Natural Science Forum / Physics / Research / October 2007

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