A rotating black hole is formed only if
q
<1, indicating that cosmic censorship holds. Stark and
Piran [290,
244] use the 3+1 formalism and the radial gauge of
Bardeen-Piran [25] to study black hole formation and gravitational wave emission
in axisymmetry. In this gauge, two metric functions used in
determining
and
can be chosen such that at large radii they tend directly to
and
(the even and odd transverse traceless amplitudes of the
gravitational waves, with 1/
r
fall-off at large radii; note that
defined in [290] has the opposite sign as that commonly used,
e.g.
in [306]). In this way, the gravitational waveform is obtained at large
radii directly in the numerical evolution. It is also easy to
compute the gravitational energy emitted, as a simple integral
over a sphere far from the source:
. Using polar slicing, black hole formation appears as a region
of exponentially small lapse, when
. The initial data consists of a nonrotating, pressure deficient
TOV solution, to which angular momentum is added by hand. The
obtained waveform is nearly independent of the details of the
collapse: It consists of a broad initial peak (since the star
adjusts its initial spherical shape to a flattened shape, more
consistent with the prescribed angular momentum), the main
emission (during the formation of the black hole), and an
oscillatory tail, corresponding to oscillations of the formed
black hole spacetime. The energy of the emitted gravitational
waves during the axisymmetric core collapse is found not to
exceed
(to which the broad initial peak has a negligible contribution).
The emitted energy scales as
, while the energy in the even mode exceeds that in the odd mode
by at least an order of magnitude.
More recently, Shibata [272] carried out axisymmetric simulations of rotating stellar
collapse in full general relativity, using a Cartesian grid, in
which axisymmetry is imposed by suitable boundary conditions. The
details of the formalism (numerical evolution scheme and gauge)
are given in [271]. It is found that rapid rotation can prevent prompt black hole
formation. When
, a prompt collapse to a black hole is prevented even for a rest
mass that is 70-80% larger than the maximum allowed mass of
spherical stars, and this depends weakly on the rotational
profile of the initial configuration. The final configuration is
supported against collapse by the induced differential rotation.
In these axisymmetric simulations, shock formation for
q
<0.5 does not result in a significant heating of the core;
shocks are formed at a spheroidal shell around the high density
core. In contrast, when the initial configuration is rapidly
rotating (
), shocks are formed in a highly nonspherical manner near high
density regions, and the resultant shock heating contributes in
preventing prompt collapse to a black hole. A qualitative
analysis in [272] suggests that a disk can form around a black hole during core
collapse, provided the progenitor is nearly rigidly rotating and
for a stiff progenitor EOS. On the other hand,
still allows for a disk formation if the progenitor EOS is soft.
At present, it is not clear how much the above conclusions depend
on the restriction to axisymmetry or on other assumptions -
3-dimensional simulations of the core collapse of such initially
axisymmetric configurations have still to be performed.
A new numerical code for axisymmetric gravitational collapse in the (2+1)+1 formalism is presented in [63].
New, fully relativistic axisymmetric simulations with coupled
hydrodynamical and spacetime evolution in the light-cone
approach, have been obtained by Siebel
et al.
[282,
281]. One of the advantages of the light-cone approach is that
gravitational waves can be extracted accurately at null infinity,
without spurious contamination by boundary conditions. The code
by Siebel
et al.
combines the light-cone approach for the spacetime evolution
with HRSC methods for the hydrodynamical evolution. In [281] it is found that gravitational waves are extracted more
accurately using the Bondi news function than by a quadrupole
formula on the null cone.
A new 2D code for axisymmetric core collapse, also using HRSC methods, has recently been introduced in [273].
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Rotating Stars in Relativity
Nikolaos Stergioulas http://www.livingreviews.org/lrr-2003-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei.mpg.de |