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Axisymmetric gravitational collapse code (with M.W.Choptuik, E.W. Hirschmann, and S.L.
Scalar field critical collapse with AMR (updated June 2002)
Black hole excision (updated May 2001)
APS April meeting 2002
Analysis, Computation and Collaboration, July 2001
CCGRRA, May 2001
Earlier flatspace AMR experiments (May 2000)
Wave equation in 2D, sharply peaked time-symmetric gaussian.
The uniform grid is not able to resolve the initial peak very well,
resulting in a slightly 'warped' wave; also the reflection at the boundaries
is not very clean.
97x97 uniform grid; d(phi)/dt: unigrid.mpg
97x97 based grid (shown here) with 2, 4:1 refinement levels (so finest
level is 1537x1537); d(phi)/dt: amr.mpg
During the AMR run between 50,000 and 150,000 gridpoints were used at
any one time, which (in principle) is quite efficient compared to
a 1537x1537 = 2,362,369 point uniform run. Though to properly test the
AMR solution one would need to compare it to a 1537x1537 uniform run. Obtaining
comparable accuracy might require lowering the maximum allowed truncation
error, which would reduce the efficiency of the AMR code.
Gravitational collapse of a minimally-coupled massless scalar field
in 2+1D AdS spacetime (with M.W.Choptuik)
Preliminary version of paper: ads.ps
Black hole formation from gaussian pulse initial data with amplitude
A=0.13305, centered at r=0.2 and having a width of 0.05 (corresponding
to APS slides below). These movies show the scalar field gradient PHI,
the curvature scalar R, and proper circumference metric element
rb as functions of the compactified, light-like coordinates (r,t).
At about t=1 a crushing, space-like curvature singularity forms,
the causal future of which is excised from the calculation. (The value
of the cosmological constant was chosen so that r(infinity)=1.0. )
Near critical evolution from gaussian initial data, A=0.13305921875,
in [ln(rb),-ln(tc-tc*)] coordinates, where rb is proper circumference
and tc is central proper time. In the critical regime the solution
is continuously self-similar (CSS) with scale-invariant variable x=rb/tc.
The movies below show various functions of the spatial gradient PHI
and time derivative PI of the scalar field, and the logarithmic
derivative of the mass aspect M. At late stages of collapse these
functions all exhibit scale-invariance, which in ln-ln coordinates appears
as unit-velocity wave propagation to the left. (But note that the output
times are not uniform in ln(tc)!)
Universality of the critical solution: Near critical evolution for
a gaussian (as above), a squared gaussian (A= 0.10060015625),
and a kink (A=0.133244140625, and note that -PHI is used for the
kink to facilitate comparison). All families were centered at r=0.2 and
had a width parameter of 0.05.
Non-compact, time-symmetric initial data: The following movie shows
the behaviour of an initially static (PI=0), `harmonic' function
PHI=A*cos(r*sqrt(-Lambda))^2 for 50 light-crossing times (we call
the function harmonic because without back reaction the solution to the
wave equation is A*sin(t*sqrt(-Lambda))*cos^2(r*sqrt(-Lambda)) : PHIrt.mpg
Slides from the APS meeting
Text slides: talk.ps
Near critical evolution, 3 families: near_crit.ps
Spacetime plots of black hole formation from a gaussian pulse,
A=0.13305, center=0.2,width=0.05 :
Quantum interest for scalar fields in Minkowski spacetime:
(161K pdf), talk
(0.9M ps) given at WORKSHOP ON BLACK HOLES II: THEORY AND MATHEMATICAL
ASPECTS, Val-Morin Quebec, June 1999.
Quasi-spherical light cones of the Kerr geometry (with W. Israel):
An operational approach to black hole entropy (with D. Vollick
and W. Israel): paper
'Templates for the solution of linear systems: Build blocks
for iterative methods' : templates.ps