Bohm and Einstein-Sasaki Metrics, Black Holes and Cosmological Event Horizons

TL;DR

<3-5 sentence high-level summary>Gibbons, Hartnoll, and Pope study Bohm metrics—countably infinite inhomogeneous Einstein metrics on spheres and sphere products—to assess the classical stability of higher‑dimensional black holes and Freund–Rubin AdS backgrounds. They develop and apply a Lichnerowicz‑operator framework, use Weyl tensor bounds and Rayleigh‑Ritz methods, and provide analytic proofs and numerical results showing negative TT Lichnerowicz modes for many Bohm metrics, implying instability of black holes built with these horizons; Einstein–Sasaki spaces evade instability due to Killing spinors, yielding stable endpoints. They also explore analytic continuations to Lorentzian geometries, constructing counterexamples to strict Cosmic Baldness while supporting the cosmic No‑Hair intuition, and discuss non‑compact Bohm metrics with infinitely many negative modes and potential links to the Dirichlet problem. The work thus delineates the landscape of stability vs. instability in higher‑dimensional gravity and informs possible endpoints and compactification strategies within string/ M‑theory contexts.

Abstract

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 x S^2 and S^3 x S^3 have negative eigenvalue modes of the Lichnerowicz operator and by numerical methods we establish that Bohm metrics on S^5 have negative eigenvalues too. We argue that all the Bohm metrics will have negative modes. These results imply that higher-dimensional black-hole spacetimes where the Bohm metric replaces the usual round sphere metric are classically unstable. We also show that the stability criterion for Freund-Rubin solutions is the same as for black-hole stability, and hence such solutions using Bohm metrics will also be unstable. We consider possible endpoints of the instabilities, and show that all Einstein-Sasaki manifolds give stable solutions. We show how Wick rotation of Bohm metrics gives spacetimes that provide counterexamples to a strict form of the Cosmic Baldness conjecture, but they are still consistent with the intuition behind the cosmic No-Hair conjectures. We show how the 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. We also argue that noncompact versions of the Bohm metrics have infinitely many negative Lichernowicz modes, and we conjecture a general relation between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet problem for Einstein's equations.

Figures (9)

• Figure 1: Relationship between Lichnerowicz modes and masses for Schwarzschild in a finite cavity.
• Figure 2: The noncompact Bohm metric in a box with a fixed value of $a/b$ at the boundary. Allowed values of $R$ are shown along with the corresponding positive, negative and zero modes from branching. It is expected that the branches will join at $a(R)/b(R)=1$.
• Figure 3: The Bohm$(2,2)_0$ (standard) Einstein metric on $S^5$. The left-hand figure shows the metric coefficients $a$ and $b$ as functions of the radial variable $t$. The function $a$ vanishes at $t=0$, and $b=b_0=1$ there. The crossover occurs at $t=t_c={{{ 1}\over { 4}}}\pi$. The right-hand figure is a parametric plot of $b$ vs. $a$.
• Figure 5: The Bohm$(2,2)_4$ Einstein metric on $S^5$. The function $b$ starts at $b_0\approx 0.053054$, and the mid-point is at $t_c\approx 1.524951$.
• Figure 6: The Bohm$_6$ Einstein metric on $S^5$. The function $b$ starts at $b_0\approx 0.010503$, and the mid-point is at $t_c\approx 1.56174$.