Static spherically symmetric Kundt vacuum solutions of higher-derivative gravities

TL;DR

This paper explores static spherically symmetric Kundt solutions in higher-derivative gravities, identifying exact solutions, analyzing singularities, and constructing gravitational wave solutions on these backgrounds.

Contribution

It provides a comprehensive analysis of Kundt solutions in quadratic and six-derivative gravity, including new exact solutions and their properties.

Findings

Identified all solutions for quadratic gravity with certain coupling constants.

Derived power series solutions using Frobenius method for special cases.

Constructed exact gravitational wave solutions on these backgrounds.

Abstract

We study static spherically symmetric Kundt solutions to the vacuum field equations of quadratic gravity with a cosmological constant, as well as specific models of six-derivative gravity. In quadratic gravity, we identify all solutions for coupling constants satisfying ${\alpha\neq3\beta}$, while the case ${\alpha=3\beta}$ is studied using the Frobenius method, where we derive the recurrence relations for the power series. In contrast, in six-derivative gravity, we focus on selected models to illustrate the variety of closed-form solutions; we also analyze possible indicial families of Frobenius solutions. For all solutions, we analyze curvature singularities and their accessibility to geodesic observers. We then construct exact gravitational-wave solutions propagating on some of these backgrounds in quadratic and six-derivative gravity. It is known that in Einstein gravity, gravitational waves on the Nariai background unavoidably contain singularities, which are interpreted as physical sources generating these gravitational waves. In contrast, in addition to singular solutions, for appropriate values of the coupling constants, higher-order gravities allow for globally smooth solutions representing gravitational waves.