Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
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=">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</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=">AAACznicjVHLSsNAFD2Nr/quunQzWARXZaYU2y6Eoi5cVrBVqCJJOurQNAnJpFBE3PoDbvWzxD/Qv/DOmIIuik5Icubcc+7MvdeLA5Vqzt8Lzszs3PxCcXFpeWV1bb20sdlNoyzxZcePgii58NxUBiqUHa10IC/iRLpDL5Dn3uDIxM9HMklVFJ7pcSyvhu5tqG6U72qieseX7VRd19gB49elMq9wzoUQzABR3+cEms1GVTSYMCFaZeSrHZXecIk+IvjIMIRECE04gIuUnh4EOGLirnBPXEJI2bjEA5bIm5FKksIldkDfW9r1cjakvcmZWrdPpwT0JuRk2CVPRLqEsDmN2XhmMxt2Wu57m9PcbUx/L881JFbjjti/fBPlf32mFo0bNGwNimqKLWOq8/Msme2KuTn7UZWmDDFxBvcpnhD2rXPSZ2Y9qa3d9Na18Q+rNKzZ+7k2w6e5JQ14MkU2HXSrFcEr4rRWbh3moy5iGzvYo3nW0cIJ2ujYjj/jBa9O2xk5D87jt9Qp5J4t/FrO0xeR8JKr</latexit><latexit sha1_base64="Ax6YwUpXMg8ifdHsa4OiFKm3HAc=">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</latexit><latexit sha1_base64="VlMQYc3oac/un4tgsrFOGXJdPvE=">AAACznicjVHLSsNAFD3GV62vqks3g0VwVWaK2HYhFHXhsoKthbZIEsc6NE1CMhFEilt/wK1+lvgH+hfeGVPQheiEJGfOPefO3Hu9OFCp5vxtxpmdm19YLCwVl1dW19ZLG5udNMoSX7b9KIiSruemMlChbGulA9mNE+mOvUBeeKNjE7+4lUmqovBc38VyMHaHobpWvquJ6p30W6m63GeHjF+WyrzCORdCMANE7YATaDTqVVFnwoRolZGvVlR6RR9XiOAjwxgSITThAC5SenoQ4IiJG+CeuISQsnGJCYrkzUglSeESO6LvkHa9nA1pb3Km1u3TKQG9CTkZdskTkS4hbE5jNp7ZzIb9Lfe9zWnudkd/L881Jlbjhti/fFPlf32mFo1r1G0NimqKLWOq8/Msme2KuTn7VpWmDDFxBl9RPCHsW+e0z8x6Ulu76a1r4+9WaViz93Nthg9zSxrwdIrsd9CpVgSviLNquXmUj7qAbexgj+ZZQxOnaKFtO/6EZ7w4LefWmTgPX1JnJvds4cdyHj8BkVCSqQ==</latexit>
0<latexit sha1_base64="oqqMJ2fBl5pKArcrHneV/gYv6q0=">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</latexit><latexit sha1_base64="oqqMJ2fBl5pKArcrHneV/gYv6q0=">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</latexit><latexit sha1_base64="oqqMJ2fBl5pKArcrHneV/gYv6q0=">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</latexit><latexit sha1_base64="oi83GgBSi8cgVIO9xBqFh5kRsWg=">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</latexit>
1<latexit sha1_base64="WhBwRGYJ0DfFnj/8xQaisLIspk0=">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</latexit><latexit sha1_base64="WhBwRGYJ0DfFnj/8xQaisLIspk0=">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</latexit><latexit sha1_base64="WhBwRGYJ0DfFnj/8xQaisLIspk0=">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</latexit><latexit sha1_base64="9D/oNYSRDJQMRwb6aHGhlYGSNuU=">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</latexit>
<latexit sha1_base64="CTYeN8XBBAFs6Bo2EJYKb0r4Ch4=">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</latexit><latexit sha1_base64="CTYeN8XBBAFs6Bo2EJYKb0r4Ch4=">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</latexit><latexit sha1_base64="CTYeN8XBBAFs6Bo2EJYKb0r4Ch4=">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</latexit><latexit sha1_base64="I/0viU9SgoJV60RLBSdU53sDeW4=">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</latexit>
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=">AAAC2HicjVHLSuRAFD1GnfE5045LN4WN4KqpEhnbhSAzG5c9YKv4oEliqYXpJCSVgUYEd4Nbf8CtfpH4B85feKpMgy5EKyQ5de49p+reG+WJKa2UjyPB6Nj4l68Tk1PTM7PfvjfmfuyUWVXEuhtnSVbsRWGpE5PqrjU20Xt5ocN+lOjd6Py3i+/+1UVpsnTbDnJ91A9PU3Ni4tCS6jXmDzul6UmxITxQBLLXaMqWlFIpJRxQaz8lwfp6e0W1hXIhribq1ckaDzjEMTLEqNCHRgpLnCBEyecAChI5uSNckCuIjI9rXGKK2opZmhkh2XN+T7k7qNmUe+dZenXMUxK+BZUCS9RkzCuI3WnCxyvv7Nj3vC+8p7vbgP+o9uqTtTgj+5FumPlZnavF4gRtX4NhTblnXHVx7VL5rribi1dVWTrk5Bw+Zrwgjr1y2GfhNaWv3fU29PEnn+lYt4/r3Ar/3S054OEUxftgZ6WlZEv9WW1u/qpHPYEFLGKZ81zDJrbQQZfeA9ziDvfBfnAV/AuuX1KDkVozjzcruHkG7lKVeQ==</latexit><latexit sha1_base64="Jdm0pLgHrdvjYbA/xUHtGCw7t4w=">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</latexit><latexit sha1_base64="Jdm0pLgHrdvjYbA/xUHtGCw7t4w=">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</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
◆
<latexit sha1_base64="h0NA9NF4j8RqKDVwkkF0JqqmasU=">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</latexit><latexit sha1_base64="h0NA9NF4j8RqKDVwkkF0JqqmasU=">AAADJ3icjVHNa9RAHH2NX2396GqPvQwuQgu6JFJQpIdaLyIeWnDbQlPKJDu7Ds0mYTKRLkv/IP8Tb978OHrTi9f22DeTLKhFdEIyb97vvZf5zSRlpisbhl/ngitXr12/Mb+wePPW7TtLnbv3dquiNqnqp0VWmP1EVirTuepbbTO1Xxolx0mm9pLjF66+906ZShf5Gzsp1eFYjnI91Km0pI46SWzViZ0OCyNORfyaxoHcCJ+JVyLWuYgTPRqtPiIc2slD0cyeXBONUch8QOdzV2vUIpwJG/vaUacb9kI/xGUQtaCLdmwXnc+IMUCBFDXGUMhhiTNIVHwOECFESe4QU3KGSPu6wikW6a2pUlRIssf8jrg6aNmca5dZeXfKv2R8DZ0CD+gpqDPE7m/C12uf7Ni/ZU99ptvbhHPSZo3JWrwl+y/fTPm/PteLxRBPfQ+aPZWecd2lbUrtT8XtXPzSlWVCSc7hAeuGOPXO2TkL76l87+5spa9/90rHunXaamv8cLvkBUd/XudlsPu4F4W9aGe9u7nVXvU8VnAfq7zPJ9jES2yjz+wP+IkznAfvg4/Bp+BLIw3mWs8yfhvBtwsLk7Rd</latexit><latexit sha1_base64="h0NA9NF4j8RqKDVwkkF0JqqmasU=">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</latexit><latexit sha1_base64="4eUv+MQPLSNbn7B2Wf5YA9Eiezk=">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</latexit>
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)
P 2<latexit sha1_base64="p28K2bItvMHRN0dZnr3byJ55+Xo=">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</latexit><latexit sha1_base64="p28K2bItvMHRN0dZnr3byJ55+Xo=">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</latexit><latexit sha1_base64="p28K2bItvMHRN0dZnr3byJ55+Xo=">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</latexit><latexit sha1_base64="cv0A4Vv/iK807B2XD8lwQbz3n6g=">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</latexit>
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
<latexit sha1_base64="1SFQWmNTRAxy5XRpAIdjtNvduqM=">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</latexit><latexit sha1_base64="1SFQWmNTRAxy5XRpAIdjtNvduqM=">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</latexit><latexit sha1_base64="1SFQWmNTRAxy5XRpAIdjtNvduqM=">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</latexit><latexit sha1_base64="UxhJh2eaJ3sIgUusWec1iPBtx9Q=">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</latexit>
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
= 0<latexit sha1_base64="gjYFRWHU/0jiirx6lDlmTggC32Y=">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</latexit><latexit sha1_base64="gjYFRWHU/0jiirx6lDlmTggC32Y=">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</latexit><latexit sha1_base64="gjYFRWHU/0jiirx6lDlmTggC32Y=">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</latexit><latexit sha1_base64="oH2OiXzUSY/Jplao2QkR1v6URwA=">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</latexit>
Thank You