Conservation of asymptotic charges from past to future null infinity: Maxwell fields

Kartik Prabhu

doi:10.1007/jhep10(2018)113

Abstract

On any asymptotically-flat spacetime, we show that the asymptotic symmetries and charges of Maxwell fields on past null infinity can be related to those on future null infinity as recently proposed by Strominger. We extend the covariant formalism of Ashtekar and Hansen by constructing a 3-manifold of both null and spatial directions of approach to spatial infinity. This allows us to systematically impose appropriate regularity conditions on the Maxwell fields near spatial infinity along null directions. The Maxwell equations on this 3-manifold and the regularity conditions imply that the relevant field quantities on past null infinity are antipodally matched to those on future null infinity. Imposing the condition that in a scattering process the total flux of charges through spatial infinity vanishes, we isolate the subalgebra of totally fluxless symmetries near spatial infinity. This subalgebra provides a natural isomorphism between the asymptotic symmetry algebras on past and future null infinity, such that the corresponding charges are equal near spatial infinity. This proves that the flux of charges is conserved from past to future null infinity in a classical scattering process of Maxwell fields. We also comment on possible extensions of our method to scattering in general relativity.

Highlights

For some symmetry on past null infinity I − equal the flux of charges for a “corresponding” symmetry on future null infinity I +? Any attempt to answer this question would first need some appropriate notion of “corresponding” i.e. some isomorphism between the asymptotic symmetries on past null infinity to the ones on future null infinity
On any asymptotically-flat spacetime, we show that the asymptotic symmetries and charges of Maxwell fields on past null infinity can be related to those on future null infinity as recently proposed by Strominger
This subalgebra provides a natural isomorphism between the asymptotic symmetry algebras on past and future null infinity, such that the corresponding charges are equal near spatial infinity

Summary

Asymptotic-flatness at null and spatial infinity

We define spacetimes which are asymptotically-flat at null and spatial infinity following [12, 14] To distinguish this from other constructions, we refer to this as an Ashtekar-Hansen structure. Since spatial infinity is represented by a single point i0 in M , various physical fields of interest, in general, will not be continuous at i0 but will admit direction-dependent limits. The Ashtekar-Hansen structure provides us with an additional universal structure at i0 given by a R-principal bundle over H , called Spi [12] This is useful in the definition of asymptotic symmetries and charges (in particular the Spi-supertranslations) at spatial infinity in general relativity. For the Maxwell case, which we consider in this paper, we will not need this additional structure

Asymptotic symmetries and charges for Maxwell fields

The space C of null and spatial directions at i0

Null-regular Maxwell fields at i0

Discussion and possible generalisations

B Rescaling function Σ in T i0

C Relation to some coordinate-based approaches

D Solutions to Maxwell equation on H and their extensions to C