Abstract
We revisit the covariant phase space formalism applied to gravitational theories with null boundaries, utilizing the most general boundary conditions consistent with a fixed null normal. To fix the ambiguity inherent in the Wald-Zoupas definition of quasilocal charges, we propose a new principle, based on holographic reasoning, that the flux be of Dirichlet form. This also produces an expression for the analog of the Brown-York stress tensor on the null surface. Defining the algebra of charges using the Barnich-Troessaert bracket for open subsystems, we give a general formula for the central — or more generally, abelian — extensions that appear in terms of the anomalous transformation of the boundary term in the gravitational action. This anomaly arises from having fixed a frame for the null normal, and we draw parallels between it and the holographic Weyl anomaly that occurs in AdS/CFT. As an application of this formalism, we analyze the near-horizon Virasoro symmetry considered by Haco, Hawking, Perry, and Strominger, and perform a systematic derivation of the fluxes and central charges. Applying the Cardy formula to the result yields an entropy that is twice the Bekenstein-Hawking entropy of the horizon. Motivated by the extended Hilbert space construction, we interpret this in terms of a pair of entangled CFTs associated with edge modes on either side of the bifurcation surface.
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S.B. Giddings, D. Marolf and J.B. Hartle, Observables in effective gravity, Phys. Rev. D 74 (2006) 064018 [hep-th/0512200] [INSPIRE].
D. Marolf, Unitarity and holography in gravitational physics, Phys. Rev. D 79 (2009) 044010 [arXiv:0808.2842] [INSPIRE].
W. Donnelly and S.B. Giddings, Diffeomorphism-invariant observables and their nonlocal algebra, Phys. Rev. D 93 (2016) 024030 [Erratum ibid. 94 (2016) 029903] [arXiv:1507.07921] [INSPIRE].
S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82 (1999) 2828 [hep-th/9812013] [INSPIRE].
A. Bagchi, S. Detournay, R. Fareghbal and J. Simón, Holography of 3D flat cosmological horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [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].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
S. Carlip, Black hole entropy from Bondi-Metzner-Sachs symmetry at the horizon, Phys. Rev. Lett. 120 (2018) 101301 [arXiv:1702.04439] [INSPIRE].
S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, Black hole entropy and soft hair, JHEP 12 (2018) 098 [arXiv:1810.01847] [INSPIRE].
A. Aggarwal, A. Castro and S. Detournay, Warped symmetries of the Kerr black hole, JHEP 01 (2020) 016 [arXiv:1909.03137] [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. 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, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
V. Chandrasekaran, É.É. Flanagan and K. Prabhu, Symmetries and charges of general relativity at null boundaries, JHEP 11 (2018) 125 [arXiv:1807.11499] [INSPIRE].
F. Hopfmüller and L. Freidel, Gravity degrees of freedom on a null surface, Phys. Rev. D 95 (2017) 104006 [arXiv:1611.03096] [INSPIRE].
F. Hopfmüller and L. Freidel, Null conservation laws for gravity, Phys. Rev. D 97 (2018) 124029 [arXiv:1802.06135] [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].
H. Adami, D. Grumiller, S. Sadeghian, M.M. Sheikh-Jabbari and C. Zwikel, T-Witts from the horizon, JHEP 04 (2020) 128 [arXiv:2002.08346] [INSPIRE].
H. Adami, M.M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo and C. Zwikel, Symmetries at null boundaries: two and three dimensional gravity cases, JHEP 10 (2020) 107 [arXiv:2007.12759] [INSPIRE].
D. Grumiller, M.M. Sheikh-Jabbari and C. Zwikel, Horizons 2020, Int. J. Mod. Phys. D 29 (2020) 2043006 [arXiv:2005.06936] [INSPIRE].
S. Carlip, Statistical mechanics and black hole entropy, gr-qc/9509024 [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].
A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, JHEP 02 (2018) 021 [arXiv:1706.05061] [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [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.
S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [INSPIRE].
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
A. Castro, A. Maloney and A. Strominger, Hidden conformal symmetry of the Kerr black hole, Phys. Rev. D 82 (2010) 024008 [arXiv:1004.0996] [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 [Living Rev. Rel. 20 (2017) 1] [arXiv:1203.3561] [INSPIRE].
L.-Q. Chen, W.Z. Chua, S. Liu, A.J. Speranza and B.d.S.L. Torres, Virasoro hair and entropy for axisymmetric Killing horizons, Phys. Rev. Lett. 125 (2020) 241302 [arXiv:2006.02430] [INSPIRE].
S. Haco, M.J. Perry and A. Strominger, Kerr-Newman black hole entropy and soft hair, arXiv:1902.02247 [INSPIRE].
M. Perry and M.J. Rodriguez, Central charges for AdS black holes, arXiv:2007.03709 [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
J. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
S. Carlip and C. Teitelboim, The off-shell black hole, Class. Quant. Grav. 12 (1995) 1699 [gr-qc/9312002] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan, A boundary term for the gravitational action with null boundaries, Gen. Rel. Grav. 48 (2016) 94 [arXiv:1501.01053] [INSPIRE].
R. Oliveri and S. Speziale, Boundary effects in general relativity with tetrad variables, Gen. Rel. Grav. 52 (2020) 83 [arXiv:1912.01016] [INSPIRE].
L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
L.-Q. Chen, The integrability of Virasoro charges for axisymmetric Killing horizons, arXiv:2009.11273 [INSPIRE].
E. Witten, Interacting field theory of open superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation, S.W. Hawking and W. Israel eds., chapter 16, Cambridge University Press, Cambridge, U.K. (1987), pg. 676.
C. Crnkovic, Symplectic geometry of the convariant phase space, Class. Quant. Grav. 5 (1988) 1557.
A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Mechanics, analysis and geometry: 200 years after Lagrange, M. Francaviglia ed., Elsevier Science Publishers B.V., The Netherlands (1991).
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
V. Shyam, Connecting holographic Wess-Zumino consistency condition to the holographic anomaly, JHEP 03 (2018) 171 [arXiv:1712.07955] [INSPIRE].
R.M. Wald, On identically closed forms locally constructed from a field, J. Math. Phys. 31 (1990) 2378.
K. Shi, X. Wang, Y. Xiu and H. Zhang, Covariant phase space with null boundaries, arXiv:2008.10551 [INSPIRE].
V. Iyer and R.M. Wald, A comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [INSPIRE].
G.A. Burnett and R.M. Wald, A conserved current for perturbations of Einstein-Maxwell space-times, Proc. Roy. Soc. Lond. A 430 (1990) 57.
L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 11 (2020) 026 [arXiv:2006.12527] [INSPIRE].
H. Bart, Quasi-local conserved charges in general relativity, Ph.D. thesis, Munich U., Munich, Germany (2019) [arXiv:1908.07504] [INSPIRE].
R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav. 23 (2006) 2927 [hep-th/0511096] [INSPIRE].
G. Compère and F. Dehouck, Relaxing the parity conditions of asymptotically flat gravity, Class. Quant. Grav. 28 (2011) 245016 [Erratum ibid. 30 (2013) 039501 [arXiv:1106.4045] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP 11 (2018) 200 [Erratum ibid. 04 (2020) 172] [arXiv:1810.00377] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, The Λ-BMS4 group of dS4 and new boundary conditions for AdS4, Class. Quant. Grav. 36 (2019) 195017 [arXiv:1905.00971] [INSPIRE].
C. Barrabès and W. Israel, Thin shells in general relativity and cosmology: the lightlike limit, Phys. Rev. D 43 (1991) 1129 [INSPIRE].
E. Poisson, A relativist’s toolkit: the mathematics of black-hole mechanics, Cambridge University Press, Cambridge, U.K. (2004).
J. Milnor, Relativity, groups and topology II, chapter 10, in Remarks on infinite-dimensional Lie groups, North-Holland Physics Publishing, The Netherlands (1984), pg. 1007.
C. Troessaert, Hamiltonian surface charges using external sources, J. Math. Phys. 57 (2016) 053507 [arXiv:1509.09094] [INSPIRE].
G. Barnich, Centrally extended BMS4 Lie algebroid, JHEP 06 (2017) 007 [arXiv:1703.08704] [INSPIRE].
E. Gourgoulhon and J.L. Jaramillo, A 3 + 1 perspective on null hypersurfaces and isolated horizons, Phys. Rept. 423 (2006) 159 [gr-qc/0503113] [INSPIRE].
T. Jacobson and G. Kang, Conformal invariance of black hole temperature, Class. Quant. Grav. 10 (1993) L201 [gr-qc/9307002] [INSPIRE].
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
L. Donnay, G. Giribet, H.A. González and M. Pino, Extended symmetries at the black hole horizon, JHEP 09 (2016) 100 [arXiv:1607.05703] [INSPIRE].
V. Chandrasekaran and K. Prabhu, Symmetries, charges and conservation laws at causal diamonds in general relativity, JHEP 10 (2019) 229 [arXiv:1908.00017] [INSPIRE].
S. Aghapour, G. Jafari and M. Golshani, On variational principle and canonical structure of gravitational theory in double-foliation formalism, Class. Quant. Grav. 36 (2019) 015012 [arXiv:1808.07352] [INSPIRE].
G. Jafari, Stress tensor on null boundaries, Phys. Rev. D 99 (2019) 104035 [arXiv:1901.04054] [INSPIRE].
L. Donnay and C. Marteau, Carrollian physics at the black hole horizon, Class. Quant. Grav. 36 (2019) 165002 [arXiv:1903.09654] [INSPIRE].
G. Compère and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
S. Carlip, Effective conformal descriptions of black hole entropy, Entropy 13 (2011) 1355 [arXiv:1107.2678] [INSPIRE].
S. Carlip, Near-horizon Bondi-Metzner-Sachs symmetry, dimensional reduction, and black hole entropy, Phys. Rev. D 101 (2020) 046002 [arXiv:1910.01762] [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161.
A. Averin, Entropy counting from a Schwarzschild/CFT correspondence and soft hair, Phys. Rev. D 101 (2020) 046024 [arXiv:1910.08061] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
L. Ciambelli, R.G. Leigh, C. Marteau and P.M. Petropoulos, Carrollstructures, null geometry and conformal isometries, Phys. Rev. D 100 (2019) 046010 [arXiv:1905.02221] [INSPIRE].
V.P. Frolov and K.S. Thorne, Renormalized stress-energy tensor near the horizon of a slowly evolving, rotating black hole, Phys. Rev. D 39 (1989) 2125 [INSPIRE].
L. Ciambelli and R.G. Leigh, Weyl connections and their role in holography, Phys. Rev. D 101 (2020) 086020 [arXiv:1905.04339] [INSPIRE].
S. Carlip, Extremal and nonextremal Kerr/CFT correspondences, JHEP 04 (2011) 076 [Erratum ibid. 01 (2012) 008] [arXiv:1101.5136] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer, New York, NY, U.S.A. (1997).
J.L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys. A 17 (1984) L385.
E. Shaghoulian, Modular forms and a generalized Cardy formula in higher dimensions, Phys. Rev. D 93 (2016) 126005 [arXiv:1508.02728] [INSPIRE].
D. Lewis, J. Marsden, R. Montgomery and T. Ratiu, The Hamiltonian structure for dynamic free boundary problems, Physica D 18 (1986) 391.
V.O. Solovev, Boundary values as Hamiltonian variables. I. New Poisson brackets, J. Math. Phys. 34 (1993) 5747 [hep-th/9305133] [INSPIRE].
K. Bering, Putting an edge to the Poisson bracket, J. Math. Phys. 41 (2000) 7468 [hep-th/9806249] [INSPIRE].
V.O. Solovev, Bering’s proposal for boundary contribution to the Poisson bracket, J. Math. Phys. 41 (2000) 5369 [hep-th/9901112] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G. Compère and J. Long, Vacua of the gravitational field, JHEP 07 (2016) 137 [arXiv:1601.04958] [INSPIRE].
C. Akers and P. Rath, Holographic Renyi entropy from quantum error correction, JHEP 05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, JHEP 10 (2019) 240 [arXiv:1811.05382] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
N. Engelhardt and A.C. Wall, Coarse graining holographic black holes, JHEP 05 (2019) 160 [arXiv:1806.01281] [INSPIRE].
W. Donnelly, Entanglement entropy and non-Abelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Int. J. Mod. Phys. D 19 (2010) 2429 [Gen. Rel. Grav. 42 (2010) 2323] [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
G. Penington, Entanglement wedge reconstruction and the information paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Edge dynamics from the path integral — Maxwell and Yang-Mills, JHEP 11 (2018) 080 [arXiv:1804.07585] [INSPIRE].
M. Geiller and P. Jai-akson, Extended actions, dynamics of edge modes, and entanglement entropy, JHEP 09 (2020) 134 [arXiv:1912.06025] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
G. Compère, Infinite towers of supertranslation and superrotation memories, Phys. Rev. Lett. 123 (2019) 021101 [arXiv:1904.00280] [INSPIRE].
D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D stress tensor for 4D gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
K. Nguyen and J. Salzer, The effective action of superrotation modes, arXiv:2008.03321 [INSPIRE].
S. Chakraborty and K. Parattu, Null boundary terms for Lanczos-Lovelock gravity, Gen. Rel. Grav. 51 (2019) 23 [Erratum ibid. 51 (2019) 47] [arXiv:1806.08823] [INSPIRE].
T. Azeyanagi, G. Compere, N. Ogawa, Y. Tachikawa and S. Terashima, Higher-derivative corrections to the asymptotic Virasoro symmetry of 4d extremal black holes, Prog. Theor. Phys. 122 (2009) 355 [arXiv:0903.4176] [INSPIRE].
E. Adjei, W. Donnelly, V. Py and A.J. Speranza, Cosmic footballs from superrotations, Class. Quant. Grav. 37 (2020) 075020 [arXiv:1910.05435] [INSPIRE].
V. Ovsienko and C. Roger, Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor-densities on S1, Indag. Math. 9 (1998) 277.
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Chandrasekaran, V., Speranza, A.J. Anomalies in gravitational charge algebras of null boundaries and black hole entropy. J. High Energ. Phys. 2021, 137 (2021). https://doi.org/10.1007/JHEP01(2021)137
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DOI: https://doi.org/10.1007/JHEP01(2021)137