The lightcone of G\"odel-like spacetimes

TL;DR

This paper explores the geometric and caustic properties of lightcones in G"odel-like spacetimes, revealing universal lensing effects and detailed caustic structures in these highly symmetric Lorentzian manifolds.

Contribution

It extends the analysis of lightcone structures and caustics from the G"odel universe to a broader family of G"odel-like spacetimes, uncovering universal lensing phenomena.

Findings

Quasi-periodic refocussing of cone generators observed in G"odel-like spacetimes.

Focal surfaces are null and generated by non-geodesic null curves.

Intrinsic invariants of the lightcone are finite at caustics.

Abstract

A study of the lightcone of the G\"odel universe is extended to the so-called G\"odel-like spacetimes. This family of highly symmetric 4-D Lorentzian spaces is defined by metrics of the form $ds^2=-(dt+H(x)dy)^2+D^2(x)dy^2+dx^2+dz^2$, together with the requirement of spacetime homogeneity, and includes the G\"odel metric. The quasi-periodic refocussing of cone generators with startling lens properties, discovered by Ozsv\'{a}th and Sch\"ucking for the lightcone of a plane gravitational wave and also found in the G\"odel universe, is a feature of the whole G\"odel family. We discuss geometrical properties of caustics and show that (a) the focal surfaces are two-dimensional null surfaces generated by non-geodesic null curves and (b) intrinsic differential invariants of the cone attain finite values at caustic subsets.