Spherically symmetric double layers in Weyl+Einstein gravity

Abstract

We study the matching conditions on singular hypersurfaces in Weyl[Formula: see text]Einstein gravity. Unlike General Relativity, the so-called quadratic gravity allows the existence of a double layer, i.e. the derivative of [Formula: see text]-function. This double layer is a purely geometrical phenomenon and it may be treated as the purely gravitational shock wave. The mathematical formalism was elaborated by Senovilla for generic quadratic gravity. We derived the matching conditions for the spherically symmetric singular hypersurface in the Weyl[Formula: see text]Einstein gravity. It was found that in the presence of the double layer, the matching conditions contain an arbitrary function. One of the consequences of such freedom is that a trace of the extrinsic curvature tensor of a singular hypersurface is necessarily equal to zero. We suggested that the [Formula: see text] and [Formula: see text] components of the surface matter energy–momentum tensor of the shell describe energy flow [Formula: see text] and momentum transfer [Formula: see text] of particles produced by the double layer itself. Moreover, the requirement of the zero trace of the extrinsic curvature tensor (mentioned above) implies that [Formula: see text], and this fact also supports our suggestion, because it means that for the observer sitting on the shell, particles will be seen created by pairs, and the sum of their momentum transfers must be zero. We found also that the spherically symmetric null double layer in the Weyl[Formula: see text]Einstein gravity does not exist at all.