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## Main Question or Discussion Point

This has bothered me for some time. In the ADM formulation, we foliate spacetime into 3+1 dimensions by creating 3 dimensional hypersurfaces via ##T = constant## along the worldline of some observer whose proper time is ##T##. This allows us to write dynamical equations for the evolution of some initial 3D spacelike hypersurface ##\Sigma##.

However, this only works for a general flat or hyperbolic manifold, because geodesics which are initially normal to ##\Sigma## will never cross. For an elliptic spacetime, those geodesics would eventually cross each other, such that the map ##\Sigma_T \rightarrow \Sigma_{T'}## would fail to be one-to-one. This means that solutions to Einstein's Equations found via the ADM formulation cannot have closed timelike curves, since every geodesic can only intersect each hypersurface once.

Therefore, what limitations does the ADM formulation of GR impose on the full set of solutions of Einstein's Equations? If we can't foliate an elliptic manifold, then doesn't that mean that we won't find any elliptic spacetime geometries using the ADM method?

However, this only works for a general flat or hyperbolic manifold, because geodesics which are initially normal to ##\Sigma## will never cross. For an elliptic spacetime, those geodesics would eventually cross each other, such that the map ##\Sigma_T \rightarrow \Sigma_{T'}## would fail to be one-to-one. This means that solutions to Einstein's Equations found via the ADM formulation cannot have closed timelike curves, since every geodesic can only intersect each hypersurface once.

Therefore, what limitations does the ADM formulation of GR impose on the full set of solutions of Einstein's Equations? If we can't foliate an elliptic manifold, then doesn't that mean that we won't find any elliptic spacetime geometries using the ADM method?