Numerical Approach to Timelike Minimal Surfaces

Project: Numerical Study of Membrane Dynamics in String and M-theory

Nov 2005
Benjamin Gutierrez

Background:

I am studying an embedding of a higher dimensional bulk spacetime into a target space, membrane located inside the bulk, where we have an indiced metric. The membrane is the a minimal (timelike) surface that describes the evolution of the membrane in time. This could be relevant for string and M-theory, because these backgrounds are explicitly time dependent. In terms of the physics, the timelike minimal surface problem comes up in both "regular" string theory and in so-called m-theory. In the m-theory case, it is the equation governing membrane motion. From the d-brane action we obtain the minimal surface functional by turning off all the physics tensors. .

Writeup: Equations of Motion, CN discretization, RNPL implementation

Maple 8 Worksheet.

TensorV6 check (Thanx to Inaki to show me this package).

TensorV6 change of variables to obtain a first order system.
t520.pdf.
Pure Wave Equation Sol. A=1.0 Phi and Pi,
t520.pdf.
No dissipation. A=1.0 Phi and Pi,
Low amplitude limit (A=0.0001) evolution for Phi and Pi, Level 2 (see write up)
A=1.0 Pi and Phi Level 2
A=1.0 Pi and Phi Level 4
A=0.005 idsign=-1 Pi and Phi Level 2

amp1:= 0.0004, amp2:= 0.0008 idsign=-1 (amp1)Pi and Phi Level 2

This is a proposal for future directions to this research:

Whats the plan: First we need to develop some intuition about these systems. Paul points out that very little is know about them, so whatever effort could be usefull. We will explore any critical dimensions present where the stability of the system changes or singularities appear. for higher dimensions i see a few technical difficulties but lets see if we can overcome them.

Higher dimensions is expected to be more stable. Lets consider is 2+1 surface in 3+1 background. How long in time can we run a thing like this? Do we see some sort of stability/decay? Can we handle co-dimension 2 cases? e.g. 2+1 surface in a 4+1 background?

We can try background metrics such as Sitter / anti-de Sitter spacetimes, which may be of interest for the string community. Explore
higher dimensions ( 3+1 in 4+1 or more).

The spherically symmetric case is interesting. Paul suggests surfaces such as
$S^n x [0,T)$ inside $R^{m +1}$, i.e. every time=constant slice is a
sphere, then it should collapse at finite time. i.e., the surface looks
like a (US) football. Is this behaviour stable under small perturbations?

Thanx a lot to:
Paul T. Allen (Department of Mathematics, University of Oregon, Eugene, OR, USA.
Inaki Olabarrieta, LSU, Baron Rouge, Lousiana, USA
Frans Pretorius, University of alberta, CA

for invaluable comments and suggestions.

References:

Allen, P., Andersson, Lars and Isenberg, James; Time-like Minimal Surfaces of General Co-dimension in Minkowski Spacetime, Private communication.
Jens Hoppe, http://arxiv.org/abs/hep-th/0206192.