On the nonlinear stability of higher dimensional triaxial Bianchi-{IX} black holes

Mihalis Dafermos, Gustav Holzegel
2006 Advances in Theoretical and Mathematical Physics  
In this paper, we prove that the five-dimensional Schwarzschild-Tangherlini solution of the Einstein vacuum equations is orbitally stable (in the fully non-linear theory) with respect to vacuum perturbations of initial data preserving triaxial Bianchi-IX symmetry. More generally, we prove that five-dimensional vacuum spacetimes developing from suitable asymptotically flat triaxial Bianchi-IX symmetric initial data and containing a trapped or marginally trapped homogeneous 3-surface necessarily
more » ... urface necessarily possess a complete null infinity I + , whose past J − (I + ) is bounded to the future by a regular event horizon H + , whose cross-sectional volume in turn satisfies a Penrose inequality, relating it to the final Bondi mass. In particular, the results of this paper give the first examples of vacuum black holes which are not stationary exact solutions. e-print archive: http://lanl.arXiv.org/abs/gr-qc/0510051 for asymptotically flat initial data possessing triaxial Bianchi-IX symmetry. This model has been recently introduced by Bizon et al. [2] . They show that vacuum solutions with this symmetry have two dynamic degrees of freedom, and the Einstein equations can be written (see [2]) as a system of nonlinear pde's on a two-dimensional Lorentzian quotient of five-dimensional spacetime by an SU (2) action with three-dimensional orbits. The system of equations thus obtained is studied numerically in [2], where analogues of critical behaviour have been discovered. Proving rigorously the kind of behaviour suggested by these numerics appears a formidable problem, beyond the scope of current techniques. Implicit in the discussion of [2], however, is the notion that there is an open set of initial data that leads to black hole formation. It is this aspect of [2] that we will formulate and rigorously prove in this paper. The main result is Theorem 1.1. Consider asymptotically flat smooth initial data (S,ḡ, K) for the vacuum Einstein equations, possessing triaxial Bianchi-IX symmetry. Let (M, g) denote the maximal Cauchy development, and let π : M → Q denote the projection map to the two-dimensional Lorentzian quotient Q. Suppose there exists an asymptotically flat spacelike Cauchy surfaceS ⊂ Q, and a point p ∈S such that π −1 (p) is trapped or marginally trapped, and (at least) one of the connected componentsS \ {p} contains an asymptotically
doi:10.4310/atmp.2006.v10.n4.a2 fatcat:nm67qfk2fvfh7aevvawo6wlgyy