The equivalence problem

Karlhede, A.

doi:10.1007/s10714-006-0292-3

Cited by 12 publications

(16 citation statements)

References 22 publications

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“…The development of computer algebra opened the way to the formulation of a procedure for testing equivalence of four-dimensional spacetimes in practice, that is, the Karlhede classification [3,4]. Finally, algorithms using both the NewmanPenrose spinor formalism and the algebraic classification of the irreducible parts of the curvature spinor were developed, enabling the implementation of the practical procedure in a computer algebra suite called classi [5,6], based on the computer algebra system for General Rela- * fcsousa@fisica.ufpb.br † jfonseca@fisica.ufpb.br ‡ cromero@fisca.ufpb.br tivity sheep [6,7,8,9,10].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Equivalence of three-dimensional spacetimes

Sousa¹,

Fonseca²,

Romero³

2008

Class. Quantum Grav.

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“…9, for a review) that the Cartan invariants completely characterize the spacetime locally (an explicit reconstruction from these invariants was given for Schwarzschild and some other solutions in [15]). The completeness of the Cartan invariants as a specification of the local geometry gives a very precise form to the remarks of Synge on the lines of "gravitational field = curvature of space-time", which Antoci et al call vague and appear to reject [2].…”

Section: Cartan Invariants and The Acceleration Invariantmentioning

confidence: 99%

“…Note that the value of each invariant at a given point of the manifold is independent of the coordinates used. The Cartan invariants for the Schwarzschild solution have been given in earlier work [15,17], and a general discussion of Petrov Type D vacua has been given in [18,19]. All type D vacua were explicitly classified, using Cartan invariants, by Åman [20].…”

Section: Cartan Invariants and The Acceleration Invariantmentioning

confidence: 99%

On singularities, horizons, invariants, and the results of Antoci, Liebscher and Mihich (Gen Relativ Gravit 38, 15 (2006) and earlier)

MacCallum

2006

Gen Relativ Gravit

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Antoci et al. have argued that the horizons of the boost-rotation, Kerr and Schwarzschild solutions are singular, having shown that a certain invariantly defined acceleration scalar blows up at the horizons. Their examples do not satisfy the usual definition of a singularity. It is argued that using the same term is seriously misleading and it is shown that such divergent functions are natural concomitants of regular horizons. In particular it is noted that the divergence is given by the special relativistic approximation to the overall metric. Earlier work on characterization of horizons by invariants is revisited, a new invariant criterion for them is proposed, and the relation of the acceleration invariant to the Cartan invariants, which are finite at the horizons and completely determine the spacetimes, is examined for the C-metric, Kerr and Schwarzschild cases. An appendix considers coordinate identifications at axes and horizons.

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“…First, it allows us to check whether a new solution to the Einstein equations is in fact already known. Consequently, it is also a method to study the metric equivalence problem, which is an alternative to the usual Cartan-Brans-Karlhede approach [37,38]. Second, it is of interest in obtaining a fully algorithmic characterization of the initial data which correspond to a given solution.…”

Section: Introductionmentioning

confidence: 99%