Lightcone reference for total gravitational energy

TL;DR

The paper addresses the problem of defining total gravitational energy using only spacetime geometry, ensuring agreement with ADM energy for asymptotically flat spacetimes, Trautman-Bondi-Sachs energy at null infinity, and Abbott-Deser energy for asymptotically AdS spacetimes. It presents the gravitational energy as a boundary value of the Brown–York Hamiltonian, with two zero-point choices: a static-time reference and a lightcone reference, yielding the explicit expression $H|_B = (8\pi)^{-1} \int_B \sqrt{\sigma}\, N\, \left(k + \sqrt{2\mathcal{R} + 4/\ell^{2}}\right)$ that reproduces the correct energies in the appropriate limits. The work demonstrates the asymptotic equivalence of the two zero-points and shows that, in the null-infinity limit, the construction yields the Trautman–Bondi–Sachs energy, providing a unified, purely geometric framework for quasilocal and total gravitational energy in both flat and AdS spacetimes. This approach offers a practical, finite, intrinsic definition of gravitational energy with a physically motivated zero-point derived from lightcone embedding, deepening connections between quasilocal and global energy concepts.

Abstract

We give an explicit expression for gravitational energy, written solely in terms of physical spacetime geometry, which in suitable limits agrees with the total Arnowitt-Deser-Misner and Trautman-Bondi-Sachs energies for asymptotically flat spacetimes and with the Abbot-Deser energy for asymptotically anti-de Sitter spacetimes. Our expression is a boundary value of the standard gravitational Hamiltonian. Moreover, although it stands alone as such, we derive the expression by picking the zero-point of energy via a ``lightcone reference.''