Osaka Journal of Mathematics

Perfect fluid spacetimes with harmonic generalized curvature tensor

Carlo Alberto Mantica, Uday Chand De, Young Jin Suh, and Luca Guido Molinari

Full-text: Open access

Abstract

We show that $n$-dimensional perfect fluid spacetimes with diver\-gen\-ce-free conformal curvature tensor and constant scalar curvature are generalized Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid Yang pure space is a GRW spacetime. We also prove that perfect fluid spacetimes with harmonic generalized curvature tensor are, under certain conditions, GRW spacetimes. As particular cases, perfect fluids with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensor are GRW spacetimes. Finally, we explore some physical consequences of such results.

Article information

Source
Osaka J. Math., Volume 56, Number 1 (2019), 173-182.

Dates
First available in Project Euclid: 16 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1547607633

Mathematical Reviews number (MathSciNet)
MR3908784

Zentralblatt MATH identifier
07055406

Subjects
Primary: 53B30: Lorentz metrics, indefinite metrics 53B50: Applications to physics
Secondary: 53C80: Applications to physics 83C15: Exact solutions

Citation

Mantica, Carlo Alberto; Chand De, Uday; Suh, Young Jin; Molinari, Luca Guido. Perfect fluid spacetimes with harmonic generalized curvature tensor. Osaka J. Math. 56 (2019), no. 1, 173--182. https://projecteuclid.org/euclid.ojm/1547607633


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References

  • L. Alías, A. Romero and M. Sánchez: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995), 71–84.
  • L. Alías, A. Romero and M. Sánchez: Compact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes; in Geometry and Topology of Submanifolds VII, River Edge NJ, USA, World Scientific, Publ., 1995, 67–70.
  • B-Y.Chen: A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 46 (2014), 1833, 5pp.
  • J.K. Beem, P.E. Ehrlich and K.L. Easley: Global Lorentzian Geometry, Pure and Applied Mathematics 202, 2nd ed., Marcel Dekker, New York, 1996.
  • M.C. Chaki and R.K. Maity: On quasi Einstein manifolds, Publ. Math. Debrecen 57 (2000), 257–306.
  • J. Chojnacka-Dulas, R. Deszcz, M. GŽogowska and M. Prvanović: On warped product manifolds satisfying some curvature conditions, J. Geom. Phys. 74 (2013), 328–341.
  • A.A. Coley: Fluid spacetimes admitting a conformal Killing vector parallel to the velocity vector, Classical Quantum Gravity 8 (1991), 955–968.
  • A. De, C. Özgür and U.C. De: On conformally flat Pseudo-Ricci Symmetric Spacetimes, Internat J. Theoret. Phys. 51 (2012), 2878–2887.
  • F. Defever and R. Deszcz: On semi Riemannian manifolds satisfying the condition R.R=Q(S,R); in Geometry and Topology of Submanifolds, III, World Scientific Publ., Singapore, 1991, 108–130.
  • R. Deszcz: On conformally flat Riemannian manifolds satisfying certain curvature conditions, Tensor (N.S.) 49 (1990), 134–145.
  • R. Deszcz, F. Dillen, L. Verstraelen and L. Vrancken: Quasi-Einstein totally real submanifolds of the nearly Kahler 6-sphere, Tohoku Math. J. 51 (1999), 461–478.
  • R. Deszcz, M. GŽogowska, M. Hotloś and Z. Sentürk: On certain quasi-Einstein semisymmetric hypersurfaces, Annu. Univ. Sci. Budapest Eötvös Sect. Math. 41 (1998), 151–164.
  • R. Deszcz, M. Hotloś and Z. Sentürk: Quasi-Einstein hypersurfaces in semi-Riemannian space forms, Colloq. Math. 89 (2001), 81–97.
  • A. Gebarowski: On nearly conformally symmetric warped product spacetimes, Soochow J. Math. 20 (1994), 61–75.
  • A. Gebarowski: Doubly warped products with harmonic Weyl conformal curvature tensor, Colloq. Math. 67 (1994), 73–89.
  • B.S. Guilfoyle and B.C. Nolan: Yang's gravitational theory, Gen. Relativity Gravitation 30 (1998), 473–495.
  • M. Gutiérrez and B. Olea: Global decomposition of a Lorentzian manifold as a generalized Robertson-Walker space, Differ. Geom. Appl. 27 (2009), 146–156.
  • S. Hervik, M. Ortaggio and L. Wylleman: Minimal tensors and purely electric and magnetic spacetimes of arbitrary dimensions, Classical Quantum Gravity 30 (2013), 165014, 50pp.
  • S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, Vol 1, Interscience, New York, 1963.
  • D. Lovelock and H. Rund: Tensors, Differential Forms and Variational Principles, Reprinted Edition, Dover, 1988.
  • C.A. Mantica and L.G. Molinari: A second-order identity for the Riemann tensor and applications, Colloq. Math. 122 (2011), 69–82.
  • C.A. Mantica and L.G. Molinari: Weakly Z-Symmetric manifolds, Acta Math. Hungar. 135 (2012), 80–96.
  • C.A. Mantica, L.G. Molinari and U.C. De: A condition for a perfect fluid spacetime to be a generalized Robertson-Walker spacetime, J. Math. Phys. 57 (2016), 022508, 6pp.
  • C.A. Mantica and Y.J. Suh: Pseudo Z-symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9 (2012), 1250004, 21pp.
  • C.A. Mantica and Y.J. Suh: Recurrent Z-forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys. 9 (2012),1250059, 26pp.
  • C.A. Mantica, Y.J. Suh and U.C. De: A note on generalized Robertson-Walker spacetimes, Int. J. Geom. Methods Mod. Phys. 13 (2016), 1650079, 9pp.
  • C.A. Mantica and L.G. Molinari: Generalized Robertson-Walker spacetimes, a survey, Int. J. Geom. Methods Mod. Phys. 14 (2017), 1730001, 27pp.
  • J. Mikeš and L. Rachůnek: Torse-forming vector fields in T-symmetric Riemannian spaces; in Steps in Differential Geometry, Proceedings of the Colloquium in Differential Geometry, Debrecen, July 25–30, 2000 (Debrecen, Hungary), edited by L. Kozma, P.T. Nagy and L. Tamassy, 219–229.
  • B. O'Neill: Semi Riemannian Geometry with applications to Relativity, Academic Press, New York, 1983.
  • M.M. Postnikov: Geometry VI, Riemannian Geometry, Encyclopaedia of Mathematical Sciences, 91, Springer-Verlag, Berlin, 2001, (translated from the 1998 Russian edition by S.A. Vakhrameev).
  • L. Rachůnek and J. Mikeš: On tensor fields semi-conjugated with torse-forming vector fields, Acta. Univ. Palacki. Olomuc., Fac. Rerum. Natur. Math 44 (2005), 151–160.
  • A. Romero, R.N. Rubio and J.J. Salamanca: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson-Walker spacetimes, Classical. Quantum Gravity. 30 (2013), 115007.
  • A. Romero, R.N. Rubio and J.J. Salamanca: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson-Walker spacetimes. Applications to uniqueness results, Int. J. Geom. Meth. Mod. Phys. 10 (2013), 1360014.
  • M. Sánchez: On the geometry of generalized Robertson-Walker spacetimes: geodesics, Gen. Relativit. Gravitation 30 (1998), 915–932.
  • M. Sánchez: On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields, Gen. Relativ. Grav. 31 (1999), 1–15.
  • R. Sharma: Proper conformal symmetries of spacetimes with divergence-free Weyl tensor, J. Math. Phys. 34 (1993), 3582–3587.
  • L.C. Shepley and A.H. Taub: Spacetimes containing perfect fluids and having a vanishing conformal divergence, Commun. Math. Phys. 5 (1967), 237–256.
  • H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Hertl: Exact solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, Cambridge University Press 2nd ed. 2003.
  • R.M. Wald: General Relativity, The University of Chicago Press, Chicago, 1984.
  • K. Yano: Concircular geometry I-IV, Proc. Imp. Acad. Tokyo 16 (1940), 195–200, 354–360, 442–448, 505–511.
  • K. Yano: On the torse-forming direction in Riemannian spaces, Proc. Imp. Acad. Tokyo, 20 (1944), 340–345.
  • K. Yano and S. Sawaki: Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom. 2 (1968), 161–184.