, Volume 9, Issue 4, pp 313-326

Conformal tensor discontinuities in general relativity

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Abstract

The postulate is made that across a given hypersurfaceN the metric and its first derivatives are continuous. This postulate is used to derive conditions which must be satisfied by discontinuities in the Riemann tensor acrossN. These conditions imply that the conformal tensor jump is uniquely determined by the stress-energy tensor discontinuity ifN is non-null (and to within an additive term of type Null ifN is lightlike). Alternatively, \([C^{\alpha \beta } _{\gamma \delta } ]\) and [R] determine \(\left[ {R_{\mu v} - \frac{1}{4}Rg_{\mu v} } \right]\) ifN is non-null. These relationships between the conformal tensor and stress-energy tensor jumps are given explicitly in terms of a three-dimensional complex representation of the antisymmetric tensors. Application of these results to perfect-fluid discontinuities is made: \([C^{\alpha \beta } _{\gamma \delta } ]\) is of type D across a fluid-vacuum boundary and across an internal, non-null shock front. \([C^{\alpha \beta } _{\gamma \delta } ]\) is of type I (non-degenerate) in general across fluid interfaces across which no matter flows, except for special cases.

The research reported herein was supported in part by the Atomic Energy Commission under contract number AT (11-1)-34, Project Agreement No. 125, while the author was at the University of California at Berkeley, and in part by National Science Foundation Grant GU-1598-University Science Development Program.