Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

TL;DR

This paper explores the relationship between regularity conditions and staticity of zero-rest-mass fields near spatial infinity, revealing that while static fields satisfy certain regularity conditions, the converse is not necessarily true, contrasting with general relativity.

Contribution

It establishes that the regularity condition involving the vanishing of the linearised Cotton spinor is satisfied by static spin-2 fields but not exclusively by them, highlighting a nuanced difference from the general relativity case.

Findings

Static spin-2 fields satisfy the regularity condition.

Not all solutions satisfying the regularity condition are static.

Contrasts with the close relation between staticity and smoothness in general relativity.

Abstract

Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich's representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised Cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of General Relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.