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

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.

The good-bad-ugly system near spatial infinity on flat spacetime

“…Remark 1. There is a sign difference in equation ( 3) respect to other discussions in the literature-see [24,25,30]-which is due to the use of a different signature convention.…”

“…While these conformal extensions are known in general only in an abstract (non-explicit) way, for the Minkowski spacetime one can write closed and explicit expressions. Although most of the expressions given in this section have been reported already-see for instance [11,22,24,25,30,32]-in the discussion presented here the emphasis is placed on making contact with the physical structures and the translation of some of the i 0 -cylinder results to the physical set-up.…”

“…consistent with the bookkeeping naming conventions of section 1.1, the metric g will be regarded as the metric of the unphysical spacetime while η as the metric of the physical spacetime. These naming conventions stem from the fact that the unphysical metric g corresponds to an explicit solution to the conformal Einstein field equations written in a gauge adapted to conformal geodesics-see [11,21,22,24,25,30,32] for further discussion and definitions. To simplify the terminology when discussing fields propagating in this geometry we will refer to it simply as the i 0 -cylinder background.…”

“…However, there is a particular case where this gauge and the construction of the cylinder at spatial infinity can be obtained in an explicit closed form: the Minkowski spacetime. This special conformal representation of the Minkowski spacetime has been used as a model to understand the behaviour of fields propagating in the vicinity of spatial infinity of asymptotically flat spacetimes and the consequences of the degeneracy of the equations in the critical sets-see [22][23][24][25][26][27][28][29]. Crucially, one of these consequences is that even linear fields propagating in this conformal representation of the Minkowski spacetime (i 0 -cylinder background) will develop logarithmic terms at the critical sets which spread out to null infinity-see [22,23,25,30].…”

“…This special conformal representation of the Minkowski spacetime has been used as a model to understand the behaviour of fields propagating in the vicinity of spatial infinity of asymptotically flat spacetimes and the consequences of the degeneracy of the equations in the critical sets-see [22][23][24][25][26][27][28][29]. Crucially, one of these consequences is that even linear fields propagating in this conformal representation of the Minkowski spacetime (i 0 -cylinder background) will develop logarithmic terms at the critical sets which spread out to null infinity-see [22,23,25,30]. Although it has been shown that the non-linearities in the conformal Einstein field equations will generate further logarithmic terms-see [31], the analysis of linear fields about the given background already serves as a basis to develop intuition into what is the minimal regularity of the field that can be be expected close to the critical sets.…”