Conformal Killing gravity: pp-waves; multiple matter sources

Abstract

Recently Harada proposed a gravitational theory which is of third order in the derivatives of the metric tensor. This attracted some attention particularly as it predicts a late-time transition from cosmological deceleration to accelerated expansion without assuming the presence of dark energy or a non-zero cosmological constant. This theory has been dubbed conformal Killing gravity (CKG). The most general exact solutions of the Harada field equations are known for a number of important physical situations: homogeneous and isotropic cosmological models, static spherically symmetric vacuum and electrovac spacetimes. These are analogues of the well-known FRWL, Schwarzschild and Reissner–Nordström metrics of general relativity (GR). In this paper a subclass of pp-waves are studied and the most general exact solution obtained together with its specialization for plane waves. The generalization from GR to Harada’s theory is straightforward. The solutions have Petrov type N or 0 and the Ricci tensor is either zero or the Segré type is [(211)] with zero eigenvalue. For any metric in CKG it is shown that more than one possible matter source can generate the solution. If the metric admits one or more Killing vectors or Killing tensors, the ambiguity in the possible matter sources increases.