Initial data sets for the Schwarzschild spacetime

TL;DR

This paper characterizes initial data sets for the Schwarzschild spacetime using a 3+1 decomposition of Weyl tensor invariants, providing necessary and sufficient conditions to identify Schwarzschild initial data.

Contribution

It introduces an invariant-based characterization and an algorithmic procedure to verify if initial data sets correspond to Schwarzschild spacetime.

Findings

Provides necessary conditions in terms of Weyl tensor concomitants.

Derives a formula for a static Killing initial data set candidate.

Establishes sufficient conditions using stationary Killing initial data.

Abstract

A characterisation of initial data sets for the Schwarzschild spacetime is provided. This characterisation is obtained by performing a 3+1 decomposition of a certain invariant characterisation of the Schwarzschild spacetime given in terms of concomitants of the Weyl tensor. This procedure renders a set of necessary conditions --which can be written in terms of the electric and magnetic parts of the Weyl tensor and their concomitants-- for an initial data set to be a Schwarzschild initial data set. Our approach also provides a formula for a static Killing initial data set candidate --a KID candidate. Sufficient conditions for an initial data set to be a Schwarzschild initial data set are obtained by supplementing the necessary conditions with the requirement that the initial data set possesses a stationary Killing initial data set of the form given by our KID candidate. Thus, we obtain an algorithmic procedure of checking whether a given initial data set is Schwarzschildean or not.