On the differentiability conditions at spacelike infinity

TL;DR

This paper proposes a weaker differentiability structure, C^{1^+}, at spacelike infinity in asymptotically flat spacetimes, allowing for a unified treatment of directions and revealing regularity properties of the Weyl tensor.

Contribution

It introduces the C^{1^+} structure at spacelike infinity, enabling consistent analysis of all directions and demonstrating the regularity of the Weyl tensor in this framework.

Findings

Weyl tensor is free from logarithmic singularities at i^0.

C^{1^+} structure treats all directions equally.

The rescaled Weyl tensor exhibits a specific symmetry.

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.