Symmetries of the Synge metric in the relativistic optics (Q2716703)

Let \((M,\gamma)\) be a (pseudo-)Riemannian manifold and \(f=f(x,y) \geq 0\) a function of a point \(x\in M\) and a one-form \(y\) on \(M\). The paper studies Synge's metric \(g=\gamma+f y\otimes y\) which is of use in relativistic geometrical optics, where \(y\) describes the velocity of the medium and \(f= 1-n^{-2}\) is given by the refractive index \(n\). The authors show that an infinitesimal symmetry of both \(\gamma\) and \(f\) is an infinitesimal symmetry of \(g\) too. If \(f\) does not depend on \(y\), i.e. \(g\) is a Kerr-Schild metric, then the converse is also true.