Abstract
Expanding around null hypersurfaces, such as generic Kerr black hole horizons, using co-rotating Kruskal-Israel-like coordinates we study the associated surface charges, their symmetries and the corresponding phase space within Einstein gravity. Our surface charges are not integrable in general. Their integrable part generates an algebra including superrotations and a BMS3-type algebra that we dub “T-Witt algebra”. The non-integrable part accounts for the flux passing through the null hypersurface. We put our results in the context of earlier constructions of near horizon symmetries, soft hair and of the program to semi-classically identify Kerr black hole microstates.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett.11 (1963) 237 [INSPIRE].
A. Einstein, Die Feldgleichungen der Gravitation, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1915 (1915) 844.
L. Donnay, G. Giribet, H.A. González and M. Pino, Supertranslations and Superrotations at the Black Hole Horizon, Phys. Rev. Lett.116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].
L. Donnay, G. Giribet, H.A. González and M. Pino, Extended Symmetries at the Black Hole Horizon, JHEP09 (2016) 100 [arXiv:1607.05703] [INSPIRE].
V. Chandrasekaran, É.É. Flanagan and K. Prabhu, Symmetries and charges of general relativity at null boundaries, JHEP11 (2018) 125 [arXiv:1807.11499] [INSPIRE].
S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, Black Hole Entropy and Soft Hair, JHEP12 (2018) 098 [arXiv:1810.01847] [INSPIRE].
D. Grumiller, A. Pérez, M.M. Sheikh-Jabbari, R. Troncoso and C. Zwikel, Spacetime structure near generic horizons and soft hair, Phys. Rev. Lett.124 (2020) 041601 [arXiv:1908.09833] [INSPIRE].
M.D. Kruskal, Maximal extension of Schwarzschild metric, Phys. Rev.119 (1960) 1743 [INSPIRE].
W. Israel, New Interpretation of the Extended Schwarzschild Manifold, Phys. Rev.143 (1966) 1016 [INSPIRE].
J.M. Bardeen and G.T. Horowitz, The Extreme Kerr throat geometry: A Vacuum analog of AdS2× S2 , Phys. Rev.D 60 (1999) 104030 [hep-th/9905099] [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett.105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys.88 (1974) 286 [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys.104 (1986) 207 [INSPIRE].
M. Henneaux and C. Troessaert, BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach, JHEP03 (2018) 147 [arXiv:1801.03718] [INSPIRE].
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys.31 (1990) 725 [INSPIRE].
V. Iyer and R.M. Wald, Some properties of N¨other charge and a proposal for dynamical black hole entropy, Phys. Rev.D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys.B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
G. Barnich and G. Compère, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav.24 (2007) F15 [gr-qc/0610130] [INSPIRE].
G. Compère and A. Fiorucci, Advanced Lectures on General Relativity, arXiv:1801.07064 [INSPIRE].
A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev.D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
A. Farahmand Parsa, H.R. Safari and M.M. Sheikh-Jabbari, On Rigidity of 3d Asymptotic Symmetry Algebras, JHEP03 (2019) 143 [arXiv:1809.08209] [INSPIRE].
G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 [arXiv:1102.4632] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
G. Compère, P. Mao, A. Seraj and M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons, JHEP01 (2016) 080 [arXiv:1511.06079] [INSPIRE].
C. Bunster, A. Gomberoff and A. Pérez, Regge-Teitelboim analysis of the symmetries of electromagnetic and gravitational fields on asymptotically null spacelike surfaces, (2018), arXiv:1805.03728 [INSPIRE].
V. Chandrasekaran and K. Prabhu, Symmetries, charges and conservation laws at causal diamonds in general relativity, JHEP10 (2019) 229 [arXiv:1908.00017] [INSPIRE].
M. Campiglia and J. Peraza, Generalized BMS charge algebra, arXiv:2002.06691 [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, arXiv:1906.08616 [INSPIRE].
M. Geiller and P. Jai-akson, Extended actions, dynamics of edge modes and entanglement entropy, arXiv:1912.06025 [INSPIRE].
G. Barnich, Black hole entropy from nonproper gauge degrees of freedom: The charged vacuum capacitor, Phys. Rev.D 99 (2019) 026007 [arXiv:1806.00549] [INSPIRE].
R.M. Wald and A. Zoupas, A General definition of ’conserved quantities’ in general relativity and other theories of gravity, Phys. Rev.D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys.B 335 (1990) 138 [INSPIRE].
G. ’t Hooft, The Black hole horizon as a quantum surface, Phys. ScriptaT 36 (1991) 247 [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole complementarity, Phys. Rev.D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
S.A. Hayward, General laws of black hole dynamics, Phys. Rev.D 49 (1994) 6467 [INSPIRE].
S. Carlip, The Statistical mechanics of the (2 + 1)-dimensional black hole, Phys. Rev.D 51 (1995) 632 [gr-qc/9409052] [INSPIRE].
A.P. Balachandran, L. Chandar and A. Momen, Edge states in gravity and black hole physics, Nucl. Phys.B 461 (1996) 581 [gr-qc/9412019] [INSPIRE].
S. Carlip, Statistical mechanics and black hole entropy, gr-qc/9509024 [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP02 (1998) 009 [hep-th/9712251] [INSPIRE].
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett.80 (1998) 904 [gr-qc/9710007] [INSPIRE].
S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett.82 (1999) 2828 [hep-th/9812013] [INSPIRE].
M. Hotta, K. Sasaki and T. Sasaki, Diffeomorphism on horizon as an asymptotic isometry of Schwarzschild black hole, Class. Quant. Grav.18 (2001) 1823 [gr-qc/0011043] [INSPIRE].
A. Ashtekar et al., Isolated horizons and their applications, Phys. Rev. Lett.85 (2000) 3564 [gr-qc/0006006] [INSPIRE].
A. Ashtekar and B. Krishnan, Dynamical horizons: Energy, angular momentum, fluxes and balance laws, Phys. Rev. Lett.89 (2002) 261101 [gr-qc/0207080] [INSPIRE].
G. ’t Hooft, The Black hole horizon as a dynamical system, Int. J. Mod. Phys.D 15 (2006) 1587 [gr-qc/0606026] [INSPIRE].
B.R. Majhi and T. Padmanabhan, Noether current from the surface term of gravitational action, Virasoro algebra and horizon entropy, Phys. Rev.D 86 (2012) 101501 [arXiv:1204.1422] [INSPIRE].
R.F. Penna, BMS invariance and the membrane paradigm, JHEP03 (2016) 023 [arXiv:1508.06577] [INSPIRE].
H. Afshar, S. Detournay, D. Grumiller and B. Oblak, Near-Horizon Geometry and Warped Conformal Symmetry, JHEP03 (2016) 187 [arXiv:1512.08233] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett.116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
G. ’t Hooft, Black hole unitarity and antipodal entanglement, Found. Phys.46 (2016) 1185 [arXiv:1601.03447] [INSPIRE].
A. Averin, G. Dvali, C. Gomez and D. Lüst, Gravitational Black Hole Hair from Event Horizon Supertranslations, JHEP06 (2016) 088 [arXiv:1601.03725] [INSPIRE].
G. Compère and J. Long, Classical static final state of collapse with supertranslation memory, Class. Quant. Grav.33 (2016) 195001 [arXiv:1602.05197] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev.D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
C. Eling and Y. Oz, On the Membrane Paradigm and Spontaneous Breaking of Horizon BMS Symmetries, JHEP07 (2016) 065 [arXiv:1605.00183] [INSPIRE].
H. Afshar, D. Grumiller and M.M. Sheikh-Jabbari, Near horizon soft hair as microstates of three dimensional black holes, Phys. Rev.D 96 (2017) 084032 [arXiv:1607.00009] [INSPIRE].
M. Mirbabayi and M. Porrati, Dressed Hard States and Black Hole Soft Hair, Phys. Rev. Lett.117 (2016) 211301 [arXiv:1607.03120] [INSPIRE].
F. Hopfmüller and L. Freidel, Gravity Degrees of Freedom on a Null Surface, Phys. Rev.D 95 (2017) 104006 [arXiv:1611.03096] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP05 (2017) 161 [arXiv:1611.09175] [INSPIRE].
H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev.D 95 (2017) 106005 [arXiv:1611.09783] [INSPIRE].
W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav.34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].
R. Bousso and M. Porrati, Soft Hair as a Soft Wig, Class. Quant. Grav.34 (2017) 204001 [arXiv:1706.00436] [INSPIRE].
A. Strominger, Black Hole Information Revisited, in Jacob Bekenstein. The Conservative Revolutionary, World Scientific, New York U.S.A. (2019), pg. 109 [arXiv:1706.07143] [INSPIRE].
E.T. Akhmedov and M. Godazgar, Symmetries at the black hole horizon, Phys. Rev.D 96 (2017) 104025 [arXiv:1707.05517] [INSPIRE].
D. Lüst, Supertranslations and Holography near the Horizon of Schwarzschild Black Holes, Fortsch. Phys.66 (2018) 1800001 [arXiv:1711.04582] [INSPIRE].
A. Blommaert, T.G. Mertens, H. Verschelde and V.I. Zakharov, Edge State Quantization: Vector Fields in Rindler, JHEP08 (2018) 196 [arXiv:1801.09910] [INSPIRE].
G. ’t Hooft, What happens in a black hole when a particle meets its antipode, arXiv:1804.05744 [INSPIRE].
W. Donnelly and S.B. Giddings, Gravitational splitting at first order: Quantum information localization in gravity, Phys. Rev.D 98 (2018) 086006 [arXiv:1805.11095] [INSPIRE].
L. Donnay, G. Giribet, H.A. González and A. Puhm, Black hole memory effect, Phys. Rev.D 98 (2018) 124016 [arXiv:1809.07266] [INSPIRE].
R.F. Penna, Near-horizon Carroll symmetry and black hole Love numbers, arXiv:1812.05643 [INSPIRE].
L. Donnay and C. Marteau, Carrollian Physics at the Black Hole Horizon, Class. Quant. Grav.36 (2019) 165002 [arXiv:1903.09654] [INSPIRE].
W. Wieland, Generating functional for gravitational null initial data, Class. Quant. Grav.36 (2019) 235007 [arXiv:1905.06357] [INSPIRE].
D. Grumiller and W. Merbis, Near horizon dynamics of three dimensional black holes, SciPost Phys.8 (2020) 010 [arXiv:1906.10694] [INSPIRE].
D. Grumiller, M.M. Sheikh-Jabbari, C. Troessaert and R. Wutte, Interpolating Between Asymptotic and Near Horizon Symmetries, JHEP03 (2020) 035 [arXiv:1911.04503] [INSPIRE].
A. Bagchi, R. Basu, A. Mehra and P. Nandi, Field Theories on Null Manifolds, JHEP02 (2020) 141 [arXiv:1912.09388] [INSPIRE].
A. Ashtekar, Black Hole evaporation: A Perspective from Loop Quantum Gravity, Universe6 (2020) 21 [arXiv:2001.08833] [INSPIRE].
K. Hajian and M.M. Sheikh-Jabbari, Solution Phase Space and Conserved Charges: A General Formulation for Charges Associated with Exact Symmetries, Phys. Rev.D 93 (2016) 044074 [arXiv:1512.05584] [INSPIRE].
M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev.D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
G. Compère, The Kerr/CFT correspondence and its extensions, Living Rev. Rel.15 (2012) 11 [arXiv:1203.3561] [INSPIRE].
A. Aggarwal, A. Castro and S. Detournay, Warped Symmetries of the Kerr Black Hole, JHEP01 (2020) 016 [arXiv:1909.03137] [INSPIRE].
É.É. Flanagan, K. Prabhu and I. Shehzad, Extensions of the asymptotic symmetry algebra of general relativity, JHEP01 (2020) 002 [arXiv:1910.04557] [INSPIRE].
H. Afshar, D. Grumiller, M.M. Sheikh-Jabbari and H. Yavartanoo, Horizon fluff, semi-classical black hole microstates — Log-corrections to BTZ entropy and black hole/particle correspondence, JHEP08 (2017) 087 [arXiv:1705.06257] [INSPIRE].
D. Grumiller and M. Riegler, Most general AdS3boundary conditions, JHEP10 (2016) 023 [arXiv:1608.01308] [INSPIRE].
D. Grumiller, W. Merbis and M. Riegler, Most general flat space boundary conditions in three-dimensional Einstein gravity, Class. Quant. Grav.34 (2017) 184001 [arXiv:1704.07419] [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
S. Pasterski, Implications of Superrotations, Phys. Rept.829 (2019) 1 [arXiv:1905.10052] [INSPIRE].
H.R. Safari and M.M. Sheikh-Jabbari, BMS4algebra, its stability and deformations, JHEP04 (2019) 068 [arXiv:1902.03260] [INSPIRE].
H. Bondi, M. van der Burg, and A. Metzner, Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems, Proc. Roy. Soc. LondonA 269 (1962) 21.
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev.128 (1962) 2851 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 2002.08346
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Adami, H., Grumiller, D., Sadeghian, S. et al. T-Witts from the horizon. J. High Energ. Phys. 2020, 128 (2020). https://doi.org/10.1007/JHEP04(2020)128
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2020)128