Classification of solutions in topologically massive gravity

TL;DR

The paper analyzes exact solutions of three-dimensional gravity with a cosmological constant and a gravitational Chern–Simons term (TMG) through the lens of algebraic curvature classification. It proves that any Petrov–Segre type D solution of TMG must be biaxially squashed AdS$_3$ and identifies two explicit families, timelike- and spacelike-squashed AdS$_3$, along with AdS pp-waves as central solution classes. A broad literature review shows that most non-Einstein TMG solutions are locally equivalent to these two families, unifying disparate coordinate descriptions under a common geometric framework. The work also sets the stage for extending the classification to other Petrov–Segre types (II, III) via Kundt spacetimes in a follow-up study.

Abstract

We study exact solutions of three-dimensional gravity with a cosmological constant and a gravitational Chern-Simons term: the theory known as topologically massive gravity. After reviewing the algebraic classification, we show that if a solution has curvature of algebraic type D, then it is biaxially squashed AdS_3. Applying the classification, we provide a comprehensive review of the literature, showing that most known solutions are locally equivalent to biaxially squashed AdS_3 or to AdS pp-waves.