Gödel Type Metrics in Three Dimensions

TL;DR

The paper analyzes Gödel-type metrics in three dimensions, showing that stationary spacetimes with constant $g_{00}$ satisfy Einstein equations with a charged (or stiff) fluid, reducing dynamics to a single first-order PDE for the fluid velocity. It extends these metrics to Topologically Massive Gravity (TMG), deriving broad families of exact solutions under the condition that the two-dimensional background has constant Gaussian curvature, and identifies precise relations between the fluid parameters and TMG constants. The work also examines geometric flows, proving that Gödel-type metrics solve Ricci flow only for zero base curvature and solve Cotton flow only when the Cotton tensor vanishes, while establishing a closed tensor algebra that enables solving a wide class of higher-curvature theories with $F_{\mu u}=f_{\mu u}$ to all orders. Together, these results provide explicit solvable backgrounds in 3D gravity, illuminate the behavior under geometric flows, and connect Gödel-type spacetimes to string-inspired higher-curvature actions. The findings offer a unified framework for exact solutions across Einstein, TMG, Ricci/Cotton flows, and higher-curvature theories in three dimensions.

Abstract

We show that the G{\" o}del type Metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. There exists only one first order partial differential equation satisfied by the components of fluid's velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the G{\" o}del Type Metrics to solve the Ricci and Cotton flow equations. When the vector field $u^μ$ is a Killing vector field we finally show that the stationary G{\" o}del Type Metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.