Superrotation charge and supertranslation hair on black holes

Journal of High Energy Physics, May 2017

Abstract It is shown that black hole spacetimes in classical Einstein gravity are characterized by, in addition to their ADM mass M, momentum \( \overrightarrow{P} \), angular momentum \( \overrightarrow{J} \) and boost charge \( \overrightarrow{K} \), an infinite head of supertranslation hair. The distinct black holes are distinguished by classical superrotation charges measured at infinity. Solutions with super-translation hair are diffeomorphic to the Schwarzschild spacetime, but the diffeomorphisms are part of the BMS subgroup and act nontrivially on the physical phase space. It is shown that a black hole can be supertranslated by throwing in an asymmetric shock wave. A leading-order Bondi-gauge expression is derived for the linearized horizon supertranslation charge and shown to generate, via the Dirac bracket, supertranslations on the linearized phase space of gravitational excitations of the horizon. The considerations of this paper are largely classical augmented by comments on their implications for the quantum theory.

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Superrotation charge and supertranslation hair on black holes

HJE Superrotation charge and supertranslation hair on Stephen W. Hawking 0 2 Malcolm J. Perry 0 2 Andrew Strominger 1 2 Wilberforce Road 2 Cambridge 2 0 DAMTP, Centre for Mathematical Sciences, University of Cambridge 1 Center for the Fundamental Laws of Nature, Harvard University 2 17 Oxford Street, Cambridge, MA , U.S.A It is shown that black hole spacetimes in classical Einstein gravity are characterized by, in addition to their ADM mass M , momentum P~ , angular momentum J~ and boost charge K~ , an infinite head of supertranslation hair. The distinct black holes are distinguished by classical superrotation charges measured at infinity. Solutions with supertranslation hair are diffeomorphic to the Schwarzschild spacetime, but the diffeomorphisms are part of the BMS subgroup and act nontrivially on the physical phase space. It is shown that a black hole can be supertranslated by throwing in an asymmetric shock wave. A leading-order Bondi-gauge expression is derived for the linearized horizon supertranslation charge and shown to generate, via the Dirac bracket, supertranslations on the linearized phase space of gravitational excitations of the horizon. The considerations of this paper are largely classical augmented by comments on their implications for the quantum theory. Black Holes; Gauge Symmetry; Nonperturbative Effects 1 Introduction 2.1 2.2 2.3 Asymptotic expansion The scattering problem Discussion 3 Asymptotic symmetries 3.1 Supertranslations 3.2 Superrotations 4 Schwarzschild supertranslations 5 Implanting supertranslation hair 2 Supertranslation and superrotation charge conservation 6 Classical superrotation charges of supertranslation hair 7 Horizon charges 7.1 7.2 7.3 Symplectic forms and linearized charges Schwarzschild charges Gauge fixing and Dirac brackets A Some useful formulae asymptotically Minkowskian spacetimes, including those in which black holes are formed and then evaporate. For each and every conserved charge, the charge on the black hole must be reduced (increased) by exactly the amount carried by any emitted (absorbed) particles [ 20?23 ]. Charge conservation is possible only if black holes themselves carry an 1Prescient early work appears in [15?19]. ? 1 ? infinite number of charges or, equivalently, have an infinite head of ?soft hair? [22]. This does not violate the classical no-hair theorems [24] because the distinct black holes are related by diffeomorphisms, albeit ?large? ones which comprise the asymptotic symmetry group and act nontrivially on the classical phase space. Soft hair has implications for the information paradox [25], since charge conservation enforces quantum correlations between the outgoing Hawking quanta and the soft hair configuration. In this paper we undertake a study the properties of the charges arising from infinitedimensional gravitational symmetries in a weak-coupling expansion. The fundamental definitions of these conserved charges will be given below in terms of simple boundary integrals near spatial infinity. As usual, integration by parts and the constraint equations can procedure is in the general case fraught with difficulties associated to the choices of slice and gauge. Quantum fluctuations of the spacetime geometry further diminish the utility of such constructions. Nevertheless, in the context of weak coupling, a perturbative analysis of charge conservation in the bulk can be informative. For example it is possible to show, to first order in the gravitational coupling, that the mass of a black hole always increases by the energy flux of radiation across its horizon. A similar picture should exist for all of the conserved charges. For the infinity of electromagnetic charges, such a picture was obtained in [22]. In this paper, while also supplying the reader with some pedagogical background, we continue the program of [22] and perturbatively analyze in some detail the infinity of so-called supertranslation and superrotation symmetries. Supertranslation (superrotation) charge conservation equates the total incoming energy at each angle to the total outgoing energy (angular momentum) at the opposing angle [2, 26]. After spelling out our notation in section 2.1, in section 2.2 we reiterate the simple origin of the infinity of conserved charges. We show that the very existence of a well-posed scattering problem in asymptotically Minkowskian general relativity requires a boundary condition which matches certain metric components at the top of I ? (past null infinity) to those at the bottom of I + (future null infinity). This immediately implies an infinite number of conserved charges, simply from the equality of all the past and future multipole moments of the matched metric data. Explicit expressions are given for the supertranslation charges arising from the matching of the Bondi mass aspect, as well as the superrotation charges arising from the matching of the angular momentum aspect. The relationship to previous (...truncated)


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Stephen W. Hawking, Malcolm J. Perry, Andrew Strominger. Superrotation charge and supertranslation hair on black holes, Journal of High Energy Physics, 2017, pp. 161, Volume 2017, Issue 5, DOI: 10.1007/JHEP05(2017)161