Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

We study physical applications of the Bohm metrics, which are infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension $5<~d<~9.$ We prove that all the Bohm metrics on $S^3\times S^2$ and $S^3\times S^3$ have negative eigenvalue modes of the Lichnerowicz operator acting on transverse traceless symmetric tensors, and by numerical methods we establish that Bohm metrics on $S^5$ have negative eigenvalues too. General arguments suggest that all the Bohm metrics will have negative Lichnerowicz modes. These results imply that generalized higher-dimensional black-hole spacetimes, in which the Bohm metric replaces the usual round sphere metric, are classically unstable. We also show that the classical stability criterion for Freund-Rubin solutions, which are products of Einstein metrics with anti–de Sitter spacetimes, is the same in all dimensions as that for black-hole stability, and hence such solutions based on the Bohm metrics will also be unstable. We consider possible end points of the instabilities, and in particular we show that all Einstein-Sasaki manifolds give stable solutions. Next, we show how analytic continuation of Bohm metrics gives Lorentzian metrics that provide counterexamples to a strict form of the cosmic baldness conjecture, but they are nevertheless consistent with the intuition behind the cosmic no-hair conjectures. We indicate how these Lorentzian metrics may be created “from nothing” in a no-boundary setting. We argue that Lorentzian Bohm metrics are unstable to decay to de Sitter spacetime. Finally, we argue that noncompact versions of the Bohm metrics have infinitely many negative Lichnerowicz modes, and we conjecture a general relationship between Lichnerowicz eigenvalues and nonuniqueness of the Dirichlet problem for Einstein’s equations.