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

Kartik Prabhu

doi:10.1007/jhep03(2019)148

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.

Highlights

Conditions” in [19]) relate the gravitational fields, it would imply infinitely many conservation laws in classical gravitational scattering in the sense that the incoming fluxes associated to the BMS group at past null infinity would equal the outgoing fluxes of the corresponding BMS group at future null infinity
We will use the methods of [31] where an analogous result was shown for Maxwell fields on any asymptotically-flat spacetime. our result can be viewed as a generalisation of [25] to include all supertranslations, or of [27] to full nonlinear general relativity, or of [28] to show the antipodal matching of the Weyl tensor and the relevant supertranslation symmetries
We introduce the subalgebra of Spi-supertranslations for which the total flux of Spi-supermomenta across spatial infinity vanishes, which gives us the antipodal matching conditions, the global diagonal symmetry algebra and the flux conservation between past and future null infinity

Summary

Asymptotic-flatness at null and spatial infinity

We define spacetimes which are asymptotically-flat at null and spatial infinity using an Ashtekar-Hansen structure [12, 14] as follows. The freedom in the choice of the conformal factor in Definition 2.1 is given by Ω → ωΩ where the function ω satisfies (1) ω > 0 on M (2) ω is smooth on M − i0 (3) ω is C>0 in spatial directions at i0 and ω|i0 = 1. It can be verified that the function in Definition 2.1 used to define a complete divergence-free normal −1na cannot be used as a conformal-rescaling at i0 since will diverge at i0 (see footnote 2, section 11.1 [37] and appendix E). Where εabcd is volume element at i0 corresponding to the metric gab, εabc is the induced volume element on H , and εab is the induced area element on some cross-section S of H with a future-pointing timelike normal ua such that habuaub = −1. We have used Υ◦ to denote the natural action of the reflection map Υ on tensor fields on H

The space C of null and spatial directions at i0

Supertranslations and supermomentum in general relativity

Null-regular spacetimes at i0

Discussion

A Null infinity in arbitrary conformal choices

C Solutions for Eab on C

D Comparison to the Compère-Dehouck supermomenta on H

E Relation to some coordinate-based approaches