Left panel: A typical gravitational collapse. The portion of the event horizon at late times is isolated. Physically, one would expect the first law to apply to 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 and of the horizon are isolated. One would expect the first law to hold on both portions although the space-time is not stationary.
A spherical star of mass undergoes collapse. Much later, a spherical shell of mass falls into the resulting black hole. While and are both isolated horizons, only is part of the event horizon.
Set-up of the general characteristic initial value formulation. The Weyl tensor component on the null surface is part of the free data which vanishes if is an IH.
Penrose diagrams of Schwarzschild–Vaidya metrics for which the mass function vanishes for . The space-time metric is flat in the past of (i.e., in the shaded region). In the left panel, as tends to infinity, vanishes and tends to a constant value . The space-like dynamical horizon , the null event horizon , and the time-like surface (represented by the dashed line) all meet tangentially at . In the right panel, for we have . Space-time in the future of is isometric with a portion of the Schwarzschild space-time. The dynamical horizon and the event horizon meet tangentially at . In both figures, the event horizon originates in the shaded flat region, while the dynamical horizon exists only in the curved region.
is a dynamical horizon, foliated by marginally trapped surfaces . is the unit time-like normal to and the unit space-like normal within to the foliations. Although is space-like, motions along can be regarded as ‘time evolution with respect to observers at infinity’. In this respect, one can think of as a hyperboloid in Minkowski space and as the intersection of the hyperboloid with space-like planes. In the figure, joins on to a weakly isolated horizon with null normal at a cross-section .
The region of space-time under consideration has an internal boundary and is bounded by two Cauchy surfaces and and the time-like cylinder at infinity. is a Cauchy surface in whose intersection with is a spherical cross-section and the intersection with is , the sphere at infinity.
The world tube of apparent horizons and a Cauchy surface intersect in a 2-sphere . is the unit time-like normal to and is the unit space-like normal to within .
Bondi-like coordinates in a neighborhood of .
The ADM mass as a function of the horizon radius 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 () and families of colored black holes () are shown. (Note that the -axis begins at rather than at .)
An initially static colored black hole with horizon is slightly perturbed and decays to a Schwarzschild-like isolated horizon , with radiation going out to future null infinity .
The ADM mass as a function of the horizon radius in theories with a built-in non-gravitational length scale. The schematic plot shows crossing of families labelled by and at .
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.
© Max Planck Society and the author(s)