Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes

Bernal, Antonio; Sánchez, Miguel

doi:10.1007/s00220-005-1346-1

Cited by 350 publications

(435 citation statements)

References 6 publications

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“…(For background material on globally hyperbolic spacetimes the reader may consult, e.g., Hawking and Ellis [11]. The fact that the original definition of global hyperbolicity is equivalent to the existence of a foliation into smooth Cauchy surfaces was completely proven only recently by Bernal and Sánchez [12].) Then M can be written as a product of a 3-dimensional manifold S, which serves as the prototype for each Cauchy surface, and a time-axis,…”

Section: A Results From Morse Theorymentioning

confidence: 99%

“…On M + , the coordinates ϕ and ϑ range over S 2 , the coordinate t ranges over R, and the coordinate r ranges over an open interval which is diffeomorphic to R; hence M + ≃ S 2 × R 2 . From now on we will consider the spacetime (M + , g), where g denotes the restriction of the Kerr-Newman metric (8) with (10) to the domain M + given by (12). For the sake of brevity, we will refer to (M + , g) as to the exterior Kerr-Newman spacetime.…”

Section: Centrifugal and Coriolis Force In The Kerr-newman Spacementioning

confidence: 99%

See 1 more Smart Citation

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

Haße

Perlick

2006

Journal of Mathematical Physics

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Consider, in the domain of outer communication M+ of a Kerr-Newman black hole, a point p (observation event) and a timelike curve γ (worldline of light source). Assume that γ (i) has no past end-point, (ii) does not intersect the caustic of the past light-cone of p, and (iii) goes neither to the horizon nor to infinity in the past. We prove that then for infinitely many positive integers k there is a past-pointing lightlike geodesic λ k of (Morse) index k from p to γ, hence an observer at p sees infinitely many images of γ. Moreover, we demonstrate that all lightlike geodesics from an event to a timelike curve in M+ are confined to a certain spherical shell. Our characterization of this spherical shell shows that in the Kerr-Newman spacetime the occurrence of infinitely many images is intimately related to the occurrence of centrifugal-plus-Coriolis force reversal.

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Section: A Results From Morse Theorymentioning

confidence: 99%

Section: Centrifugal and Coriolis Force In The Kerr-newman Spacementioning

confidence: 99%

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

Haße

Perlick

2006

Journal of Mathematical Physics

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“…There has been some imprecision in the literature concerning the proof of the smoothness of the splitting of globally hyperbolic spacetimes. See [27,28,29] for a survey of this question, and a complete proof of the smoothness of the splitting M ≈ S × R.…”

Section: N5 Domains Of Dependence and Geodesic Laminationsmentioning

confidence: 99%

Notes on a paper of Mess

et al. 2007

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“…There exists a global time function t on M. The level surfaces Σ t , t ∈ R, of the function t define a foliation of M, all Σ t being Cauchy surfaces and homemorphic to a given smooth 3−manifold Σ (see Geroch [25]). Geroch's theorem does not say anything about the regularity of the leaves Σ t ; the time function is only proved to be continuous and they are thus simply understood as topological submanifolds of M. A regularization procedure for the time function can be found in [8], [9]. In the concrete cases which we consider in this paper the time function is smooth and all the leaves are diffeomorphic to Σ.…”

Section: Spin Structuresmentioning

confidence: 99%

Creation of fermions by rotating charged black holes

Häfner¹

2009

Mémoires Soc. Math. France

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:This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the space-time outside the star is given by the Kerr-Newman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic behavior of certain timelike geodesics in the Kerr-Newman metric. The Dirac equation is then written using coordinates and a Newman-Penrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields in a globally hyperbolic space-time is described. We formulate and prove a theorem about the Hawking effect in this setting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result for solutions of a characteristic Cauchy problem for Dirac fields in the Kerr-Newman space-time. In an appendix we construct explicitly a Penrose compactification of block I of the Kerr-Newman space-time based on simple null geodesics. Résumé :Ce travail est dédiéà l'étude mathématique de l'effet Hawking pour des fermions dans le cadre de l'effondrement d'uneétoile chargée en rotation. On démontre qu'un observateur localisé loin de l'étoile et au repos par rapport aux coordonnées de BoyerLindquist observe l'émergence d'unétat thermal quand son temps propre tend vers l'infini. On introduit d'abord un modèle de l'effondrement de l'étoile. On suppose que l'espace-tempsà l'extérieur de l'étoile est donné par la métrique de Kerr-Newman. Les hypothèses sur le comportement asymptotique de la surface de l'étoile sont inspirées par le comportement asymptotique de certaines géodésiques de type temps dans la métrique de Kerr-Newman. L'équation de Dirac est alorsécrite en utilisant des coordonnées et une tétrade de Newman-Penrose adaptésà l'effondrement. Ce système de coordonnées et cette tétrade sont basés sur des géodésiques qu'on appelle des géodésiques simples isotropes. La quantification des champs de Dirac dans un espace-temps globalement hyperbolique est décrite. On formule un théorème sur l'effet Hawking dans ce cadre. La preuve du théorème contient une estimation de vitesse minimale pour les champs de Dirac légèrement plus forte que les estimations usuelles ainsi qu'un résultat d'existence et d'unicité pour les solutions d'un problème caractéristique pour les champs de Dirac dans l'espace-temps de Kerr-Newman. Dans un appendice, nous construisons explicitement la compactification de Penrose du bloc I de l'espace-temps de Kerr-Newman qui est basée sur les géodésiques simples isotropes.2000 Mathematics Subject Classification : 35P25, 35Q75, 58J45, 83C47, 8...

show abstract

Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes

Cited by 350 publications

References 6 publications

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

A Morse-theoretical analysis of gravitational lensing by a Kerr-Newman black hole

Notes on a paper of Mess

Creation of fermions by rotating charged black holes

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