General definition of “conserved quantities” in general relativity and other theories of gravity

Abstract

In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define "conserved quantities" in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired "conserved quantities" are not, in general, conserved!) In this paper we give a prescription for defining "conserved quantities" by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, 1 although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations. 2

BMS charge algebra

“…These charges are very similar and should be compared to those proposed earlier in [5] in the context of a closely related, but slightly different approach to asymptotically flat spacetimes.…”

“…This is a notoriously difficult task as the surface charges are non-conserved and non-integrable at null infinity [5]. It is the purpose of the present paper to fill this gap.…”

“…Whereas there is no issue with integrability in the linearized theory, in the full interacting theory with prescribed asymptotics, the expressions are one-forms on solution space indexed by asymptotic symmetries and one has to face the question whether these oneforms are integrable, i.e., whether one can construct suitable "Hamiltonians" for them [5]. This explains the notation δ / in (2.1).…”

Asymptotic symmetries on Killing horizons

“…In Section 4, we analyze the Poisson bracket algebra of the conserved charges conjugate to asymptotic Killing vectors, by using the covariant phase space formalism developed by Wald and his collaborators [18,19]. Unfortunately, however, it will turn out that the Poisson bracket algebra does not acquire non-trivial central charges.…”

Gravitational Chern-Simons terms and black hole entropy. Global aspects

“…The method used is very akin to the one used by Wald, [5] (see also [6][7][8][9][10][11]), the covariant phase space formalism. In order to bypass formal obstacles in the derivation we were obliged in [2] to use a particular coordinate system (a Kruskal-type system of coordinates), which led us to a formula for the entropy expressed in such coordinates.…”