G\"odel Type Metrics in Three Dimensions

TL;DR

This paper explores G"odel type metrics in three dimensions, demonstrating their solutions to Einstein-perfect fluid, topologically massive gravity, Ricci and Cotton flow equations, and certain electromagnetic-curvature coupled field equations, under various conditions.

Contribution

It establishes the conditions under which G"odel type metrics solve multiple gravitational field equations in three dimensions, including Einstein, topologically massive gravity, and electromagnetic-curvature models.

Findings

G"odel type metrics satisfy Einstein-perfect fluid equations with arbitrary 2D backgrounds.

They also solve topologically massive gravity equations with constant negative curvature backgrounds.

Stationary G"odel metrics with Killing vectors solve equations with arbitrary electromagnetic and curvature interactions.

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 $u^{\mu}$ 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.