Symmetries and charges of general relativity at null boundaries

Published for SISSA by Springer Received: August 2, 2018 Accepted: November 8, 2018 Published: November 21, 2018 Venkatesa Chandrasekaran,a Éanna É. Flanaganb,c and Kartik Prabhuc a Center for Theoretical Physics and Department of Physics, University of California, Berkeley, CA 94720, U.S.A. b Department of Physics, Cornell University, Ithaca, NY 14853, U.S.A. c Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE), Cornell University, Ithaca, NY 14853, U.S.A. E-mail: ven , , Abstract: We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger. Keywords: Black Holes, Classical Theories of Gravity, Space-Time Symmetries ArXiv ePrint: 1807.11499 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2018)125 JHEP11(2018)125 Symmetries and charges of general relativity at null boundaries Contents 1 Introduction 1.1 Notation and conventions 1 4 4 4 6 7 8 11 11 15 3 Review of the local geometry of null hypersurfaces 3.1 Foundations 3.2 Geometric fields defined on a null hypersurface 3.3 Divergence operator 3.4 Stationary regions of null hypersurfaces 3.5 Orthonormal basis formalism 16 16 17 19 20 21 4 Universal intrinsic structure of a null hypersurface 4.1 Definition of intrinsic structure 4.2 Symmetry group of a complete intrinsic structure 4.3 Symmetry algebra of a complete intrinsic structure 4.4 Preferred subalgebra for stationary regions of a null hypersurface: Killing supertranslations 4.5 Symmetry groups of null hypersurfaces with boundaries 21 21 23 24 5 General relativity with a null boundary: covariant phase space 5.1 Definition of field configuration space 5.2 Symmetry algebra of the field configuration space 5.3 Boundary conditions on the variation of the metric 29 30 32 34 6 Global and localized charges for a null boundary component 6.1 Noether charge 6.2 Variation of Noether charge 6.3 Global charges that generate boundary symmetries 6.4 Localized (Wald-Zoupas) charges and fluxes 6.5 Charges and fluxes for specific symmetry generators 6.6 Stationary regions of the null surface 36 36 37 38 39 41 42 7 Global conservation laws involving black holes 42 –i– 28 28 JHEP11(2018)125 2 Review of the covariant phase space formalism 2.1 Definitions of field configuration space and covariant phase space 2.2 Definitions of currents 2.3 Definition of presymplectic form on covariant phase space 2.4 Global charges that generate boundary symmetries 2.5 Boundary symmetry algebras of linearized diffeomorphisms 2.6 Localized (Wald-Zoupas) charges, fluxes and conservation laws 2.7 Potential ambiguities in global and localized charges 45 45 47 48 9 Discussion, applications and future directions 9.1 Recap 9.2 Black holes: localized conservation laws and horizon memory 9.3 The limit to future null infinity 9.4 Generalizations 48 48 49 50 51 A Orthonormal basis formalism for null surfaces A.1 Review of structures associated with a choice of auxiliary null vector A.2 Geometric fields on an orthonormal basis A.3 Expressions for charges 52 52 53 53 B Gauge fixing in the definition of field configuration space 54 C Characterization of trivial diffeomorphisms at a null boundary 55 D Consistency check of symmetry algebra 56 E Choice of reference solution 56 F Consistency of two expressions for flux of localized charge 58 G Symplectic currents on black holes horizons 59 H Alternative definition of field configuration space and associated symmetry algebra 60 1 Introduction It is well known that gauge transformations of a diffeomorphism invariant theory can become genuine symmetries of the theory at boundaries of the spacetime. In general relativity, diffeomorphisms of asymptotically flat spacetimes that preserve the fall-off conditions for the metric near null infinity yield the standard BMS group [1–3]. Similarly, in QED there exists an infinite set of symmetries at null infinity comprised of large gauge transformations [4, 5]. Associated to the various symmetries are global conserved charges which act as generators of the symmetries [6, 7]. There are in addition localized charges such as Bondi mass which quantify the amount of charge in subregions of the spacetime boundary, which can be calculated using a variety of formalisms [6, 8, 9]. –1– JHEP11(2018)125 8 Algebra of symmetry generator charges and central charges 8.1 Algebra of symmetry generator charges in general contexts 8.2 Symmetry algebra of global charges at event horizons 8.3 Symmetry algebras of localized charges 1 Our symmetry group does not coincide exactly with any of the several different groups in refs. [10– 13, 19, 21], since we preserve a particular geometric structure on the null surface which defines our field configuration space, and other authors preserve other quantities such as the near horizon geometry. –2– JHEP11(2018)125 More recently, it has been found that stationary black holes also possess an infinite number of symmetries beyond the usual horizon Killing symmetries [10–20] (see [21] for older work on this topic, and [22] for the electromagnetic case). The new symmetries are diffeomorphisms which preserve the near horizon geometry under specific gauge conditions, and a subclass of them are similar to the supertranslations at null infinity. These horizon supertranslations give rise to contributions to the global charges associated with supertranslations, in addition to the contribution from null infinity. In [14–16] it was suggested that this enlarged group of horizon symmetries and its associated charges and conservation laws play a role in how information is released as a black hole evaporates, and may lead to a resolution of the information loss paradox (see also [23, 24]). At the least, a complete (...truncated)