Isolated and Dynamical Horizons and Their Applications

Abstract

Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modelled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity, and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity, suggested a phenomenological model for hairy black holes, provided novel techniques to extract physics from numerical simulations, and led to new laws governing the dynamics of black holes in exact general relativity.

Figure 1

Left panel: A typical gravitational collapse. The portion Δ of the event horizon at late times is isolated. Physically, one would expect the first law to apply to Δ even though the entire space-time is not stationary because of the presence of gravitational radiation in the exterior region. Right panel: Space-time diagram of a black hole which is initially in equilibrium, absorbs a finite amount of radiation, and again settles down to equilibrium. Portions Δ₁ and Δ₂ of the horizon are isolated. One would expect the first law to hold on both portions although the space-time is not stationary.

Figure 2

A spherical star of mass M undergoes collapse. Much later, a spherical shell of mass δM the resulting black hole. While Δ₁ and Δ₂ are both isolated horizons, only Δ₂ is part of the event horizon.

Figure 3

Set-up of the general characteristic initial value formulation. The Weyl tensor component Ψ₀ on the null surface Δ is part of the free data which vanishes if Δ is an IH.

Figure 4

Penrose diagrams of Schwarzschild-Vaidya metrics for which the mass function M(v) vanishes for v ≤ 0 [137]. The space-time metric is flat in the past of v = 0 (i.e., in the shaded region). In the left panel, as v tends to infinity, vanishes and tends to a constant value M₀. The space-like dynamical horizon H, the null event horizon E, and the time-like surface r = 2M₀ (represented by the dashed line) all meet tangentially at i⁺. In the right panel, for v ≤ v₀ we have . Space-time in the future of v = v₀ is isometric with a portion of the Schwarzschild space-time. The dynamical horizon H and the event horizon E meet tangentially at v = v₀. In both figures, the event horizon originates in the shaded flat region, while the dynamical horizon exists only in the curved region.

Figure 5

H is a dynamical horizon, foliated by marginally trapped surfaces S. is the unit time-like normal to H and the unit space-like normal within H to the foliations. Although H is space-like, motions along an be regarded as ‘time evolution with respect to observers at infinity’. In this respect, one can think of H as a hyperboloid in Minkowski space and S as the intersection of the hyperboloid with space-like planes. In the figure, H joins on to a weakly isolated horizon Δ with null normal at a cross-section S₀.

Figure 6

The region of space-time under consideration has an internal boundary Δ and is bounded by two Cauchy surfaces M₁ and M₂ and the time-like cylinder τ_∞ at infinity. M is a Cauchy surface in whose intersection with Δ is a spherical cross-section S and the intersection with τ_∞ is S_∞, the sphere at infinity.

Figure 7

The world tube of apparent horizons and a Cauchy surface M intersect in a 2-sphere S. T^a is the unit time-like normal to M and R^a is the unit space-like normal to S within M.

Figure 8

Bondi-like coordinates in a neighborhood of Δ.

Figure 9

The ADM mass as a function of the horizon radius RΔ of static spherically symmetric solutions to the Einstein-Yang-Mills system (in units provided by the Yang-Mills coupling constant). Numerical plots for the colorless (n = 0) and families of colored black holes (n = 1, 2) are shown. (Note that the y-axis begins at M = 0.7 rather than at M = 0.)

Figure 10

An initially static colored black hole with horizon Δ_in is slightly perturbed and decays to a Schwarzschild-like isolated horizon Δ_fin, with radiation going out to future null infinity .

Figure 11

The ADM mass as a function of the horizon radius R_Δ in theories with a built-in non-gravitational length scale. The schematic plot shows crossing of families labelled by n = 1 and .

Figure 12

Quantum horizon. Polymer excitations in the bulk puncture the horizon, endowing it with quantized area. Intrinsically, the horizon is flat except at punctures where it acquires a quantized deficit angle. These angles add up to endow the horizon with a 2-sphere topology.