Energy conditions for non-timelike thin shells

Abstract

We study energy conditions for non-timelike thin shells in arbitrary $n(\ge 3)$ dimensions. It is shown that the induced energy-momentum tensor $t_{\mu\nu}$ on a shell $\Sigma$ is of the Hawking-Ellis type I if $\Sigma$ is spacelike and either of type I, II, or III if $\Sigma$ is null. Then, we derive simple equivalent representations of the standard energy conditions for $t_{\mu\nu}$. In particular, on a spacelike shell or on a null shell with non-vanishing surface current, $t_{\mu\nu}$ inevitably violates the dominant energy condition. If the surface pressure on the null shell is vanishing in addition, $t_{\mu\nu}$ is of type III and violates all the standard energy conditions. Those fully general results are obtained without imposing a spacetime symmetry and can be used in any theory of gravity. Lastly, several applications of the main results are presented in general relativity in four dimensions.