[Paper Review] Exact solutions of topologically massive gravity

David D. K. Chow, C.N. Pope+1 more|arXiv (Cornell University)|Jun 19, 2009

Black Holes and Theoretical Physics24 references5 citations

TL;DR

This paper presents a comprehensive classification of exact solutions in three-dimensional topologically massive gravity (TMG), identifying biaxially squashed AdS3 and Kundt spacetimes as central solution families. It establishes that algebraic types III and II Kundt solutions exist, and shows that all constant scalar invariant Kundt solutions are deformations of AdS3 or biaxially squashed AdS3, including AdS pp-waves, with most known solutions locally equivalent to these forms.

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. General techniques and classifications of solutions are reviewed. We show that biaxially squashed AdS3 solutions follow from assuming algebraic type D curvature for which the one-dimensional eigenspace of the traceless Ricci tensor is aligned with a Killing vector. We find the general Kundt solutions: spacetimes admitting an expansion-free geodesic null congruence. These provide the first examples of solutions with algebraic types III and II. We find that a Kundt solution of algebraic type D must be biaxial spacelike-squashed AdS3. We present explicitly the Kundt solutions for which all scalar polynomial curvature invariants are constant, and find that all Kundt solutions with algebraic types D, N and III have this property. These constant scalar invariant Kundt solutions are deformations of round or biaxial spacelike-squashed AdS3, and include AdS pp-waves. We also provide a comprehensive review of the literature, showing that most known solutions are locally equivalent to biaxially squashed AdS3 or AdS pp-waves. Contents

Motivation & Objective

• To systematically classify exact solutions in three-dimensional topologically massive gravity (TMG), a theory combining Einstein-Hilbert and Chern-Simons terms.
• To identify and characterize biaxially squashed AdS3 solutions through algebraic type D curvature and alignment with a Killing vector.
• To construct and analyze general Kundt solutions, which admit an expansion-free geodesic null congruence, and classify them by algebraic type.
• To determine the conditions under which scalar polynomial curvature invariants are constant in Kundt spacetimes.
• To unify and review existing solutions, showing that most known solutions are locally equivalent to biaxially squashed AdS3 or AdS pp-waves.

Proposed method

• Assumes algebraic type D curvature and aligns the one-dimensional eigenspace of the traceless Ricci tensor with a Killing vector to derive biaxially squashed AdS3 solutions.
• Applies the Kundt metric ansatz to construct solutions admitting a geodesic, expansion-free null congruence.
• Uses algebraic classification of curvature tensors to categorize solutions by Weyl and Ricci tensor types (e.g., III, II, D, N).
• Imposes the condition that all scalar polynomial curvature invariants are constant to identify special subclasses of Kundt solutions.
• Performs a comprehensive literature review to establish local equivalence of known solutions to biaxially squashed AdS3 or AdS pp-waves.
• Employs differential geometric techniques, including Ricci and Weyl tensor analysis, in three dimensions with a cosmological constant and Chern-Simons term.

Experimental results

Research questions

• RQ1 What are the general conditions under which biaxially squashed AdS3 solutions arise in topologically massive gravity?
• RQ2 Do Kundt solutions with algebraic types III and II exist in three-dimensional TMG, and what are their geometric properties?
• RQ3 Which Kundt solutions possess constant scalar polynomial curvature invariants, and how are they related to known backgrounds like AdS3 or pp-waves?
• RQ4 To what extent are known exact solutions in TMG locally equivalent to biaxially squashed AdS3 or AdS pp-waves?
• RQ5 How does the alignment of the traceless Ricci tensor's eigenspace with a Killing vector constrain the geometry of solutions?

Key findings

• Biaxially squashed AdS3 solutions are derived from the assumption of algebraic type D curvature with the traceless Ricci tensor's one-dimensional eigenspace aligned with a Killing vector.
• The first examples of Kundt solutions with algebraic types III and II are constructed, expanding the known class of exact solutions in TMG.
• A Kundt solution of algebraic type D must be biaxially spacelike-squashed AdS3, establishing a strong geometric constraint.
• All constant scalar invariant Kundt solutions—of types D, N, and III—are shown to be deformations of round or biaxial spacelike-squashed AdS3, including AdS pp-waves.
• The literature review confirms that most known exact solutions in TMG are locally equivalent to either biaxially squashed AdS3 or AdS pp-waves, unifying their classification.