List of Figures

	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 $δM$ falls into 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 $v ≤ 0$ [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.
	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 $S0$ .
	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.
	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 $M$ .
	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 $Δin$ is slightly perturbed and decays to a Schwarzschild-like isolated horizon $Δ fin$ , with radiation going out to future null infinity $+ ℐ$ .
	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 $inter RΔ = RΔ$ .
	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.