An algorithm for determining whether a space-time is homothetic

Koutras, Andreas; Skea, Jim

doi:10.1016/s0010-4655(98)00132-5

Cited by 11 publications

(12 citation statements)

References 11 publications

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“…In the case of stationary black holes, the Cartan invariants have been shown to detect the event horizon [13,25], thereby providing an alternative set of invariants which are easier to compute than the related SPIs. We believe that by exploring the relationship between J D and the Cartan invariants, additional conformally covariant invariants may be produced and this will give insight into the equivalence of spacetimes under the conformal group [28][29][30].…”

Section: Discussionmentioning

confidence: 99%

Curvature invariant characterization of event horizons of four-dimensional black holes conformal to stationary black holes

McNutt

2017

Phys. Rev. D

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We introduce three approaches to generate curvature invariants that transform covariantly under a conformal transformation of a four dimensional spacetime. For any black hole conformally related to a stationary black hole, we show how a set of conformally covariant invariants can be combined to produce a conformally covariant invariant that detects the event horizon of the conformally related black hole. As an application we consider the rotating dynamical black holes conformally related to the Kerr-NUT-(Anti)-de Sitter spacetimes and construct an invariant that detects the conformal Killing horizon along with a second invariant that detects the conformal stationary limit surface.In addition, we present necessary conditions for a dynamical black hole to be conformally related to a stationary black hole and apply these conditions to the ingoing Kerr-Vaidya and Vaidya black hole solutions to determine if they are conformally related to stationary black holes for particular choices of the mass function. While two of the three approaches cannot be generalized to higher dimensions, we discuss the existence of a conformally covariant invariant that will detect the event horizon for any higher dimensional black hole conformally related to a stationary black hole which admits at least two conformally covariant invariants, including all vacuum spacetimes.

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Section: Discussionmentioning

confidence: 99%

Curvature invariant characterization of event horizons of four-dimensional black holes conformal to stationary black holes

McNutt

2017

Phys. Rev. D

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“…CA can check (Joly 1987 {Sheep}) or find a metric’s isometry group (Karlhede and MacCallum 1982 ; Araujo and Skea 1988a , b ; Araujo et al. 1992 ; Grebot and Wolf 1994 {Sheep, Reduce}) and other symmetries such as homothety (McIntosh and Steele 1991 ; Koutras and Skea 1998 ; Vaz and Collinson 1993 {Sheep, Reduce}) or conformal motions. Some applications are described in McLenaghan and van den Bergh ( 1993 ) {Maple}, O’Connor and Prince ( 1998 ) and Hickman and Yazdan ( 2017 ) {Dimsym, Exterior} and in the next subsection.…”

Section: Applicationsmentioning

confidence: 99%

Computer algebra in gravity research

MacCallum

2018

Living Rev Relativ

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The complicated nature of calculations in general relativity was one of the driving forces in the early development of computer algebra (CA). CA has become widely used in gravity research (GR) and its use can be expected to grow further. Here the general nature of computer algebra is discussed, along with some aspects of CA system design; features particular to GR’s requirements are considered; information on packages for CA in GR is provided, both for those packages currently available and for their predecessors; and applications of CA in GR are outlined.

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“…Although the Karlhede algorithm is more efficient than the original procedures proposed by Cartan or Brans [4], it may need to go as far as to R 7 [21], and as a consequence, for some spacetimes, long complicated calculations are required, which usually need computer support, e.g., using the programme CLASSI [1], or the Maple-based GRTensor programme [18]. Subsequently, it has been shown how the Karlhede algorithm can be exploited to determine the structure of the isometry group of the spacetime, as well as subclasses within the spacetime which have additional isometries [22]; more recently the scheme has been the basis for an algorithm which determines whether a spacetime admits a homothetic Killing vector [25].…”

Section: Karlhede Algorithm For Invariant Classification Of Metricsmentioning

confidence: 99%

“…Koutras and Skea [25] have given an algorithm to determine whether a spacetime admits a homothetic vector 11 . This algorithm was designed to exploit an invariant classification using the Karlhede algorithm, but it is easy to see that it can be used for any invariant classification, as given below: (1) Use an existing classification algorithm to provide an invariant classification of the spacetime, R n .…”

Section: Homothetic Killing Vector In Intrinsic Coordinate Versionmentioning

confidence: 99%

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Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics

2009

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Abstract.Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the * This is an expanded version of the publication [3]. Also some typos and numerical coefficients have been corrected compared to the published version. 1 different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we are able to obtain complete agreement with the results obtained in intrinsic coordinates. In particular, we have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.

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An algorithm for determining whether a space-time is homothetic

Cited by 11 publications

References 11 publications

Curvature invariant characterization of event horizons of four-dimensional black holes conformal to stationary black holes

Curvature invariant characterization of event horizons of four-dimensional black holes conformal to stationary black holes

Computer algebra in gravity research

Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics

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