# Conformal geodesics in spherically symmetric vacuum spacetimes with Cosmological constant

## TL;DR

This paper studies conformal geodesics in Schwarzschild-de Sitter and Schwarzschild-anti de Sitter spacetimes, showing how initial data can generate global coordinate systems or reveal limitations in their coverage.

## Contribution

It demonstrates the conditions under which conformal geodesics form caustic-free congruences covering maximal extensions, and distinguishes between the global coordinate applicability in different spacetime families.

## Key findings

- Conformal geodesics can cover the entire maximal extension in Schwarzschild-de Sitter.
- Global conformal Gaussian systems exist for Schwarzschild-de Sitter but not for Schwarzschild-anti de Sitter.
- Initial data can be chosen to avoid caustics in the geodesic congruences.

## Abstract

An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild-de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the case of the Schwarzschild-anti de Sitter family, the natural parameter of the curves only covers a restricted time span so that these global conformal Gaussian systems do not exist.