List of Figures

View Image 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.
View Image Figure 2:
A spherical star of mass M undergoes collapse. Much later, a spherical shell of mass δM falls into the resulting black hole. While Δ1 and Δ2 are both isolated horizons, only Δ2 is part of the event horizon.
View Image 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.
View Image Figure 4:
Penrose diagrams of Schwarzschild–Vaidya metrics for which the mass function M (v ) vanishes for v ≤ 0 [138]. 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, M˙ vanishes and M 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 M˙ = 0. 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 = v 0. In both figures, the event horizon originates in the shaded flat region, while the dynamical horizon exists only in the curved region.
View Image Figure 5:
H is a dynamical horizon, foliated by marginally trapped surfaces S. a ^τ is the unit time-like normal to H and ^ra the unit space-like normal within H to the foliations. Although H is space-like, motions along ^ra can 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 a ℓ¯ at a cross-section S0.
View Image Figure 6:
The region of space-time ℳ 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 ℳ whose intersection with Δ is a spherical cross-section S and the intersection with τ∞ is S∞, the sphere at infinity.
View Image Figure 7:
The world tube of apparent horizons and a Cauchy surface M intersect in a 2-sphere S. Ta is the unit time-like normal to M and Ra is the unit space-like normal to S within M.
View Image Figure 8:
Bondi-like coordinates in a neighborhood of Δ.
View Image 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.)
View Image Figure 10:
An initially static colored black hole with horizon Δin is slightly perturbed and decays to a Schwarzschild-like isolated horizon Δ fin, with radiation going out to future null infinity + ℐ.
View Image 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 n = 2 at inter RΔ = RΔ.
View Image 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.