# Einstein's field equation on orbifolds

I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).

Here, by an orbifold I mean the "stacky" quotient of, for example, a compact Lie group $G$ acting smoothly on a manifold $M$ and with finite stabilizers...hence the stacky quotient will be (the Morita equivalence class of) its translation (Lie) groupoid $[M\times G\rightrightarrows M]$, in contrast to the "coarse" orbit space $M/G$ which is an object with singularities.

For this kind of stacky objects there is a theory of connections and curvature for $S^1$-bundles (related with Maxwell's equations, I guess)...I´m just curious if there exist more "classical" notions of curvature which allow to state Einstein´s field equation in this context...of course, if this is the case it would be great to think about its meaning from the perspective of singularities as well.

• If by an orbifold you mean a quotient of a, say, lorentzian manifold $(M,g)$ by a discrete group $\Gamma$ acting via isometries, then $M/\Gamma$ with the induced metric is locally isometric to $(M,g)$. So if $(M,g)$ satisfies Einstein's equations, so will the manifold of smooth points on $M/\Gamma$. Is your question then about what happens at the singular locus? – José Figueroa-O'Farrill Sep 23 '16 at 8:54
• General Relativity is a mathematical theory developed to model a physical phenomenon (gravity). If you want to ask about what happens at the singular locus, you are in a realm that is already outside the classical domain of GR; so if you want to get a reasonable answer you probably want to say something about what the singular points in your orbifold are supposed to be modelling. – Willie Wong Sep 23 '16 at 17:04
• Alternatively, if your interest is purely mathematical, you should say what aspects of Einstein's field equations (or their solutions) you want to reproduce on orbifolds. Then people may be able to come up with viable expressions. – Willie Wong Sep 23 '16 at 17:07