Killing initial data revisited

Abstract

Considering the Killing vector fields on a Lorentzian manifold in terms of their lapse and shift along a space-like hypersurface, we give a new definition of the Killing Initial Data (KID) of Beig and Chruściel [Class. Quantum Grav. 14-1A, A83 (1996)]. We define, on these new KID’s, the bracket operation induced by the usual Lie bracket of Killing vector fields on the Lorentzian manifold, and study the conditions for our new KID’s to form a Lie algebra. The interesting fact is that these conditions only depend on data along the hypersurface.