G

#### Greg Egan

##### Guest

>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.