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Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

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Abstract

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.

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References

  1. Aldrovandi, R., Beltrán Almeida, J.P., Pereira, J.G.: de Sitter special relativity. Class. Quantum Gravity 24, 1385–1404 (2007). arXiv:gr-qc/0606122v2

    Article  MATH  ADS  Google Scholar 

  2. Aldrovandi, R., Beltrán Almeida, J.P., Pereira, J.G.: Cosmological term and fundamental physics. Int. J. Mod. Phys. D13, 2241–2248 (2004). arXiv:gr-qc/0405104v1

    ADS  Google Scholar 

  3. Aldrovandi, R., Beltrán Almeida, J.P., Pereira, J.G.: A singular conformal spacetime. J. Geom. Phys. 56, 1042–1056 (2006). arXiv:gr-qc/0403099v2

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. da Rocha, R., Capelas de Oliveira, E.: AdS geometry, projective embedded coordinates and associated isometry groups. Int. J. Theor. Phys. 45, 562–575 (2006). arXiv:math-ph/0309040v2

    Article  Google Scholar 

  5. Di Francesco, P., Mathieu, P., Senechal, D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, Berlin (1996)

    Google Scholar 

  6. Coimbra-Araujo, C.H., da Rocha, R., Pedron, I.T.: Anti-de Sitter curvature radius constrained by quasars in brane-world scenarios. Int. J. Mod. Phys. D 14, 1883–1898 (2005). arXiv:astro-ph/0505132v4

    Article  MATH  ADS  Google Scholar 

  7. Kaku, M.: Strings, Conformal Fields, and M-Theory. Graduate Texts in Contemporary Physics. Springer, Berlin (2000)

    MATH  Google Scholar 

  8. Weinberg, S.: A model of leptons. Phys. Rev. Lett. 19, 1264 (1967)

    Article  ADS  Google Scholar 

  9. Barut, A.O., Bracken, A.J.: Zitterbewegung and the internal geometry of the electron. Phys. Rev. A 23, 2454–2462 (1981)

    ADS  MathSciNet  Google Scholar 

  10. Prasad, R.: De Sitter model for elementary particles. Il Nuovo Cimento A 44, 299–309 (1966)

    Article  ADS  Google Scholar 

  11. Prasad, R.: The de Sitter model for elementary particles with nonstatic frame. Nuovo Cimento A 52, 972 (1967)

    Article  ADS  Google Scholar 

  12. Börner, G., Dürr, H.P.: Classical and quantum fields in de Sitter space. Il Nuovo Cimento A 64, 669–712 (1969)

    Article  MATH  ADS  Google Scholar 

  13. Arcidiacono, G.: Relatività e Cosmologia. Veschi, Roma (1987)

    Google Scholar 

  14. Arcidiacono, G.: The de Sitter universe and mechanics. Gen. Relativ. Gravit. 8, 865 (1977)

    Article  MATH  ADS  Google Scholar 

  15. Capelas de Oliveira, E., da Rocha, R.: On conformal d’Alembert-like equation. Electron. J. Theor. Phys. 5, 21–32 (2008). arXiv:math-ph/0502011

    Google Scholar 

  16. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998). arXiv:hep-th/9711200

    MATH  ADS  MathSciNet  Google Scholar 

  17. Maldacena, J.M.: J. Int. Theor. Phys. 38, 1113 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Witten, E.: Anti-de Sitter spacetime and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998). arXiv:hep-th/9802150

    MATH  MathSciNet  Google Scholar 

  19. Susskind, L., Lindesay, J.: An Introduction to Black Holes, Information and the String Theory Revolution. The Holographic Universe. World Scientific, London (2005)

    MATH  Google Scholar 

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da Rocha, R., Capelas  Oliveira, E. Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes. Int J Theor Phys 48, 127–138 (2009). https://doi.org/10.1007/s10773-008-9788-9

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  • DOI: https://doi.org/10.1007/s10773-008-9788-9

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