Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Hyperbolic Metrics on Riemann Surfaces and Space-Like CMC-1 Surfaces in de Sitter 3-Space Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

Conformally Standard Stationary SpaceTimes and Fermat Metrics

verfasst von : Miguel Angel Javaloyes

Erschienen in: Recent Trends in Lorentzian Geometry

Verlag: Springer New York

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this chapter, we collect several results for conformally standard stationary spacetimes \((S \times \mathbb{R},g)\) obtained in terms of a Finsler metric of Randers type on the orbit manifold S that we call Fermat metric. This metric is obtained by applying the relativistic Fermat principle and it turns out that it encodes all the causal aspects of the space time.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel The Geometry of Collapsing Isotropic Fluids
Nächstes Kapitel Can We Make a Finsler Metric Complete by a Trivial Projective Change?
Literatur
1.
Abbondandolo, A., Schwarz, M.: A smooth pseudo-gradient for the Lagrangian action functional. Adv. Nonlinear Stud. 9(4), 597–623 (2009)MathSciNetMATH
2.
Abramowicz, M., Carter, B., Lasota, J.-P.: Optical reference geometry for stationary and static dynamics. Gen. Relativ. Gravit. 20, 1172–1183 (1988)MathSciNetCrossRef
3.
Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics. Springer, New York (2000)MATHCrossRef
4.
Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66(3), 377–435 (2004)MathSciNetMATH
5.
Bartolo, R., Candela, A.M., Caponio, E.: Normal geodesics connecting two non-necessarily spacelike submanifolds in a stationary spacetime. Adv. Nonlinear Stud. 10(4), 851–866 (2010)MathSciNetMATH
6.
Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, 2nd edn. Marcel Dekker, New York (1996)
7.
Bernal A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Comm. Math. Phys. 243(3), 461–470 (2003)MathSciNetMATHCrossRef
8.
Biliotti L., Javaloyes, M.Á.: t-periodic light rays in conformally stationary spacetimes via Finsler geometry. Houston J. Math. 37(1), 127–146 (2011)
9.
Caponio, E.: Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold, these proceedings, Recent Trends in Lorentzian Geometry, Springer Proceedings in Mathematics & Statistics (2013)
10.
Caponio, E.: The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 26(2), 265–274 (2010)
11.
Caponio, E., Germinario, A., Sánchez, M.: Geodesics on convex regions of stationary spacetimes and Finslerian Randers spaces. arXiv:1112.3892v1 [math.DG] (2011)
12.
Caponio, E., Javaloyes, M.A.: A remark on the Morse Theorem about infinitely many geodesics between two points. arXiv:1105.3923v1 [math.DG] (2011)
13.
Caponio, E., Javaloyes, M.Á., Masiello, A.: Finsler geodesics in the presence of a convex function and their applications. J. Phys. A 43(13), 135207, 15 (2010)
14.
Caponio, E., Javaloyes, M.Á., Masiello, A.: Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3), 857–876 (2010)MathSciNetMATHCrossRef
15.
Caponio, E., Javaloyes, M.Á., Masiello A.: On the energy functional on Finsler manifolds and applications to stationary spacetimes. Math. Ann. 351(2), 365–392 (2011)MathSciNetMATHCrossRef
16.
Caponio, E., Javaloyes, M.A., Sánchez, M.: On the interplay between Lorentzian Causality and Finsler metrics of Randers type. Rev. Mat. Iberoamericana 27(3), 919–952 (2011)MATHCrossRef
17.
Dirmeier, A., Plaue, M., Scherfner, M.: Growth conditions, Riemannian completeness and Lorentzian causality. J. Geom. Phys. 62(3), 604–612 (2011)MathSciNet
18.
Dowker J., Kennedy, G.: Finite temperature and boundary effects in static space-times. J. Phys. A: Math. Gen. 11, 895–920 (1978)MathSciNetCrossRef
19.
Flores J.L., Herrera, J.: The c-boundary construction of spacetimes: application to stationary kerr spacetime, these proceedings, Recent Trends in Lorentzian Geometry, Springer Proceedings in Mathematics & Statistics (2013).
20.
Flores, J.L., Herrera, J., Sánchez, M.: Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds. arXiv:1011.1154v2 [math.DG], in Mem. Amer. Math. Soc. (to appear)
21.
Flores, J.L., Herrera, J., Sánchez, M.: On the final definition of the causal boundary and its relation with the conformal boundary. Adv. Theor. Math. Phys. 15(4), 991–1058 (2011)MathSciNetMATH
22.
Fortunato, D., Giannoni, F., Masiello, A.: A Fermat principle for stationary space-times and applications to light rays. J. Geom. Phys. 15(2), 159–188 (1995)MathSciNetMATHCrossRef
23.
Giambò, R., Giannoni, F., Piccione, P.: Genericity of nondegeneracy for light rays in stationary spacetimes. Comm. Math. Phys. 287(3), 903–923 (2009)MathSciNetMATHCrossRef
24.
Giambò, R., Javaloyes, M.A.: Addendum to: Genericity of nondegeneracy for light rays in stationary spacetimes. Comm. Math. Phys. 295(1), 289–291 (2010)MathSciNetMATHCrossRef
25.
Gibbons, G., Perry, M.: Black holes and thermal green functions. Proc. Roy. Soc. London, Ser. A 358, 467–494 (1978)MathSciNet
26.
Gibbons, G.W., Herdeiro, C.A.R., Warnick, C.M., Werner, M.C.: Stationary metrics and optical Zermelo-Randers-Finsler geometry. Phys. Rev. D 79(4), 044022, 21 (2009)
27.
Ingarden, R.S.: On the geometrically absolute optical representation in the electron microscope. Trav. Soc. Sci. Lett. Wrocław. Ser. B. (45), 60 (1957)
28.
Javaloyes, M.A., Lichtenfelz, L., Piccione, P.: Almost isometries of non-reversible metrics with applications to stationary spacetimes. arXiv:1205.4539v1 [math.DG] (2012)
29.
Javaloyes, M.A., Sánchez, M.: A note on the existence of standard splittings for conformally stationary spacetimes. Class. Quantum Grav. 25(16), 168001, 7 (2008)
30.
Javaloyes, M.A., Sánchez, M.: On the definition and examples of Finsler metrics. arXiv:1111.5066v2 [math.DG] (2011), in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)
31.
Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Wiley Classics Library, vol. I. Wiley, New York (1996) Reprint of the 1963 original, A Wiley-Interscience Publication
32.
Kovner, I.: Fermat principle in gravitational fields. Astrophys. J. 351, 114–120 (1990)CrossRef
33.
Landau, L., Lifshitz, E.: The classical theory of fields, 2nd edn. Pergamon Press, Addison-Wesley, Oxford, Reading, MA (1962)MATH
34.
Levi-Civita, T.: La teoria di Einstein e il principio di Fermat. Nuovo Cimento 16, 105–114 (1918)MATHCrossRef
35.
Levi-Civita, T.: The Absolute Differential Calculus. Blackie & Son Limited, London (1927)MATH
36.
Lichnerowicz, A.: Théories relativistes de la gravitation et de l’électromagnétisme. Relativité générale et théories unitaires. Masson et Cie, Paris (1955)MATH
37.
Lichnerowicz, A., Thiry, Y.: Problèmes de calcul des variations liés à la dynamique classique et à la théorie unitaire du champ. C. R. Acad. Sci. Paris 224, 529–531 (1947)MathSciNetMATH
38.
Masiello, A.: An alternative variational principle for geodesics of a Randers metric. Adv. Nonlinear Stud. 9(4), 783–801 (2009)MathSciNetMATH
39.
Masiello, A., Piccione, P.: Shortening null geodesics in Lorentzian manifolds. Applications to closed light rays. Differ. Geom. Appl. 8(1), 47–70 (1998)MathSciNetMATH
40.
Matsumoto, M.: On C-reducible Finsler spaces. Tensor (N.S.) 24, 29–37 (1972) Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, Vol. I
41.
Matsumoto, M.: On Finsler spaces with Randers’ metric and special forms of important tensors. J. Math. Kyoto Univ. 14, 477–498 (1974)MathSciNetMATH
42.
Mawhin, J., Willem, M.: Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, vol. 74. Springer, New York (1989)
43.
44.
Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes. In: Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., pp. 299–358. Eur. Math. Soc., Zürich (2008)
45.
46.
O’Neill, B.: Semi-Riemannian geometry. Pure and Applied Mathematics, vol. 103. Academic [Harcourt Brace Jovanovich Publishers], New York (1983) With applications to relativity
47.
48.
Perlick, V.: On Fermat’s principle in general relativity. II. The conformally stationary case. Class. Quantum Grav. 7(10), 1849–1867 (1990)MathSciNetMATHCrossRef
49.
50.
Pham, Q.: Inductions électromagnétiques en rélativité général et principe de Fermat. Arch. Ration. Mech. Anal. 1, 54–80 (1957)MathSciNetCrossRef
51.
Randers, G.: On an asymmetrical metric in the fourspace of general relativity. hys. Rev. 59(2), 195–199 (1941)
52.
Sánchez, M.: Some remarks on causality theory and variational methods in Lorenzian manifolds. Conf. Semin. Mat. Univ. Bari (265), ii+12 (1997)
53.
Shibata, C., Shimada, H., Azuma, M., Yasuda, H.: On Finsler spaces with Randers’ metric. Tensor (N.S.) 31(2), 219–226 (1977)
54.
Synge, J.: An alternative treatment of Fermat’s principle for a stationary gravitational field. Philos. Mag. J. Sci. 50, 913–916 (1925)MATHCrossRef
55.
56.
Weyl, H.: Zur gravitationstheorie. Ann. Phys. (Berlin) 54, 117–145 (1917)
57.
58.
Zaustinsky, E.: Spaces with non-symmetric distance. Mem. Amer. Math. Soc. 34, 1–91 (1959)MathSciNet
Metadaten
Titel
Conformally Standard Stationary SpaceTimes and Fermat Metrics
verfasst von
Miguel Angel Javaloyes
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Recent Trends in Lorentzian Geometry

Print ISBN: 978-1-4614-4896-9

Electronic ISBN: 978-1-4614-4897-6

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-4897-6_9