Killing and homothetic initial data for general hypersurfaces

Abstract

In this paper we present a collection of general identities relating the deformation tensor of an arbitrary vector field η with the tensor on an abstract hypersurface of any causal character. As an application we establish necessary conditions on for the existence of a homothetic Killing vector on the spacetime where is embedded. The sufficiency of these conditions is then analysed in three specific settings. For spacelike hypersurfaces, we recover the well-known homothetic KID equations in the language of hypersurface data. For two intersecting null hypersurfaces, we generalize a previous result by Chruściel and Paetz, valid for Killings, to the homothetic case and, moreover, demonstrate that the equations can be formulated solely in terms of the initial data for the characteristic Cauchy problem, i.e. without involving a priori spacetime quantities. This puts the characteristic KID problem on equal footing with the spacelike KID problem. Furthermore, we highlight the versatility of the formalism by addressing the homothetic KID problem for smooth spacelike-characteristic initial data. Other initial value problems, such as the spacelike-characteristic with corners, can be approached similarly.