Black Holes in Six-dimensional Conformal Gravity

Abstract: We study conformally-invariant theories of gravity in six dimensions. In four dimensions, there is a unique such theory that is polynomial in the curvature and its derivatives, namely Weyl-squared, and furthermore all solutions of Einstein gravity are also solutions of the conformal theory. By contrast, in six dimensions there are three independent conformally-invariant polynomial terms one could consider. There is a unique linear combination (up to overall scale) for which Einstein metrics are also solutions, and this specific theory forms the focus of our attention in this paper. We reduce the equations of motion for the most general spherically-symmetric black hole to a single 5th-order differential equation. We obtain the general solution in the form of an infinite series, characterised by 5 independent parameters, and we show how a finite 3-parameter truncation reduces to the already known Schwarzschild-AdS metric and its conformal scaling. We derive general results for the thermodynamics and the first law for the full 5-parameter solutions. We also investigate solutions in extended theories coupled to conformally-invariant matter, and in addition we derive some general results for conserved charges in cubic-curvature theories in arbitrary dimensions.