Frans Pretorius

Newer Caltech web page
Department of Physics and Astronomy
University of British Columbia
6224 Agricultural Road
Vancouver BC,   V6T 1Z1,   Canada
Office :     Hennings 408
Main Office: (604) 822-3853
Fax: (604) 822-5324

Current research

Axisymmetric gravitational collapse code (with M.W.Choptuik, E.W. Hirschmann, and S.L. Liebling)

Scalar field critical collapse with AMR (updated June 2002)

Black hole excision (updated May 2001)

Recent Talks

APS April meeting 2002

Analysis, Computation and Collaboration, July 2001 (source)

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 (340K)

97x97 based grid (shown here) with 2, 4:1 refinement levels (so finest level is 1537x1537);  d(phi)/dt: amr.mpg (346k)

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:

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 (2.6M)

Slides from the APS meeting

Other research

Quantum interest for scalar fields in Minkowski spacetime:
paper (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):  paper (169K pdf)
An operational approach to black hole entropy (with D. Vollick and W. Israel):  paper (122K pdf)