Extremal Rotating Black Holes in the Near-Horizon Limit: Phase Space and Symmetry Algebra

We construct the NHEG phase space , the classical phase space of Near-Horizon Extremal Geometries with fixed angular momenta and entropy, and with the largest symmetry algebra. We focus on vacuum solutions to d dimensional Einstein gravity. Each element in the phase space is a geometry with SL(2,R)×U(1)d−3 isometries which has vanishing SL(2,R) and constant U(1) charges. We construct an on-shell vanishing symplectic structure, which leads to an infinite set of symplectic symmetries. In four spacetime dimensions, the phase space is unique and the symmetry algebra consists of the familiar Virasoro algebra, while in d>4 dimensions the symmetry algebra, the NHEG algebra , contains infinitely many Virasoro subalgebras. The nontrivial central term of the algebra is proportional to the black hole entropy. The conserved charges are given by the Fourier decomposition of a Liouville-type stress-tensor which depends upon a single periodic function of d−3 angular variables associated with the U(1) isometries. This phase space and in particular its symmetries can serve as a basis for a semiclassical description of extremal rotating black hole microstates.

Note:

Published in PLB, 5 pages

symmetry: algebra
black hole: rotation
symmetry: symplectic
algebra: Virasoro
black hole: entropy
SL(2,R)
angular momentum
semiclassical
gravitation
microstate

References(37)

Figures(0)

[1]

The Four laws of black hole mechanics

- Commun.Math.Phys. 31 (1973) 161-170
•
DOI:
- 10.1007/BF01645742

[1]

Black holes and entropy

Jacob D. Bekenstein
(
- Princeton U. and
- Texas U.
)

- Phys.Rev.D 7 (1973) 2333-2346
•
DOI:
- 10.1103/PhysRevD.7.2333

[1]

Particle Creation by Black Holes

S.W. Hawking
(
- Cambridge U., DAMTP and
- Caltech
)

- Commun.Math.Phys. 43 (1975) 199-220,
- Commun.Math.Phys. 46 (1976) 206 (erratum),
•
DOI:
- 10.1007/BF02345020

[2]

665 • S. L. Braunstein, S. Pirandola and K. Zyczkowski 101301 • A. Almheiri, D. Marolf, J. Polchinski and J. Sully

L. Susskind

- Nature Phys. 2 (2006) 10

[3]

Highlights and Discoveries from the Chandra X-ray Observatory

- Rept.Prog.Phys. 77 (2014) 066902
•
e-Print:
- 1405.7847
•
DOI:
- 10.1088/0034-4885/77/6/066902

[4]

The Spin of the Near-Extreme Kerr Black Hole GRS 1915+105

et al.

- Astrophys.J. 652 (2006) 518-539
•
e-Print:
- astro-ph/0606076
•
DOI:
- 10.1086/508457

[4]

Confirmation Via the Continuum-Fitting Method that the Spin of the Black Hole in Cygnus X-1 is Extreme

et al.

- Astrophys.J. 790 (2014) 1, 29
•
e-Print:
- 1308.4760
•
DOI:
- 10.1088/0004-637X/790/1/29

[5]

The Extreme Kerr throat geometry: A Vacuum analog of AdS(2) x S**2

- Phys.Rev.D 60 (1999) 104030
•
e-Print:
- hep-th/9905099
•
DOI:
- 10.1103/PhysRevD.60.104030

[6]

Near-horizon symmetries of extremal black holes

Hari K. Kunduri
(
- Nottingham U.
)
,
James Lucietti
(
- Durham U., Dept. of Math.
)
,
Harvey S. Reall
(
- Nottingham U.
)

- Class.Quant.Grav. 24 (2007) 4169-4190
•
e-Print:
- 0705.4214
•
DOI:
- 10.1088/0264-9381/24/16/012

[6]

All vacuum near horizon geometries in arbitrary dimensions

Stefan Hollands
(
- Cardiff U.
)
,
Akihiro Ishibashi
(
- KEK, Tsukuba
)

- Annales Henri Poincare 10 (2010) 1537-1557
•
e-Print:
- 0909.3462
•
DOI:
- 10.1007/s00023-010-0022-y

[7]

Classification of near-horizon geometries of extremal black holes

- Living Rev.Rel. 16 (2013) 8
•
e-Print:
- 1306.2517
•
DOI:
- 10.12942/lrr-2013-8

[8]

Some properties of Noether charge and a proposal for dynamical black hole entropy

Vivek Iyer
(
- Chicago U., EFI and
- Chicago U.
)
,
Robert M. Wald
(
- Chicago U., EFI and
- Chicago U.
)

- Phys.Rev.D 50 (1994) 846-864
•
e-Print:
- gr-qc/9403028
•
DOI:
- 10.1103/PhysRevD.50.846

[9]

NHEG Mechanics: Laws of Near Horizon Extremal Geometry (Thermo)Dynamics

K. Hajian
(
- IPM, Tehran and
- Sharif U. of Tech.
)
,
A. Seraj
(
- IPM, Tehran
)
,
M.M. Sheikh-Jabbari
(
- IPM, Tehran
)

- JHEP 03 (2014) 014
•
e-Print:
- 1310.3727
•
DOI:
- 10.1007/JHEP03(2014)014

[10]

Rotating attractors

Dumitru Astefanesei
(
- Harish-Chandra Res. Inst.
)
,
Kevin Goldstein
(
- Tata Inst.
)
,
Rudra P. Jena
(
- Tata Inst.
)
,
Ashoke Sen
(
- Harish-Chandra Res. Inst.
)
,
Sandip P. Trivedi
(
- Tata Inst.
)

- JHEP 10 (2006) 058
•
e-Print:
- hep-th/0606244
•
DOI:
- 10.1088/1126-6708/2006/10/058

[11]

Wiggling Throat of Extremal Black Holes

G. Compère
(
- Brussels U. and
- Intl. Solvay Inst., Brussels
)
,
K. Hajian
(
- IPM, Tehran and
- Sharif U. of Tech.
)
,
A. Seraj
(
- Brussels U. and
- IPM, Tehran and
- Intl. Solvay Inst., Brussels
)
,
M.M. Sheikh-Jabbari
(
- IPM, Tehran
)

- JHEP 10 (2015) 093,
- JHEP 10 (2015) 093
•
e-Print:
- 1506.07181
•
DOI:
- 10.1007/JHEP10(2015)093

[12]

Microscopic origin of the Bekenstein-Hawking entropy

Andrew Strominger
(
- UC, Santa Barbara
)
,
Cumrun Vafa
(
- Harvard U.
)

- Phys.Lett.B 379 (1996) 99-104
•
e-Print:
- hep-th/9601029
•
DOI:
- 10.1016/0370-2693(96)00345-0

[13]

Extremal black holes and elementary string states

Ashoke Sen
(
- ICTP, Trieste and
- Tata Inst.
)

- Mod.Phys.Lett.A 10 (1995) 2081-2094
•
e-Print:
- hep-th/9504147
•
DOI:
- 10.1142/S0217732395002234

[14]

The Large $N$ limit of superconformal field theories and supergravity

Juan Martin Maldacena
(
- Harvard U.
)

- Int.J.Theor.Phys. 38 (1999) 1113-1133 (reprint),
- Adv.Theor.Math.Phys. 2 (1998) 231-252
•
e-Print:
- hep-th/9711200
•
DOI:
- 10.4310/ATMP.1998.v2.n2.a1