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Re: This Week's Finds in Mathematical Physics (Week 232)

  1. Nov 4, 2006 #1
    Ralph Hartley wrote:

    >Consider a polyhedron inscribed in a sphere of radius 1, centered at the
    >origin. Let the surface of the polyhedron inherit the metric from R^3
    >(which will be flat except at the vertexes).
    >
    >For any point p other than the origin, let p_1 be the intersection of
    >the polyhedron with the ray from the origin through p. Let t(p) = |p|/|p_1|.
    >
    >The metric (on R^3-O) ds^2 = -dt(p)^2 + dp_1^2 is flat except at the
    >rays from the origin through the vertexes, and any [space]like surface has
    >total deficit 4Pi.


    That's a really elegant construction, but (at least in the static case) I
    think you can get rid of your Big Bang at t=0. Just take the Cartesian
    product of the real line R with the polyhedron, with its inherited
    metric, and put ds^2 = -dt^2 + dp_1^2, where now p_1 is the projection
    from the Cartesian pair (t,p_1).

    This would generalise to "polyhedra" formed by triangulating any compact
    2-manifold and putting flat metrics on the triangles. The total deficit
    in the general case will be 2pi*chi, where chi is the Euler
    characteristic of the manifold.
     
  2. jcsd
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