Equivalence of three-dimensional spacetimes

Abstract

A solution to the equivalence problem in three-dimensional gravity is given and a practically useful method to obtain a coordinate invariant description of local geometry is presented. The method is a nontrivial adaptation of Karlhede invariant classification of spacetimes of general relativity. The local geometry is completely determined by the curvature tensor and a finite number of its covariant derivatives in a frame where the components of the metric are constants. The results are presented in the framework of real two-component spinors in three-dimensional spacetimes, where the algebraic classifications of the Ricci and Cotton-York spinors are given and their isotropy groups and canonical forms are determined. As an application we discuss Gödel-type spacetimes in three-dimensional General Relativity. The conditions for local space and time homogeneity are derived and the equivalence of three-dimensional Gödel-type spacetimes is studied and the results are compared with previous works on four-dimensional Gödel-type spacetimes.

On scalar curvature invariants in three dimensional spacetimes

“…We are interested in constructing a minimal set of algebraically independent scalar curvature invariants formed by the contraction of the Riemann (Ricci) tensor and its covariant derivatives up to some maximum order of differentiation q in 3D Lorentzian spacetimes. The question of the bound (maximal value) for q in 3D was addressed in [17]. In 3D spacetimes, the Weyl tensor vanishes and the canonical frame of the Karlhede algorithm [18] is aligned with principal directions of the Ricci tensor rather than the Weyl tensor.…”

“…Hence the equivalence method suggests a possible approach to determine a basis of all scalar curvature invariants. The equivalence problem in 3D was first considered in [17]. The maximal order of covariant derivative required for the invariant classification of a 3D Lorentzian pseudo-Riemannian manifold (q ≤ 5 [19]) is relevant in determining the worst case scenarios for implementing the equivalence algorithm.…”

“…It may be possible to provide a minimal basis on a case by case basis; some cases may have a smaller q and will then have fewer algebraically independent scalar invariants. The results have been applied to the equivalence of 3D Godeltype spacetimes [17,22]) and Segre type {(1, 1), 1} spacetimes with no isotropy group [12].…”

Polarization modes of gravitational waves in three-dimensional massive gravities

“…Since the higher-order Einstein equation becomes a lower-order equation when using the linearized Ricci tensor R L instead of a metric tensor h , this work will provide another approach in addition to the conventional metric-perturbation theory. Let us first introduce a triad 1 [7][8][9] of real vectors fk; n; mg which are related to the Cartesian tetrad vectors fe t ; e x ; e z g in three dimensions with metric signature ðÀ; þ; þÞ as…”

“…Therefore, the Riemann tensor with six independent components can be decomposed into the Ricci tensor and Ricci scalar. On the other hand, by using the formalism of real two-component spinors [9], the Ricci spinor È ABCD can be expressed in terms of the Ricci tensor as…”