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Ricci Flow in Physics 
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#1
Oct1707, 05:00 AM

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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_ijR_target_ij) the Einstin equation can be written as: R g_ij = 2 (R_ijT_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 spacetime. So, can we generalize Ricci flow for nonpositive metrics? Does it have any other applications in physics? Gerard 


#2
Oct2107, 05:00 AM

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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 spacetime. > > So, can we generalize Ricci flow for nonpositive metrics? Of course the Ricci flow *equation* is welldefined 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 ndimensional rectangular grid of spacetime points in the Lorentzian case, with periodic boundary conditions, say (i.e., spacetime is an ndimensional torus). A Lorentzian metric assigns to each grid point a symmetric (n x n)matrix g with 1 negative eigenvalue and n1 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 (n1,1) 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 


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