Abstract
The Brown-York stress tensor provides a means for defining quasilocal gravitational charges in subregions bounded by a timelike hypersurface. We consider the generalization of this stress tensor to null hypersurfaces. Such a stress tensor can be derived from the on-shell subregion action of general relativity associated with a Dirichlet variational principle, which fixes an induced Carroll structure on the null boundary. The formula for the mixed-index tensor Tij takes a remarkably simple form that is manifestly independent of the choice of auxiliary null vector at the null surface, and we compare this expression to previous proposals for null Brown-York stress tensors. The stress tensor we obtain satisfies a covariant conservation equation with respect to any connection induced from a rigging vector at the hypersurface, as a result of the null constraint equations. For transformations that act covariantly on the boundary structures, the Brown-York charges coincide with canonical charges constructed from a version of the Wald-Zoupas procedure. For anomalous transformations, the charges differ by an intrinsic functional of the boundary geometry, which we explicity verify for a set of symmetries associated with finite null hyper-surfaces. Applications of the null Brown-York stress tensor to symmetries of asymptotically flat spacetimes and celestial holography are discussed.
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G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [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].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
M. Campiglia and A. Laddha, New symmetries for the gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [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].
E.E. Flanagan, K. Prabhu and I. Shehzad, Extensions of the asymptotic symmetry algebra of general relativity, JHEP 01 (2020) 002 [arXiv:1910.04557] [INSPIRE].
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, The Weyl BMS group and Einstein’s equations, JHEP 07 (2021) 170 [arXiv:2104.05793] [INSPIRE].
S. Hollands, A. Ishibashi and D. Marolf, Counter-term charges generate bulk symmetries, Phys. Rev. D 72 (2005) 104025 [hep-th/0503105] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616] [INSPIRE].
S. Carlip, Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82 (1999) 2828 [hep-th/9812013] [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].
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].
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].
V. Chandrasekaran and A.J. Speranza, Anomalies in gravitational charge algebras of null boundaries and black hole entropy, JHEP 01 (2021) 137 [arXiv:2009.10739] [INSPIRE].
E.E. Flanagan, Order-unity correction to Hawking radiation, Phys. Rev. Lett. 127 (2021) 041301 [arXiv:2102.04930] [INSPIRE].
E.E. Flanagan, Infrared effects in the late stages of black hole evaporation, JHEP 07 (2021) 137 [arXiv:2102.13629] [INSPIRE].
S. Pasterski and H. Verlinde, HPS meets AMPS: how soft hair dissolves the firewall, JHEP 09 (2021) 099 [arXiv:2012.03850] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
M. Geiller, Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity, Nucl. Phys. B 924 (2017) 312 [arXiv:1703.04748] [INSPIRE].
A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, JHEP 02 (2018) 021 [arXiv:1706.05061] [INSPIRE].
M. Geiller, Lorentz-diffeomorphism edge modes in 3d gravity, JHEP 02 (2018) 029 [arXiv:1712.05269] [INSPIRE].
M. Geiller and P. Jai-akson, Extended actions, dynamics of edge modes, and entanglement entropy, JHEP 09 (2020) 134 [arXiv:1912.06025] [INSPIRE].
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].
W. Donnelly, L. Freidel, S.F. Moosavian and A.J. Speranza, Gravitational edge modes, coadjoint orbits, and hydrodynamics, JHEP 09 (2021) 008 [arXiv:2012.10367] [INSPIRE].
J.D. Brown and J.W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [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].
V. Chandrasekaran, É.É. Flanagan, I. Shehzad and A.J. Speranza, A general framework for gravitational charges and holographic renormalization, arXiv:2111.11974 [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 046002 [hep-th/9903203] [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].
M. Henneaux, Zero Hamiltonian signature spacetimes, Bull. Soc. Math. Belg. Ser. A 31 (1979) 47.
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A 47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].
C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [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].
M. Henneaux and P. Salgado-ReboLledó, Carroll contractions of Lorentz-invariant theories, JHEP 11 (2021) 180 [arXiv:2109.06708] [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].
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].
F. Hopfmüller and L. Freidel, Gravity degrees of freedom on a null surface, Phys. Rev. D 95 (2017) 104006 [arXiv:1611.03096] [INSPIRE].
R. Oliveri and S. Speziale, Boundary effects in general relativity with tetrad variables, Gen. Rel. Grav. 52 (2020) 83 [arXiv:1912.01016] [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].
M. Mars and J.M.M. Senovilla, Geometry of general hypersurfaces in space-time: junction conditions, Class. Quant. Grav. 10 (1993) 1865 [gr-qc/0201054] [INSPIRE].
R. Geroch, Asymptotic structure of space-time, in Asymptotic structure of space-time, F.P. Esposito and L. Witten eds., Plenum Press, New York, NY, U.S.A. (1977).
A. Ashtekar, Asymptotic quantization: based on 1984 Naples lectures, Bibliopolis, Naples, Italy (1987).
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].
G. Daŭtcourt, Characteristic hypersurfaces in general relativity. I, J. Math. Phys. 8 (1967) 1492.
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. Damour, Surface effects in black-hole physics, in Marcel Grossmann meeting: general relativity, (1982), pg. 587.
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.
A.M. Grant, K. Prabhu and I. Shehzad, The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity, arXiv:2105.05919 [INSPIRE].
R.M. Wald, On identically closed forms locally constructed from a field, J. Math. Phys. 31 (1990) 2378.
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav. 23 (2006) 2927 [hep-th/0511096] [INSPIRE].
Y. Kosmann-Schwarzbach and B.E. Schwarzbach, The Noether theorems: invariance and conservation laws in the twentieth century, Springer, New York, NY, U.S.A. (2011).
T. Jacobson, Initial value constraints with tensor matter, Class. Quant. Grav. 28 (2011) 245011 [arXiv:1108.1496] [INSPIRE].
M.D. Seifert and R.M. Wald, A general variational principle for spherically symmetric perturbations in diffeomorphism covariant theories, Phys. Rev. D 75 (2007) 084029 [gr-qc/0612121] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
S. Hollands, A. Ishibashi and D. Marolf, Comparison between various notions of conserved charges in asymptotically AdS-spacetimes, Class. Quant. Grav. 22 (2005) 2881 [hep-th/0503045] [INSPIRE].
J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Perfect fluids, SciPost Phys. 5 (2018) 003 [arXiv:1710.04708] [INSPIRE].
X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective, J. Math. Phys. 59 (2018) 072503 [arXiv:1505.03739] [INSPIRE].
L. Ciambelli, R.G. Leigh, C. Marteau and P.M. Petropoulos, Carroll structures, null geometry and conformal isometries, Phys. Rev. D 100 (2019) 046010 [arXiv:1905.02221] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field theories with conformal Carrollian symmetry, JHEP 05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
J. Hartong, Holographic reconstruction of 3D flat space-time, JHEP 10 (2016) 104 [arXiv:1511.01387] [INSPIRE].
L. Ciambelli and C. Marteau, Carrollian conservation laws and Ricci-flat gravity, Class. Quant. Grav. 36 (2019) 085004 [arXiv:1810.11037] [INSPIRE].
L. Ciambelli, C. Marteau, A.C. Petkou, P.M. Petropoulos and K. Siampos, Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids, Class. Quant. Grav. 35 (2018) 165001 [arXiv:1802.05286] [INSPIRE].
L. Ciambelli, C. Marteau, A.C. Petkou, P.M. Petropoulos and K. Siampos, Flat holography and Carrollian fluids, JHEP 07 (2018) 165 [arXiv:1802.06809] [INSPIRE].
J.D. Brown, S.R. Lau and J.W. York, Jr., Energy of isolated systems at retarded times as the null limit of quasilocal energy, Phys. Rev. D 55 (1997) 1977 [gr-qc/9609057] [INSPIRE].
I.S. Booth, Metric based Hamiltonians, null boundaries, and isolated horizons, Class. Quant. Grav. 18 (2001) 4239 [gr-qc/0105009] [INSPIRE].
F. Hopfmüller and L. Freidel, Null conservation laws for gravity, Phys. Rev. D 97 (2018) 124029 [arXiv:1802.06135] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
M. Rangamani, Gravity and hydrodynamics: lectures on the fluid-gravity correspondence, Class. Quant. Grav. 26 (2009) 224003 [arXiv:0905.4352] [INSPIRE].
R.F. Penna, BMS3 invariant fluid dynamics at null infinity, Class. Quant. Grav. 35 (2018) 044002 [arXiv:1708.08470] [INSPIRE].
L. Ciambelli, C. Marteau, P.M. Petropoulos and R. Ruzziconi, Gauges in three-dimensional gravity and holographic fluids, JHEP 11 (2020) 092 [arXiv:2006.10082] [INSPIRE].
R. Fareghbal and A. Naseh, Flat-space energy-momentum tensor from BMS/GCA correspondence, JHEP 03 (2014) 005 [arXiv:1312.2109] [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].
É.É. Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95 (2017) 044002 [arXiv:1510.03386] [INSPIRE].
G. Compère, R. Oliveri and A. Seraj, The Poincaré and BMS flux-balance laws with application to binary systems, JHEP 10 (2020) 116 [arXiv:1912.03164] [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].
W. Vogel, Über lineare Zusammenhänge in singulären Riemannschen Räumen (in German), Arch. Math. 16 (1965) 106.
L. Ciambelli and R.G. Leigh, unpublished notes.
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Chandrasekaran, V., Flanagan, É.É., Shehzad, I. et al. Brown-York charges at null boundaries. J. High Energ. Phys. 2022, 29 (2022). https://doi.org/10.1007/JHEP01(2022)029
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DOI: https://doi.org/10.1007/JHEP01(2022)029