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)