Gregory J. Galloway

A Generalization of Hawking's black hole topology theorem to higher dimensions

Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S2 × S1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

Topological Censorship and Higher Genus Black Holes

Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti‐de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for the horizon topology of black holes. We find that the genera of horizons are controled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the nonspherical horizon topologies of locally anti‐de Sitter black holes. More specifically, letD be the domain of outer communications of a boundary at infinity ‘‘scri.’’ We show that the principle of topological censorship ~PTC!, which is that every causal curve in D having end points on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D: i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces ~Cauchy surfaces or analogues thereof! in the domain of outer communications of any four-dimensional spacetime obeying the PTC. From this, we establish that the sum of the genera of the cross sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes. @S0556-2821~99!08218-1#

The AdS/CFT correspondence and topological censorship

Abstract In this Letter we consider results on topological censorship, previously obtained by the authors in Phys. Rev. D 60 (1999) 104039, in the context of the AdS/CFT correspondence. These, and further, results are used to examine the relationship of the topology of an asymptotically locally anti-de Sitter spacetime (of arbitrary dimension) to that of its conformal boundary-at-infinity (in the sense of Penrose). We also discuss the connection of these results to results in the Euclidean setting of a similar flavor obtained by Witten and Yau in Adv. Theor. Math. Phys. 3 (1999).

MATHEMATICAL GENERAL RELATIVITY: A SAMPLER

We provide an introduction to selected recent advances in the mathematical understanding of Einsteins theory of gravitation.

Maximum principles for null hypersurfaces and null splitting theorems

A maximum principle for C^0 null hypersurfaces is obtained and used to derive a splitting theorem for spacetimes which contain null lines. As a consequence of this null splitting theorem, it is proved that an asymptotically simple vacuum (Ricci flat) spacetime which contains a null line is isometric to Minkowski space.

On the topology of the domain of outer communication

It is shown that the topological censorship theorem of Friedman, Schleich and Witt implies, in the general setting of their result, that the domain of outer communication is simply connected. This improves recent related results of Chrusciel and Wald and of Jacobson and Venkataramani.

On the topology and area of higher-dimensional black holes

Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher-dimensional analogues of some well known results for black holes in 3 + 1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking’s black hole topology theorem for asymptotically flat (� = 0) black hole spacetimes, and Gibbons’ and Woolgar’s genus-dependent, lower entropy bound for topological black holes in asymptotically locally antide Sitter ( �< 0) spacetimes. In higher dimensions the genus is replaced by the so-called σ -constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

Abstract.The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.

The Cosmological Time Function

Let be a time-oriented Lorentzian manifold and d the Lorentzian distance on M. The function is the cosmological time function of M, where as usual p< q means that p is in the causal past of q. This function is called regular iff for all q and also along every past inextendible causal curve. If the cosmological time function of a spacetime is regular it has several pleasant consequences: (i) it forces to be globally hyperbolic; (ii) every point of can be connected to the initial singularity by a rest curve (i.e. a timelike geodesic ray that maximizes the distance to the singularity); (iii) the function is a time function in the usual sense; in particular, (iv) is continuous, in fact, locally Lipschitz and the second derivatives of exist almost everywhere.

Closed timelike geodesics

On montre que chaque classe de t-homotopie stable libre de courbes fermees du genre temps dans une variete de Lorentz compacte contient une courbe la plus longue qui doit etre une geodesique fermee du genre temps