Spacetime Splitting, Admissible Coordinates and Causality

To confront relativity theory with observation, it is necessary to split spacetime into its temporal and spatial components. The timelike threading approach involves fundamental observers that are at rest in space; indeed, this (1+3) splitting implies restrictions on the gravitational potentials $(g_{\mu \nu})$. On the other hand, the spacelike slicing approach involves (3+1) splittings of any congruence of observers with corresponding restrictions on $(g^{\mu \nu})$. These latter coordinate conditions exclude closed timelike curves (CTCs) within any such coordinate patch. While the threading coordinate conditions can be naturally integrated into the structure of Lorentzian geometry and constitute the standard coordinate conditions in general relativity, this circumstance does not extend to the slicing coordinate conditions. From this viewpoint, the existence of CTCs is not, in principle, prohibited by classical general relativity.