Re: This Week's Finds in Mathematical Physics (Week 232) (1 Viewer)

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Greg Egan

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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.
 

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