Rimannian multi-sheeted structure of space-time. Whatcan be observable? Only correlation functions!

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Conformal field theory as a starting pointfor 2D Quantum Gravity. A.B.Zamolodchikovand Al.B.Zamolodchikov (1995), J.Teshner (2003). Con-formal Bootstrap. Multipoint correlation functions asobservable for 2D Quantum Gravity.

4D Kerr-Schild Gravity ⇒ S2 ⊕ (1 + 1).

2D Conformal field theory on S2 = G(Y ) ⊗ G(Y ) ⇒Twistorial Complex Null Planes ⇒ a way to 4D Quan-tum Gravity!

Left and Right complex null planes.

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