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Figure 1:
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. |
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Figure 2:
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. |
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Figure 3:
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. |
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Figure 4:
Penrose diagrams of
Schwarzschild-Vaidya metrics for which the mass function
vanishes for [137]. 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. |
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Figure 5:
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 . |
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Figure 6:
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. |
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Figure 7:
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 . |
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Figure 8:
Bondi-like coordinates in
a neighborhood of . |
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Figure 9:
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 .) |
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Figure 10:
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 . |
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Figure 11:
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 . |
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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. |
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