Spherically symmetric double layers in Weyl+Einstein gravity

Abstract

The main results are the following. We derived the matching conditions for the spherically symmetric singular hypersurface (in our case it is equivalent to the world line) in the Weyl$+$Einstein gravity. It was found, that the residual extrinsic curvature tensor of this surface must be continuous on the singular hypersurface. The result is the same as that found by Senovilla, it is dictated by the very possibility to have the double layer, but the jump in the normal derivative of the radius may be not zero. It was found that in the presence of the double layer, the matching conditions contain an arbitrary function, and this result is quite new and very important. One of the consequences of such freedom is that the trace of the extrinsic curvature tensor of the singular hypersurface is necessarily equal to zero. We suggested the physical interpretation for the $S_0^n$ and $S_n^n$ components of the surface matter energy-momentum tensor of the shell. In General Relativity they are zero by virtue of the Einstein equations. In the quadratic gravity they are not necessarily zero. Our interpretation is that these components describe the energy flow $(S^{0n})$ and the momentum transfer $(S^{nn})$ of the particles produced by the double layer itself. Moreover, the requirement of the zero trace of the extrinsic curvature tensor (mentioned above) implies that $S_n^n=0$, and this fact also support our suggestion, because it means that for the observer sitting on the shell, the particles will be seen created by pairs, and the sum of their momentum transfers must zero. We derived also the matching conditions for the null hypersurface, and this is, again, quite new. We found that the null-double layer in the Weyl$+$ Einstein gravity does not exist at all.