Ricci Flow in Physics

  1. 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
     
  2. jcsd
  3. 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
     
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