Initial data sets for the Schwarzschild spacetime

Abstract

A characterization of initial data sets for the Schwarzschild spacetime is provided. This characterization is obtained by performing a $3+1$ decomposition of a certain invariant characterization 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.