On the differentiability conditions at spacelike infinity
Magnus Herberthson
TL;DR
This work addresses how to formulate differentiability at spacelike infinity $i^0$ for asymptotically flat spacetimes by proposing a $C^{1^+}$ structure that equalizes all directions but permits logarithmic metric divergences. It shows that, despite such divergences, the Weyl tensor can have direction-dependent regular limits, preserving the physically meaningful content of the gravitational field, and it derives antipodal symmetry properties for the rescaled Weyl data. Through two Schwarzschild completions, a Bianchi-identity analysis, and an electromagnetic-field study, the paper clarifies which aspects of curvature and field data remain well-behaved at $i^0$ and how these results influence interpretations of angular momentum and logarithmic ambiguities in conformal completions. The findings offer a robust framework for understanding the asymptotic gravitational and electromagnetic fields in a setting that tolerates milder differentiability at spacelike infinity, with potential implications for defining conserved quantities and radiation content.
Abstract
We consider space-times which are asymptotically flat at spacelike infinity, i^0. It is well known that, in general, one cannot have a smooth differentiable structure at i^0, but have to use direction dependent structures. Instead of the oftenly used C^{>1}-differentiabel structure, we suggest a weaker differential structure, a C^{1^+} structure. The reason for this is that we have not seen any completions of the Schwarzschild space-time which is C^{>1} in both spacelike and null directions at {i^0}. In a C^{1^+} structure all directions can be treated equal, at the expense of logarithmic singularities at {i^0}. We show that, in general, the relevant part of the curvature tensor, the Weyl part, is free from these singularities, and that the (rescaled) Weyl tensor has a certain symmetry.