Ricci Flow in Physics

Gerard Westendorp

I'm playing with Ricci flow at the moment. Ricci flow is quite a buzzy
word these days, since it was used recently to prove the Poincarre
conjecture. Since the Poincarre conjecture, as one of the 7 Clay
institute millenium problems, was famously difficult to prove, you'd
think Ricci flow would be something very difficult too. But a discrete
version of it is actually easy enough for me to understand, and even
implement it in a computer program.

The idea of 'discrete Ricci flow', is basically to gradually deform
(flow) a mesh, so that it converges to a correctly curved mesh of a
predetermined manifold. To find out how to deform the mesh locally, you
figure out the local Ricci tensor, and plug it into a formula to stretch
your mesh.
The idea is especially simple 2D surfaces, and beautifully visual. Check
out this:

http://www.cs.sunysb.edu/~vislab/papers/RicciFlow.pdf

Anyway, it struck me that the formula for Ricci flow looks much like the
Einstein field equation. While Ricci flow may be expressed:

d_t g_ij = -2 (R_ij-R_target_ij)

the Einstin equation can be written as:

R g_ij = -2 (R_ij-T_ij)

One way to look at it is to say that Ricci flow can be used to converge
a manifold to a solution of the Einstein equation, by setting

R_target_ij = T_ij -R

One thing I don't understand yet is how to treat the minus signs in the
metric of general relativity. The Ricci flow I'm doing right now has a
positive metric, it is curved space, but not space-time.

So, can we generalize Ricci flow for non-positive metrics? Does it have
any other applications in physics?

Gerard

Marc Nardmann

Gerard Westendorp wrote:

> I'm playing with Ricci flow at the moment. [...] a discrete version of
> it is actually easy enough for me to understand, and even implement it
> in a computer program.
>
> The idea of 'discrete Ricci flow', is basically to gradually deform
> (flow) a mesh, so that it converges to a correctly curved mesh of a
> predetermined manifold. To find out how to deform the mesh locally,
> you figure out the local Ricci tensor, and plug it into a formula to
> stretch your mesh.
> [...] Anyway, it struck me that the formula for Ricci flow looks much
> like the Einstein field equation. [...]
>
> One thing I don't understand yet is how to treat the minus signs in
> the metric of general relativity. The Ricci flow I'm doing right now
> has a positive metric, it is curved space, but not space-time.
>
> So, can we generalize Ricci flow for non-positive metrics?

Of course the Ricci flow *equation* is well-defined for Lorentzian
metrics, too. The question is whether *solutions* exist and if they
exist, what properties they have; in particular, whether the metrics of
the flow converge in a suitable sense to solutions of the Einstein
equation. This is where the Riemannian (positive definite) case should
be quite different from the Lorentzian one: one should not expect that
the Lorentzian Ricci flow tends to produce constant Ricci curvature as
the flow parameter t increases. In constrast, the usual Riemannian Ricci
flow has (for suitable start metrics) properties quite similar to the
heat equation: you start from a certain temperature distribution, and as
the flow (time) parameter increases, the temperature distribution
becomes more and more homogeneous; it converges to a constant
temperature. This property of the heat equation and (with
qualifications) of the Riemannian Ricci flow depends crucially on the
metric being positive definite, and on the parameter t flowing to larger
(instead of smaller) values.

You can study this numerically: For simplicity, use an n-dimensional
rectangular grid of spacetime points in the Lorentzian case, with
periodic boundary conditions, say (i.e., spacetime is an n-dimensional
torus). A Lorentzian metric assigns to each grid point a symmetric (n x
n)-matrix g with 1 negative eigenvalue and n-1 positive eigenvalues. The
Ricci curvature of g assigns to each grid point an (n x n)-matrix Ric,
which is computed by the usual formula; you just compute the first
partial derivatives of g_{ij} via differences of g_{ij}at neighbouring
grid points, and the second derivatives as partial derivatives of these
partial derivatives. Now use the Ricci flow formula to compute stepwise
how the metric g changes. (You have to check after each step whether the
metric has become singular, i.e. whether the new field of matrices has
still signature (n-1,1) at each point. You should probably choose the
initial metric so that its values do not vary too much between
neighbouring points; otherwise singularities might form very early.) If
you start with a generic (i.e. not very special) Lorentzian metric g, I
expect that the flow does *not* bring the metric closer to the condition
Ric = cg globally (even if the initial metric is already close); and I
expect that it does not matter in this respect whether the parameter t
flows to larger or smaller values. But I have not tried this, so you
might want to check it.

-- Marc Nardmann