Kerr geodesics on conformal projection diagrams

Abstract: We provide a rigorous representation of timelike geodesics in the maximally extended Kerr spacetime, using projection diagrams introduced by Chruściel, Ölz, and Szybka. In contrast to standard conformal diagrams, projection diagrams allow for plotting general causal curves (not necessarily confined to the two-dimensional surface associated with the diagram), preserving their causal characteristics. Our analysis utilizes two systems of horizon-penetrating coordinates (both ingoing and outgoing) to ensure a regular description of geodesics crossing the event and Cauchy horizons. On the other hand, at most four causal blocks can be joined smoothly by a Kruskal-Szekeres-type construction proposed for projection diagrams. Consequently, depicting a trajectory spanning several causal blocks may require more than one diagram, as the conformal factor either vanishes or diverges at the block boundaries.