Regularity of Horizons and The Area Theorem

In a space-time with cosmological constant $\Lambda>0$ and matter satisfying the dominant energy condition, the area of a black or white hole cannot exceed $4\pi/\Lambda$. This applies to event horizons where defined, i.e. in an asymptotically deSitter space-time, and to outer trapping horizons (cf. apparent horizons) in any space-time. The bound is attained if and only if the horizon is identical to that of the degenerate `Schwarzschild-deSitter' solution. This yields a topo...

On a class of dynamical spacetimes which are asymptotic as $t\to\infty$ to a stationary spacetime containing a horizon $\mathcal{H}_0$, we show the existence of a unique null hypersurface $\mathcal{H}$ which is asymptotic to $\mathcal{H}_0$. This is a special case of a general unstable manifold theorem for perturbations of flows which translate in time and have a normal sink at an invariant manifold in space. Examples of horizons $\mathcal{H}_0$ to which our result applies in...

This article introduces the subject of quasi-local horizons at a level suitable for physics graduate students who have taken a first course on general relativity. It reviews properties of trapped surfaces and trapped regions in some simple examples, general properties of trapped surfaces including their stability properties, the definitions and some applications of dynamical-, trapping-, and isolated-horizons.

We treat two aspects of the physics of stationary black holes. First we prove that the proportionality, d(energy) ~ d(area) for arbitrary perturbations (``extended first law''), follows directly from an extremality theorem drawn from earlier work. Second we consider quantum fluctuations in the shape of the horizon, concluding heuristically that they exhibit a fractal character, with order lambda fluctuations occurring on all scales lambda below M^{1/3} in natural units.

We present some of the recent results and open questions on the causality problem in General Relativity. The concept of singularity is intimately connected with future trapped surface and inner event horizon formation. We offer a brief overview of the Hawking Penrose singularity theorems and discuss a few open problems concerning the future Cauchy development (domain of dependence), breakdown criteria and energy conditions for the horizon stability. A key question is whether ...

We establish the conjectured area-angular momentum-charge inequality for stable apparent horizons in the presence of a positive cosmological constant, and show that it is saturated precisely for extreme Kerr-Newman-de Sitter horizons. As with previous inequalities of this type, the proof is reduced to minimizing an `area functional' related to a harmonic map energy; in this case maps are from the 2-sphere to the complex hyperbolic plane. The proof here is simplified compared ...

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it...

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.

In this work we analyse asymptotically flat, spherically symmetric spacetimes in which an event horizon is present without any trapped surfaces. We identify two types of such spacetimes, each related to the asymptotic behaviour (in time) of one of the two degrees of freedom of the metric. We study the causal structure of both types, showing that one almost always has a Cauchy horizon beyond which it is extendable, while the other is inextendable but has two separate future nu...

We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides...