Rotating Universes from the Gödel Solution to Modern Modified Gravity

The Gödel universe remains one of the sharpest exact spacetime tests of general relativity. It is homogeneous, stationary, filled by rotating pressureless matter and a negative cosmological constant, and it admits closed timelike curves through every event. These propertiesmake it simultaneously a usefulmodel, a counterexample to overly simple notions of cosmic time, and a benchmark for alternative theories of gravity. This review surveys the main line of development from Gödel’s 1949 solution and the homogeneous Gödel-type metrics to expanding rotating cosmologies, Einstein–Cartan and gauge-gravity models, string-inspired and supergravity solutions, and recent modified-gravity constructions. Particular attention is given to the distinction between rotation and causality violation, to observational limits on global vorticity and shear, and to the status of stability analyses. The central lesson is conservative: cosmic rotation is a legitimate relativistic degree of freedom, but the causal pathology of the original G¨odel solution is not generic. Whether chronology is violated depends on the global metric functions, matter sector, energy conditions, and boundary conditions of the theory under consideration.