Geometrical origin of the Kodama vector

Abstract

It has been known that warped-product spacetimes such as spherically symmetric ones admit the Kodama vector. This vector provides a locally conserved current made by contraction of the Einstein tensor, even though there is no Killing vector. In addition, a quasilocal mass, Birkhoff's theorem and various properties are closely related to the Kodama vector. Recently, it is shown that the notion of the Kodama vector can be extended to three-dimensional axisymmetric spacetimes even if the spacetimes are not warped product. This implies that warped product may not be a necessary condition for a spacetime to admit the Kodama vector. We show properties of the Kodama vector originate from the conformal Killing-Yano 2-form. In particular, the well-known spacetimes that admit the Kodama vector have a closed conformal Killing-Yano 2-form. Furthermore, we show the Kodama vector provides local conserved currents for each order of the Lovelock tensor as well as the Einstein tensor.