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
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