1, 2). Denis Dobkowski-Ryłko, Jerzy Lewandowski, Tomasz Pawłowski (2018);

3). JL, Adam Szereszewski (2018); 4). DDR, Wojtek Kamiński, JL, AS (2018);

Uniwersytet Warszawski

Isolated horizons of the Petrov type D

Null surface stationary to the 2nd order

`

`µ`µ = 0

3d null surface in 4d spacetime

Ltgµ⌫ = 0

[Lt,rµ] = 0

LtRµ⌫↵� = 0

t tt= `

HM

}M Gµ⌫ + ⇤gµ⌫ = 0Assumption about :

Exists:

Such that

�2

Intrinsic geometry

`

gµ⌫ , rµ

µ, ⌫ � M

gab, ra

- spacetime geometry and derivative

- rotation potential- surface gravity

3

Ha, b�H

- zeroth law of thermodynamics

ra`b = !a`

b

` = !a`a

ra` = L`!a

Gµ⌫ + ⇤gµ⌫ = 0)determine

= 0

` 6= 0Assumption:

(gab,!a) Rµ⌫↵�, rµ, ra determine gµ⌫

Data on a 2d slice

gAB

`

4

!A

g0AB = gAB

!0A

g0AB !0A = !A + f,A

(!A, gAB) (!a, gab)! Rµ⌫↵�, rµ, ! gµ⌫

S

S0<latexit sha1_base64="xXLDwKb4xzU5J59GpK/dmCQFyRU=">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</latexit><latexit sha1_base64="xXLDwKb4xzU5J59GpK/dmCQFyRU=">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</latexit><latexit sha1_base64="xXLDwKb4xzU5J59GpK/dmCQFyRU=">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</latexit><latexit sha1_base64="91VoiCyvOOq/81xdFLppbHkqNh8=">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</latexit>

Gaussian curvature:

Scalar invariants:

Data on a 2d slice: summary

2d-surface

(gAB ,!A)Endowed with

S

rA

modulo

combined:Od! =:

K

dArea

!0A = !A + f,A

- the corresponding derivative rAgBC = 0 r[ArB]f = 0

:= �1

2(K + iO)

5

The Weyl tensor in Newman-Penrose formalism- Spacetime Weyl tensor in the null frame formalism may be

expressed by the following complex valued N-P components:

- Four components are constant along the null generators of H:

- Additionally we assume:

- The components and vanish due to vanishing of the expansion and shear of :

- is related to the complex invariant:

0 = C4141, 1 = C4341 2 = C4123,

3 = C3432, 4 = C3232<latexit sha1_base64="1i4wYtNU2w5KMgGP2cfOa64jfKg=">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</latexit><latexit sha1_base64="1i4wYtNU2w5KMgGP2cfOa64jfKg=">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</latexit><latexit sha1_base64="1i4wYtNU2w5KMgGP2cfOa64jfKg=">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</latexit><latexit sha1_base64="rmlYckii/SFfcXitgvXrEQt5Ofg=">AAADXHicjVFba9RAFD7ZVK1bq2sFX3wZXBQf6jKTLHb7IBT74uMKbltoypKk0zY0NyaTQln6O/xp4ovP+iMEvznNeqXohCRnvss5c+YkdZ41VspPXs9fuXX7zurd/tq99fsPBg839pqqNamepVVemYMkbnSelXpmM5vrg9rouEhyvZ+c7zp+/0KbJqvK9/ay1kdFfFpmJ1kaW0DzwYdo2mRzGYnnryOxO1+M1VhdbYro5+N4FQnQzIfg/2ADZtkchM5cVmVbJNqIKBJ9loQ/CoTjMPi7wPi6AAsCCMR8MJQjKaVSSrhAbb2SCLa3J4GaCOUorCF1a1oNPlJEx1RRSi0VpKkkizinmBo8h6RIUg3siBbADKKMeU1X1Ie3hUpDEQM9x/cUu8MOLbF3ORt2p6iS4zVwCnoGTwWdQeyqCeZbzuzQm3IvOKc72yX+SZerAGrpDOi/fEvl//pcL5ZOaMI9ZOipZsR1l3ZZWr4Vd3LxS1cWGWpgLj4GbxCn7Fzes2BPw727u42Z/8JKh7p92mlb+upOiQEvpyhuDvaCkZIj9S4Y7rzpRr1KT+gpvcA8t2iH3tKUZsj9zRt6m97L3md/xV/z16+lPa/zPKLflv/4O7NNuIM=</latexit>

D I = 0, I = 0, 1, 2, 3<latexit sha1_base64="uhjrPF7egDIdaNDDZs8cJsrkbic=">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</latexit><latexit sha1_base64="uhjrPF7egDIdaNDDZs8cJsrkbic=">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</latexit><latexit sha1_base64="uhjrPF7egDIdaNDDZs8cJsrkbic=">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</latexit><latexit sha1_base64="xSYGcLBqYJcl9aPC4m+V5h9pOGA=">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</latexit>

D 4 = 0<latexit sha1_base64="Ax6YwUpXMg8ifdHsa4OiFKm3HAc=">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</latexit><latexit sha1_base64="Ax6YwUpXMg8ifdHsa4OiFKm3HAc=">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</latexit><latexit sha1_base64="Ax6YwUpXMg8ifdHsa4OiFKm3HAc=">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</latexit><latexit sha1_base64="VlMQYc3oac/un4tgsrFOGXJdPvE=">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</latexit>

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2<latexit sha1_base64="4cUfPs2sgFQBzFh0oIyKGk8m5gc=">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</latexit><latexit sha1_base64="4cUfPs2sgFQBzFh0oIyKGk8m5gc=">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</latexit><latexit sha1_base64="4cUfPs2sgFQBzFh0oIyKGk8m5gc=">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</latexit><latexit sha1_base64="Bpyzgi7yC/IHHcKKRMAdy8MowjM=">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</latexit>

2 = +⇤

6<latexit sha1_base64="Is/UVzVgv1m51v7dNERt8KX6Xlo=">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</latexit><latexit sha1_base64="Is/UVzVgv1m51v7dNERt8KX6Xlo=">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</latexit><latexit sha1_base64="Is/UVzVgv1m51v7dNERt8KX6Xlo=">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</latexit><latexit sha1_base64="C0HY6VyshcMCJWecsxh5+GcZxqI=">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</latexit>

6

0 = 1 = 0<latexit sha1_base64="Jdm0pLgHrdvjYbA/xUHtGCw7t4w=">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</latexit><latexit sha1_base64="Jdm0pLgHrdvjYbA/xUHtGCw7t4w=">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</latexit><latexit sha1_base64="Jdm0pLgHrdvjYbA/xUHtGCw7t4w=">AAAC2HicjVHLSuRAFD1GnfE5045LN4WN4KqpEhnbhSAzG5c9YKv4oEliqYXpJCSVgUYEd4Nbf8CtfpH4B85feKpMgy5EKyQ5de49p+reG+WJKa2UjyPB6Nj4l68Tk1PTM7PfvjfmfuyUWVXEuhtnSVbsRWGpE5PqrjU20Xt5ocN+lOjd6Py3i+/+1UVpsnTbDnJ91A9PU3Ni4tCS6jXmDzul6UmxITxQBLLXaMqWlFIpJRxQaz8lwfp6e0W1hXIhribq1ckaDzjEMTLEqNCHRgpLnCBEyecAChI5uSNckCuIjI9rXGKK2opZmhkh2XN+T7k7qNmUe+dZenXMUxK+BZUCS9RkzCuI3WnCxyvv7Nj3vC+8p7vbgP+o9uqTtTgj+5FumPlZnavF4gRtX4NhTblnXHVx7VL5rribi1dVWTrk5Bw+Zrwgjr1y2GfhNaWv3fU29PEnn+lYt4/r3Ar/3S054OEUxftgZ6WlZEv9WW1u/qpHPYEFLGKZ81zDJrbQQZfeA9ziDvfBfnAV/AuuX1KDkVozjzcruHkG7lKVeQ==</latexit><latexit sha1_base64="KlcNHYY3EqDhRaEaMg8P0Vqc00s=">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</latexit>

Possible Petrov types The spacetime Weyl tensor at is determined by the data

Theorem:The possible Petrov types of H are: I, II, D, III, N, O

+⇤

6= 0

+⇤

66= 0

, O

) generically II, unless…

K =⇤

3d! = 0

wherein:

,

H

(S, gAB ,!A)

7

The Petrov type D equation

gAB = mAmB + mAmB

At H the spacetime Weyl tensor is of the Petrov type Diff the following two conditions are satisfied:

Theorem:

+⇤

66= 0

mAmBrArB ( +⇤

6)�

13 = 0

We use a null 2-frame

AreaBC = i(mBmC � mCmB)d

8

In local conformally flat coordinates

gABdxAdxB =

2

P 2dzdz

@z(P2@z( +

⇤

6)�

13 ) = 0

mA@A = P@z

The Petrov type D equation:

9

The Petrov type D equation as integrability condition for the near horizon geometry equation

Suppose (gAB ,!A) satisfy the NHG equation, namely

r(A!B) + !A!B +1

2(⇤�K)gAB = 0

Theorem:

Then they also satisfy the Petrov type D equation:

mAmBrArB( +1

6⇤)�

13 = 0

10

Non-twisting of the second principal null direction of the Weyl tensor

gAB

`

!A

Suppose (gAB ,!A) satisfy the NHG equation, namely r(A!B) + !A!B +

1

2(⇤�K)gAB = 0

Theorem:

Then the null vector

Sn0

n0

orthogonal to the corresponding slice S

is a double principal direction of the spacetime Weyl

tensor at H.In that case there exists another

symmetry t’ that is extremal 11

No-hair theorem for axisymmetric solutions to the Petrov type D equation.

The family of axisymmetric solutions of the Petrov type D equation with (or without) cosmological constant defined on a topological sphere can be parametrized by two numbers (A,J): the area and angular momentum, respectively. They can take the following values:

Theorem

Lewandowski, Pawlowski (2003) for ⇤ = 0

for ⇤ > 0 : J 2✓�1,1

◆for A 2

✓0,

12⇡

⇤

◆and |J | 2

0,

A

16⇡

r⇤ A

12⇡� 1

◆for A 2

✓12⇡

⇤,1

◆

<latexit sha1_base64="XYX4DsNZvg1xKa9TxWBh927i4/E=">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</latexit><latexit sha1_base64="XYX4DsNZvg1xKa9TxWBh927i4/E=">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</latexit><latexit sha1_base64="XYX4DsNZvg1xKa9TxWBh927i4/E=">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</latexit><latexit sha1_base64="wnxOIlTeV+k4tJhLuK6I/P85yu4=">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</latexit>

for ⇤ < 0 : J 2✓�1,1

◆and A 2

✓0,1

◆

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12

Embeddability of the axisymmetric solutions

Every solution defines a type D isolated horizon whose intrinsic geometry coincides with the intrinsic geometry of a non-extremal Killing horizon contained in one of the following spacetimes:

1). Kerr - (anti) de Sitter;

2). Schwarzschild - (anti) de Sitter ;

3). Near horizon limit spacetime near an extremal horizon contained either in the K(a)dS or S(a)dS spacetime;

The Petrov type D equation on of genus > 0

gABdxAdxB =

2

P 2dzdz

@z(P2@z( +

⇤

6)�

13 ) = 0

mA@A = P@z

The Petrov type D equation:

)

) F (z)@z

is a globally defined holomorphic vector field

S

) F (z) = const

F (z) = 0

if genus =1

if genus >1

14

@z� +

⇤

6

�� 13 =

F (z)

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The Petrov type D equation on of genus > 0

Theorem. Suppose is a compact 2-surface of genus >0. The only solutions to the Petrov type D equation with a cosmological constant are such that

S

⇤ (g,!)

d! = 0 K = const 6= ⇤

3and

Remark: There are no rotating solutions

15

S

A bifurcated Petrov type D horizon: data

S

HH

0

g0AB = gAB

0 =

16

!0A = �!A

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M. J. Cole, I. Racz, J. A. Valiente Kroon, 2018

J. Lewandowski, A. Szereszewski, 2018

A bifurcated Petrov type D horizon: equations

The Petrov

mAmBrArB( +1

6⇤)�

13 = 0

type D equations

and for H’:

mAmBrArB( +1

6⇤)�

13 = 0

for H:

hold simultaneously on S

17

In local conformally flat coordinates

gABdxAdxB =

2

P 2dzdz

@z(P2@z( +

⇤

6)�

13 ) = 0

mA@A = P@z

@z(P2@z( +

⇤

6)�

13 ) = 0 @z( +

⇤

6)�

13 =

G(z)

P 2

@z( +⇤

6)�

13 =

F (z)

P 2

@z

✓F (z)

P 2

◆@z

✓G(z)

P 2

◆=

)

)

)

) L�gAB = 0 � := F (z)@z � G(z)@z

L� d! = 0)

18

Suppose (gAB ,!A) defined on satisfy the Petrov type D

Theorem:

The axial symmetry without the rigidity theorem

mAmBrArB( +1

6⇤)�

13 = 0

and the conjugate onemAmBrArB( +

1

6⇤)�

13 = 0

Then, there is a vector field at such that

S

equation

� S

L�gAB = 0 L�d! = 0and

�A = Re/Im

✓dAreaAB@A( +

⇤

6)�

13

◆

19

Summary

• The type D equation:

• Non-twisting of the second double principal vector if:

• All the axisymmetric solutions of the type D eq. on topological sphere parametrized by (A, J);

• All solutions on genus>0 derived (non-rotating);

• The extra (axial) symmetry in the case of bifurcated horizon;

• Open problems: existence of non-axisymmetric solutions on topological sphere

r(A!B) + !A!B +1

2(⇤�K)gAB = 0

mAmBrArB

✓�

1

2K �

1

2iO +

⇤

6

◆� 13

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Thank You