# Conformal tensor discontinuities in general relativity

- Received:

DOI: 10.1007/BF01654285

- Cite this article as:
- Shepley, L.C. Commun.Math. Phys. (1968) 9: 313. doi:10.1007/BF01654285

- 1 Citations
- 49 Downloads

## Abstract

The postulate is made that across a given hypersurface*N* 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 across*N*. These conditions imply that the conformal tensor jump is uniquely determined by the stress-energy tensor discontinuity if*N* is non-null (and to within an additive term of type Null if*N* is lightlike). Alternatively,\([C^{\alpha \beta } _{\gamma \delta } ]\) and [*R*] determine\(\left[ {R_{\mu v} - \frac{1}{4}Rg_{\mu v} } \right]\) if*N* 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.