The physical setup is described in Figure 5. The outgoing null hypersurfaces extend to future null
infinity
on a compactified numerical grid. Consequently, there is no need for either an artificial outer
boundary condition or an interior extraction worldtube. The outgoing radiation is computed in the
coordinates of an observer in an inertial frame at infinity, thus avoiding any gauge ambiguity in the
waveform.
The first calculations were carried out with nonzero data for on
and zero data on
[65
]
(so that no ingoing radiation entered the system). The resulting simulations were highly accurate and
tracked the quasi-normal ringdown of a perturbation consisting of a compact pulse through 10 orders of
magnitude and tracked the final power law decay through an additional 6 orders of magnitude. The
measured exponent of the power law decay varied from
, at the beginning of the tail, to
near the end, in good agreement with the predicted value of
for a quadrupole
wave [198].
The accuracy of the perturbative solutions provide a virtual exact solution for carrying out convergence
tests of the nonlinear PITT null code. In this way, the error in the Bondi news function computed by the
PITT code was calibrated for perturbative data consisting of either an outgoing pulse on or an
ingoing pulse on
. For the outgoing pulse, clean second order convergence was confirmed until late
times in the evolution, when small deviations from second order arise from accumulation of roundoff and
truncation error. For the Bondi news produced by the scattering of an ingoing pulse, clean
second order convergence was again confirmed until late times when the pulse approached the
black hole horizon. The late time error arises from loss of resolution of the pulse (in the
radial direction) resulting from the properties of the compactified radial coordinate used in the
code. This type of error could be eliminated by using characteristic the AMR techniques under
development [197
].
The characteristic Teukolsky code has been used to study radiation from axisymmetric white holes
and black holes in the close approximation. The radiation from an axisymmetric fissioning
white hole [65] was computed using the Weyl data on
supplied by the conformal horizon
model described in Section 4.3, with the fission occurring along the axis of symmetry. The close
approximation implies that the fission takes place far in the future, i.e. in the region of
above
the black hole horizon
. The data have a free parameter
which controls the energy
yielded by the white hole fission. The radiation waveform reveals an interesting dependence on
the parameter
. In the large
limit, the waveform consists of a single pulse, followed by
ringdown and tail decay. The amplitude of the pulse scales quadratically with
and the width
decreases with
. As
is reduced, the initial pulse broadens and develops more structure. In
the small
limit, the amplitude scales linearly with
and the shape is independent of
.
Since there was no incoming radiation, the above model gave the physically appropriate boundary
conditions for a white hole fission (in the close approximation). From a time reversed view point, the system
corresponds to a black hole merger with no outgoing radiation at future null infinity, i.e. the analog of an
advanced solution with only ingoing but no outgoing radiation. In the axisymmetric case studied, the
merger corresponds to a head-on collision between two black holes. The physically appropriate boundary
conditions for a black hole merger correspond to no ingoing radiation on and binary black hole data
on
. Because
and
are disjoint, the corresponding data cannot be used directly to
formulate a double null characteristic initial value problem. However, the ingoing radiation at
supplied by the advanced solution for the black hole merger could be used as Stage I of a two
stage approach to determine the corresponding retarded solution. In Stage II, this ingoing
radiation is used to generate the analogue of an advanced minus retarded solution. A pure retarded
solution (with no ingoing radiation but outgoing radiation at
can then be constructed by
superposition. The time reflection symmetry of the Schwarzschild background is key to carrying out this
construction.
This two stage strategy has been carried out by Husa, Zlochower, Gómez, and Winicour [151]. The
superposition of the Stage I and II solutions removes the ingoing radiation from
while modifying the
close approximation perturbation of
, essentially making it ring. The amplitude of the radiation
waveform at
has a linear dependence on the parameter
, which in this black hole scenario governs
the energy lost in the inelastic merger process. Unlike the fission waveforms, there is very little
-dependence in their shape and the amplitude continues to scale linearly even for large
. It is not
surprising that the retarded waveforms from a black hole merger differs markedly from the retarded
waveforms from a white hole merger. The white hole process is directly visible at
whereas the merger
waveform results indirectly from the black holes through the preceding collapse of matter or
gravitational energy that formed them. This explains why the fission waveform is more sensitive to the
parameter
which controls the shape and timescale of the horizon data. However, the weakness of
the dependence of the merger waveform on
is surprising and has potential importance for
enabling the design of an efficient template for extracting a gravitational wave signal from
noise.
In the purely vacuum approach to the binary black hole problem, the stars which collapse to form the black
holes are replaced either by imploding gravitational waves or some past singularity as in the Kruskal
picture. This avoids hydrodynamic difficulties at the expense of a globally complicated initial value problem.
The imploding waves either emanate from a past singularity, in which case the time-reversed application of
cosmic censorship implies the existence of an anti-trapped surface; or they emanate from , which
complicates the issue of gravitational radiation content in the initial data and its effect on the outgoing
waveform. These complications are avoided in the two stage approach adopted in the close
approximation studies described in Section 4.4.1, where advanced and retarded solutions in
a Schwarzschild background can be rigorously identified and superimposed. Computational
experiments have been carried out to study the applicability of this approach in the nonlinear
regime [113].
From a time reversed viewpoint, the first stage is equivalent to the determination of the outgoing
radiation from a fission of a white hole in the absence of ingoing radiation, i.e. the physically appropriate
“retarded” waveform from a white hole fission. This fission problem can be formulated in terms of data on
the white hole horizon and data representing the absence of ingoing radiation on a null
hypersurface
which emanates from
at an early time. The data on
is provided by the
conformal horizon model for a fissioning white hole. This allows study of a range of models
extending from the perturbative close approximation regime, in which the fission occurs inside a
black hole event horizon, to the nonlinear regime of a “bare” fission visible from
. The
study concentrates on the axisymmetric spinless fission (corresponding in the time reversed view
to the head-on collision of non-spinning black holes). In the perturbative regime, the news
function agrees with the close approximation waveforms. In the highly nonlinear regime, a bare
fission was found to produce a dramatically sharp radiation pulse, which then undergoes a
damped oscillation. Because the black hole fission is visible from
, it is a more efficient
source of gravitational waves than a black hole merger and can produce a higher fractional mass
loss!
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