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.

35 pages, 4 figures, 2 tables. v3: Corrected the case of $J=0$, $\mu =0$ in Proposition 2 to Type I, which was categorized as Type II in the published version. Main results on energy conditions remain unchanged

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