Killing vectors of FLRW metric (in comoving coordinates) and zero modes of the scalar Laplacian

Abstract

Based on an examination of the solutions to the Killing Vector equations for the FLRW-metric in co moving coordinates , it is conjectured and proved that the components(in these coordinates) of Killing Vectors, when suitably scaled by functions, are \emph{zero modes} of the corresponding \emph{scalar} Laplacian. The complete such set of zero modes(infinitely many) are explicitly constructed for the two-sphere. They are parametrised by an integer n. For $n\,\ge\,2$, all the solutions are \emph{irregular} (in the sense that they are neither well defined everywhere nor are \emph{square-integrable}). The associated 2-d vectors are also \emph{not normalisable}. The $n=0$ solutions being constants (these correspond to the zero angular momentum solutions) are regular and normalizable. Not all of the $n=1$ solutions are regular but the associated vectors are normalizable. Of course, the action of scalar Laplacian coordinate independent significance only when acting on scalars. However, our conclusions have an unambiguous meaning as long as one works in this coordinate system. As an intermediate step, the covariant Laplacians(vector Laplacians) of Killing vectors are worked out for four-manifolds in two different ways, both of which have the novelty of not explicitly needing the connections. It is further shown that for certain maximally symmetric sub-manifolds(hypersurfaces of one or more constant comoving coordinates) of the FLRW-spaces also, the scaled Killing vector components are zero modes of their corresponding scalar Laplacians. The Killing vectors for the maximally symmetric four-manifolds are worked out using the elegant embedding formalism originally due to Schr\"odinger . Some consequences of our results are worked out. Relevance to some very recent works on zero modes in AdS/CFT correspondences , as well as on braneworld scenarios is briefly commented upon.