Spacetime splitting, admissible coordinates, and causality

Abstract

To confront relativity theory with observation, it is necessary to split spacetime into its temporal and spatial components. The (1+3) timelike threading approach involves restrictions on the gravitational potentials $(g_{\mu \nu})$, while the (3+1) spacelike slicing approach involves restrictions on $(g^{\mu \nu})$. These latter coordinate conditions protect chronology 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. We explore the influence of chronology violation on wave motion. In particular, we consider the propagation of radiation parallel to the rotation axis of stationary G\"odel-type universes characterized by parameters $\eta > 0$ and $\lambda > 0$ such that for $\eta < 1$ ($\eta >1$) chronology is protected (violated). We show that in the WKB approximation such waves can freely propagate only when chronology is protected.