Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LORENTZIAN MANIFOLDS: A CHARACTERIZATION WITH SEMICONFORMAL CURVATURE TENSOR

  • Received : 2018.06.03
  • Accepted : 2019.02.20
  • Published : 2019.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we characterize semiconformally flat spacetimes and a spacetime with harmonic semiconformal curvature tensor. At first in a semiconformally flat perfect fluid spacetime we obtain a state equation and prove that in particular for dimension n = 4, the spacetime represents a model for incoherent radiation. Next we prove that perfect fluid spacetime with harmonic semiconformal curvature tensor is of Petrov type I, D or O and the spacetime is a GRW spacetime. As a consequence we obtain several corollaries.

Keywords

References

  1. L. Alias, A. Romero, and M. Sanchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995), no. 1, 71-84. https://doi.org/10.1007/BF02105675
  2. L. Alias, A. Romero, and M. Sanchez, Compact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, In Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 67-70, World Sci. Publ., River Edge, NJ, 1995.
  3. A. Barnes, On shear free normal flows of a perfect fluid, General Relativity and Gravitation 4 (1973), no. 2, 105-129. https://doi.org/10.1007/bf00762798
  4. J. P. Bourguigon, Codazzi tensor fields and curvature operators, Global Diff. geometry and global analysis, (Proceedings, Berlin 1979), 249-250, Lecture Notes in Maths. 838, Spriger Verlag, Berlin, New York, 2006.
  5. M. Brozos-Vazquez, E. Garcia-Rio, and R. Vazquez-Lorenzo, Some remarks on locally conformally at static space-times, J. Math. Phys. 46 (2005), no. 2, 022501, 11 pp. https://doi.org/10.1063/1.1832755
  6. M. C. Chaki and S. Ray, Space-times with covariant-constant energy-momentum tensor, Internat. J. Theoret. Phys. 35 (1996), no. 5, 1027-1032. https://doi.org/10.1007/BF02302387
  7. B.-Y. Chen, A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 46 (2014), no. 12, Art. 1833, 5 pp. https://doi.org/10.1007/s10714-014-1833-9
  8. A. A. Coley, Fluid spacetimes admitting a conformal Killing vector parallel to the velocity vector, Classical Quantum Gravity 8 (1991), no. 5, 955-968. https://doi.org/10.1088/0264-9381/8/5/019
  9. A. De, C. Ozgur, and U. C. De, On conformally flat almost pseudo-Ricci symmetric spacetimes, Internat. J. Theoret. Phys. 51 (2012), no. 9, 2878-2887. https://doi.org/10.1007/s10773-012-1164-0
  10. U. C. De and L. Velimirovic, Spacetimes with semisymmetric energy-momentum tensor, Internat. J. Theoret. Phys. 54 (2015), no. 6, 1779-1783. https://doi.org/10.1007/ s10773-014-2381-5
  11. R. Deszcz, M. Glogowska, M. Hotlos, and Z. Senturk, On certain quasi-Einstein semisymmetric hypersurfaces, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 41 (1998), 151-164.
  12. R. Deszcz, M. Hotlos, and Z. Sentur, Quasi-Einstein hypersurfaces in semi-Riemannian space forms, Colloq. Math. 89 (2001), no. 1, 81-97. https://doi.org/10.4064/cm89-1-6
  13. K. L. Duggal and R. Sharma, Symmetries of Spacetimes and Riemannian Manifolds, Mathematics and its Applications,487, Kluwer Academic Publishers, Dordrecht, 1999. https://doi.org/10.1007/978-1-4615-5315-1
  14. B. S. Guilfoyle and B. C. Nolan, Yang's gravitational theory, Gen. Relativity Gravitation 30 (1998), no. 3, 473-495. https://doi.org/10.1023/A:1018815027071
  15. S. Guler and S. Altay Demirbag, On Ricci symmetric generalized quasi Einstein space-times, Miskolc Math. Notes 16 (2015), no. 2, 853-868. https://doi.org/10.18514/MMN.2015.1447
  16. S. Guler and S. Altay Demirbag, A study of generalized quasi-Einstein spacetimes with applications in general relativity, Int. J. Theor. Phys. 55 (2016), 548-562. https://doi.org/10.1007/s10773-015-2692-1
  17. G. S. Hall and A. D. Rendall, Uniqueness of the metric from the Weyl and energy-momentum tensors, J. Math. Phys. 28 (1987), no. 8, 1837-1839. https://doi.org/10.1063/1.527443
  18. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, London, 1973.
  19. Y. Ishii, On conharmonic transformations, Tensor (N.S.) 7 (1957), 73-80.
  20. J. Kim, On pseudo semiconformally symmetric manifolds, Bull. Korean Math. Soc. 54 (2017), no. 1, 177-186. https://doi.org/10.4134/BKMS.b151007
  21. J. Kim, A type of conformal curvature tensor, to appear in Far East J. Math. Soc.
  22. D. Lovelock and H. Rund, Tensor, Differential Forms, and Variational Principles, Wiley-Interscience, New York, 1975.
  23. S. Mallick and U. C. De, Spacetimes admitting $W_2$-curvature tensor, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 4, 1450030, 8 pp. https://doi.org/10.1142/S0219887814500303
  24. S. Mallick, Y. J. Suh, and U. C. De, A spacetime with pseudo-projective curvature tensor, J. Math. Phys. 57 (2016), no. 6, 062501, 10 pp. https://doi.org/10.1063/1.4952699
  25. C. A. Mantica and L. G. Molinari, A second-order identity for the Riemann tensor and applications, Colloq. Math. 122 (2011), no. 1, 69-82. https://doi.org/10.4064/cm122-1-7
  26. C. A. Mantica and L. G. Molinari, Weakly Z-symmetric manifolds, Acta Math. Hungar. 135 (2012), no. 1-2, 80-96. https://doi.org/10.1007/s10474-011-0166-3
  27. C. A. Mantica, L. G. Molinari, and U. C. De, A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time, J. Math. Phys. 57 (2016), no. 2, 022508, 6 pp. https://doi.org/10.1063/1.4941942
  28. C. A. Mantica and Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 1, 1250004, 21 pp. https://doi.org/10.1142/S0219887812500041
  29. C. A. Mantica and Y. J. Suh, Pseudo-Z symmetric space-times, J. Math. Phys. 55 (2014), no. 4, 042502, 12 pp. https://doi.org/10.1063/1.4871442
  30. M. M. Postnikov, Geometry VI, translated from the 1998 Russian edition by S. A. Vakhrameev, Encyclopaedia of Mathematical Sciences, 91, Springer-Verlag, Berlin, 2001. https://doi.org/10.1007/978-3-662-04433-9
  31. S. Ray, On a type of general relativistic perfect fluid space-time with divergence-free projective curvature tensor, Tensor (N.S.) 61 (1999), no. 2, 176-180.
  32. M. Sanchez, On the geometry of generalized Robertson-Walker spacetimes: geodesics, Gen. Relativity Gravitation 30 (1998), no. 6, 915-932. https://doi.org/10.1023/A:1026664209847
  33. M. Sanchez, On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields, J. Geom. Phys. 31 (1999), no. 1, 1-15. https://doi.org/10.1016/S0393-0440(98)00061-8
  34. R. Sharma, Proper conformal symmetries of space-times with divergence-free Weyl con-formal tensor, J. Math. Phys. 34 (1993), no. 8, 3582-3587. https://doi.org/10.1063/1.530046
  35. L. C. Shepley and A. H. Taub, Space-times containing perfect fluids and having a vanishing conformal divergence, Comm. Math. Phys. 5 (1967), no. 4, 237-256. https://doi.org/10.1007/BF01646477
  36. H. Stephani, General Relativity, translated from the German by Martin Pollock and John Stewart, Cambridge University Press, Cambridge, 1982.
  37. F. Zengin, M-projectively flat spacetimes, Math. Rep. (Bucur.) 14(64) (2012), no. 4, 363-370.