Conservation of asymptotic charges from past to future null infinity: Supermomentum in general relativity

TL;DR

This work proves Strominger's antipodal matching conjecture for BMS supertranslations and their associated supermomenta in a broad class of asymptotically flat spacetimes by constructing the cylinder $\mathscr C$ of null and spatial directions at spatial infinity $i^0$ and imposing null-regularity. Through direction-dependent limits of the Weyl tensor and a reflection map $\Upsilon$, the authors show antipodal matching of Weyl-field data and identify a totally fluxless Spi-subalgebra $\mathfrak s^{\times}$ that yields a diagonal, flux-conserving symmetry between past and future null infinities. The approach unifies null and spatial infinity within the Ashtekar-Hansen framework, yielding conservation laws for BMS-supermomenta in classical gravitational scattering under suitable timelike infinity conditions. The results extend prior linearized and restricted nonlinear matching, provide a covariant geometric basis for the diagonal symmetry, and illuminate connections to Beig-Schmidt and Compère-Dehouck formalisms, with implications for soft theorems and broader aspects of gravitational scattering. Open questions remain regarding the existence of a large data class satisfying the null-regular conditions and extensions to angular momentum and black hole information contexts.

Abstract

We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl tensor components on past null infinity are antipodally matched to those on future null infinity. The subalgebra of totally fluxless supertranslations near spatial infinity provides a natural isomorphism between the BMS-supertranslations on past and future null infinity. This proves that the flux of the supermomenta is conserved from past to future null infinity in a classical gravitational scattering process provided additional suitable conditions are satisfied at the timelike infinities.

Figures (2)

• Figure 1: The (non-compact) unit-hyperboloid $\mathscr H$ in $Ti^0$ representing spatial directions $\vec{\eta}$ at $i^0$.
• Figure 2: The space $\mathscr C$ of null and spatial directions $\vec{N}$ at $i^0$. The boundaries $\mathscr N^\pm \cong \mathbb S^2$, diffeomorphic to the space of generators of $\mathscr I^\pm$ respectively, represent the space of null directions. $\mathscr C \!\setminus\! \mathscr N^\pm$ is the space of rescaled spatial directions conformally diffeomorphic to the unit-hyperboloid $\mathscr H$. $\mathscr C$ depends on the choice of the rescaling function $\Sigma$ (defined below) and need not be a cylinder of unit radius in $Ti^0$ --- we have drawn a "wiggly" cylinder to emphasise this.

Theorems & Definitions (19)

• Definition 2.1: Ashtekar-Hansen structure Ash-in-Held
• Remark 2.1: Freedom in the conformal factor AHAsh-in-Held
• Remark 2.2: Choices of conformal factor
• Definition 2.2: Rescaling function $\Sigma$
• Remark 2.3: Freedom in the rescaling function
• Remark 3.1: Relation to previously obtained formulae
• Remark 3.2: Translation vectors at $i^0$
• Remark 3.3: Supertranslation vector fields near $i^0$
• Definition 4.1: Null-regular spacetime at $i^0$
• Remark 4.1: Finiteness of flux through $\mathscr I$