An intrinsic characterization of the Schwarzschild metric

Abstract

An intrinsic algorithm that exclusively involves conditions on the metric tensor and its differential concomitants is presented to identify every typeD static vacuum solution. In particular, the necessary and sufficient explicit and intrinsic conditions are given for a Lorentzian metric to be the Schwarzschild solution.

Initial data sets for the Schwarzschild spacetime

“…Now, in order to answer the more subtle question whether the necessary conditions are also sufficient ones, one has to confront in one way or the other the issue of their propagation. In our context the propagation of the conditions resulting from the 3+1 splitting of the characterisation of [9] would imply that the initial data under question are indeed Schwarzschildean. If the conditions do not propagate, then we may need to add extra conditions in order to construct Schwarzschild initial data.…”

“…In particular, we want to avoid the use of the two global ingredients used in [26]: the asymptotic flatness and the non-vanishing of the ADM mass. Our starting point is a certain invariant characterisation of the Schwarzschild spacetime in terms of concomitants of the Weyl tensor which was given in [9]. Necessary conditions for a pair (h ij , K ij ) satisfying the vacuum constraint equations to be Schwarzschild data are obtained by making a 3+1 splitting of the characterisation of reference [9].…”

“…Our starting point is a certain invariant characterisation of the Schwarzschild spacetime in terms of concomitants of the Weyl tensor which was given in [9]. Necessary conditions for a pair (h ij , K ij ) satisfying the vacuum constraint equations to be Schwarzschild data are obtained by making a 3+1 splitting of the characterisation of reference [9]. Now, in order to answer the more subtle question whether the necessary conditions are also sufficient ones, one has to confront in one way or the other the issue of their propagation.…”

“…If the conditions do not propagate, then we may need to add extra conditions in order to construct Schwarzschild initial data. This issue is dealt with by observing that the invariant characterisation of [9] also yields a formula for the static Killing vector of the Schwarzschild spacetime. This formula is again given in terms of concomitants of the Weyl tensor, and can be split in lapse and shift terms.…”

“…The article is structured as follows: in section II we detail the local invariant characterisation of the Schwarzschild spacetime presented in [9]. This will be our starting point.…”