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Frans Pretorius

Newer Caltech web page

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Current research

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__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)

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Recent Talks

APS April meeting 2002
Analysis, Computation and Collaboration, July 2001
(source)

CCGRRA, May 2001

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__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.

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__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
(2.6M)
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Slides from the APS meeting

**Text slides: ** talk.ps
**Near critical evolution**, 3 families: near_crit.ps
(3M)

**Spacetime plots of black hole formation** from a gaussian pulse,
A=0.13305, center=0.2,width=0.05 :

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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)

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Resources

'Templates for the solution of linear systems: Build blocks
for iterative methods' : templates.ps

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Links