The theory of spherically symmetric thin shells in conformal gravity

Abstract

The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy-momentum tensor of the thin shell is proportional to Dirac delta-function. We constructed the theory of the spherically symmetric thin shells for three types of gravitational theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl) gravity where the gravitational part of the total Lagrangian is just the square of the Weyl tensor; (3) Weyl+Einstein gravity. The results are compared with these in General Relativity (Israel equations). We considered in details the shells immersed in the vacuum. Some peculiar properties of such shells are found. In particular, for the traceless (= massless) shells it is shown that their dynamics can not be derived from the matching conditions and, thus, is completely arbitrary. On the contrary, in the case of the Weyl+Einstein gravity the trajectory of the same type of shell is completely restored even without knowledge of the outside solution.

Spherically symmetric double layers in Weyl+Einstein gravity

“…The above consideration is especially important in the case of the spherical symmetry because we are able to use the radius of the sphere (more exactly, its square) as the conformal factor (see Refs. 55,56,57,58,59 for more details), and it is this radius as a function of the proper time on the shell, that determines completely its trajectory in the space-time.…”