Null infinity as an open Hamiltonian system

Abstract

When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time-dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there is a natural IR regulator. To remove the IR regulator, we introduce a double null foliation together with an adapted Newman-Penrose null tetrad. Both null directions are surface orthogonal. We study the falloff conditions for such specific null foliations and take the limit to null infinity. At null infinity, we recover the Bondi mass and the usual covariant phase space for the two radiative modes at the full non-perturbative level. Apart from technical results, the framework gives two important physical insights. First of all, it explains the physical significance of the corner term that is added in the Wald-Zoupas framework to render the quasi-conserved charges integrable. The term to be added is simply the derivative of the Hamiltonian with respect to the background fields that drive the time-dependence of the Hamiltonian. Secondly, we propose a new interpretation of the Bondi mass as the thermodynamical free energy of gravitational edge modes at future null infinity. The Bondi mass law is then simply the statement that the free energy always decreases on its way towards thermal equilibrium.

Coadjoint representation of the BMS group on celestial Riemann surfaces

“…The most interesting question is to understand in detail how such effective actions for the sector captured by the coadjoint representation interacts with the radiative degrees of freedom, as described in [66,67] and more recently in [68], see also [69,70] in this context.…”

Gravitational SL(2, ℝ) algebra on the light cone

“…In addition, ∂M 0 = C 0 and ∂M 1 = C 1 . To introduce the boundary conditions and the corresponding boundary field theory along N, we proceed as in [9]. First of all, we note that on a null surface, there always exists a spinor-valued two-form η Aab and a (commuting) spinor A such that…”

“…The boundary action is now constructed in the same way as in reference [9] from the boundary fields (η Aab , A , l a , m a ) and additional auxiliary variables κ a , ω a and N A ab . The combined action for the bulk plus boundary field theory is given by…”

“…The first and perhaps simplest possibility is to restrict ourselves to asymptotically flat spacetimes (Λ = 0) and quantise the asymptotic phase space alone. Consider an auxiliary double null foliation {N r } r>0 in a vicinity of future (past) null infinity I ± such that I ± = lim r→∞ N r , see [9]. Given such a foliation, the radiative phase space at null infinity can be obtained from an asymptotic r → ∞ limit of the quasi-local radiative phase space introduced above.…”

“…Our approach will be quasi-local [3][4][5][6]. This is to say that we describe the gravitational field as an open (dissipative) system in a finite box [7][8][9]. The box has a boundary, and we choose it to be null.…”