Three-dimensional space-times

Hall, Graham; Morgan, Thomas A.; Perjés, Zoltán

doi:10.1007/bf00759150

Cited by 81 publications

(78 citation statements)

References 6 publications

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“…The Segre type [12] has two eigenvectors, a null k a and a space-like m a , with eigenvalues α and γ, respectively. The Segre type [3] has one null eigenvector k a with eigenvalue α [34,35]. For Segre type [11,1] the eigenvectors forms a Lorentz frame…”

Section: Segre Typementioning

confidence: 99%

Equivalence of three-dimensional spacetimes

Sousa¹,

Fonseca²,

Romero³

2008

Class. Quantum Grav.

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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.

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Section: Segre Typementioning

confidence: 99%

Equivalence of three-dimensional spacetimes

Sousa¹,

Fonseca²,

Romero³

2008

Class. Quantum Grav.

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“…The geometrical setting for the seminull coframe basis with varying conventions has been introduced previously [20][21][22][23] in the literature. The notation and most of the conventions used below follow those of Aliev and Nutku introduced in a spinor formulation of TMG.…”

Section: Geometrical Preliminarymentioning

confidence: 99%

Impulsive gravitational waves in general massive 3D gravity

Baykal

Dereli

2017

Phys. Rev. D

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“…To explore this relationship, we work with a three-dimensional Riemannian version of the Newman-Penrose formalism [8]; ultimately, we have in mind results analogous to the well-known Sachs equations [12] and the Goldberg-Sachs theorem [5] from four-dimensional Lorentzian (spacetime) geometry. We point out that a four-dimensional Riemannian version of the Goldberg-Sachs theorem (not employing the Newman-Penrose formalism) has already been undertaken in [1]; moreover, in three dimensions, the Riemannian NewmanPenrose formalism used here is essentially identical to the one for stationary spacetimes studied in [10] (see also [6]). Indeed, a secondary goal of this paper is to offer up the Newman-Penrose formalism as an avenue of pursuit in the study of vector flows in three-dimensional Riemannian geometry.…”

Section: Introductionmentioning

confidence: 99%

The Newman–Penrose formalism for Riemannian 3-manifolds

Aazami

2015

Journal of Geometry and Physics

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Abstract. We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field k with geodesic flow, if an integral curve of k is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if k is complete, hypersurfaceorthogonal, and satisfies Ric(k, k) ≥ 0, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length k with geodesic flow satisfies Ric(k, k) > 0, then k cannot be hypersurface-orthogonal, thus recovering a result in [7]. Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields k that satisfy the condition R(k, ·, ·, ·) = 0. When the scalar curvature is a nonzero constant, we show that a hypersurface-orthogonal unit vector field k satisfies R(k, ·, ·, ·) = 0 if and only if it is a Killing vector field.

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Three-dimensional space-times

Cited by 81 publications

References 6 publications

Equivalence of three-dimensional spacetimes

Equivalence of three-dimensional spacetimes

Impulsive gravitational waves in general massive 3D gravity

The Newman–Penrose formalism for Riemannian 3-manifolds

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