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A singularity theorem based on spatial averages

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Abstract

Inspired by Raychaudhuri’s work, and using the equation named after him as a basic ingredient, a new singularity theorem is proved. Open non-rotating Universes, expanding everywhere with a non-vanishing spatial average of the matter variables, show severe geodesic incompletness in the past. Another way of stating the result is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies.

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References

  1. A K Raychaudhuri, Phys. Rev. Lett. 80, 654 (1998)

    Article  MATH  ADS  Google Scholar 

  2. J M M Senovilla, Phys. Rev. Lett. 81, 5032 (1998)

    Article  MATH  ADS  Google Scholar 

  3. A K Raychaudhuri, Phys. Rev. 98, 1123 (1955)

    Article  MATH  ADS  Google Scholar 

  4. A K Raychaudhuri, Phys. Rev. 106, 172 (1957)

    Article  ADS  Google Scholar 

  5. J M M Senovilla, Phys. Rev. Lett. 64, 2219 (1990)

    Article  MATH  ADS  Google Scholar 

  6. J M M Senovilla, Phys. Rev. D53, 1799 (1996)

    ADS  Google Scholar 

  7. J M M Senovilla, Gen. Relativ. Gravit. 30, 701 (1998)

    Article  MATH  ADS  Google Scholar 

  8. R Penrose, Phys. Rev. Lett. 14, 57 (1965)

    Article  MATH  ADS  Google Scholar 

  9. S W Hawking and R Penrose, Proc. R. Soc. London A314, 529 (1970)

    ADS  Google Scholar 

  10. S W Hawking and G F R Ellis, The large scale structure of spacetime (Cambridge University Press, Cambridge, 1973)

    Google Scholar 

  11. J Maddox, Nature (London) 345, 201 (1990)

    Article  ADS  Google Scholar 

  12. P Joshi, Phys. Rev. Lett. 67, 2109 (1991)

    Article  MATH  ADS  Google Scholar 

  13. J M M Senovilla, Phys. Rev. Lett. 67, 2110 (1991)

    Article  ADS  Google Scholar 

  14. F J Chinea, L Fernández-Jambrina and J M M Senovilla, Phys. Rev. D45, 481 (1992)

    ADS  Google Scholar 

  15. E Ruiz and J M M Senovilla, Phys. Rev. D45, 1995 (1992)

    ADS  Google Scholar 

  16. A Feinstein and J M M Senovilla, Class. Quant. Grav. 6, L89 (1989)

    Article  ADS  Google Scholar 

  17. N van den Bergh and J E F Skea, Class. Quant. Grav. 9, 527 (1992)

    Article  MATH  ADS  Google Scholar 

  18. L K Patel and N K Dadhich, Class. Quant. Grav. 10, L85 (1993)

    Article  ADS  Google Scholar 

  19. N K Dadhich, L K Patel and R Tikekar, Pramana — J. Phys. 44, 303 (1995)

    ADS  Google Scholar 

  20. N K Dadhich, R Tikekar and L K Patel, Curr. Sci. 65, 694 (1993)

    Google Scholar 

  21. N K Dadhich, in Inhomogeneous cosmological models edited by A Molina and J M M Senovilla (World Scientific, Singapore, 1995) pp. 63–74

    Google Scholar 

  22. M Mars, Phys. Rev. D51, R3989 (1995)

    ADS  Google Scholar 

  23. P S Letelier, J. Math. Phys. 20, 2078 (1979)

    Article  ADS  Google Scholar 

  24. J B Griffiths and J Bicák, Class. Quant. Grav. 12, L81 (1995)

    Article  ADS  Google Scholar 

  25. M Mars, Class. Quant. Gravit. 12, 2831 (1995)

    Article  MATH  ADS  Google Scholar 

  26. M Mars and J M M Senovilla, Class. Quant. Grav. 14, 205 (1997)

    Article  MATH  ADS  Google Scholar 

  27. Recall that all Robertson-Walker geometries are cylindrically symmetric

  28. L Fernández-Jambrina, Class. Quant. Grav. 14, 3407 (1997)

    Article  MATH  ADS  Google Scholar 

  29. N K Dadhich and L K Patel, Gen. Relativ. Gravit. 29, 179 (1997)

    Article  MATH  ADS  Google Scholar 

  30. A Barnes, Gen. Relativ. Gravit. 30, 515 (1998)

    Article  MATH  ADS  Google Scholar 

  31. N K Dadhich, J. Astrophys. Astron. 18, 343 (1997)

    ADS  Google Scholar 

  32. N K Dadhich and M A H MacCallum, Report on the Workshop on Classical General Relativity, in Gravitation and Cosmology (Proceedings of ICGC-95) edited by S Dhurandhar and T Padmanabhan (Kluwer Academic, Dordrecht, 1997) pp. 279–296

    Google Scholar 

  33. A Saa, Phys. Rev. Lett. 81, 5031 (1998)

    Article  MATH  ADS  Google Scholar 

  34. I thank Dr Narayan Banerjee for helping with AKR’s e-mails

  35. A K Raychaudhuri, Phys. Rev. Lett. 81, 5033 (1998)

    Article  MATH  ADS  Google Scholar 

  36. N K Dadhich and A K Raychaudhuri, Mod. Phys. Lett. A14, 2135 (1999)

    ADS  Google Scholar 

  37. L Fernández-Jambrina and L M González-Romero, Class. Quant. Grav. 16, 953 (1999)

    Article  MATH  ADS  Google Scholar 

  38. L Fernández-Jambrina, J. Math. Phys. 40, 4028 (1999)

    Article  MATH  ADS  Google Scholar 

  39. M Giovannini, Phys. Rev. D59, 083511 (1999)

  40. M Gasperini and G Veneziano, Phys. Rep. 373, 1 (2003)

    Article  ADS  Google Scholar 

  41. A K Raychaudhuri, Mod. Phys. Lett. A15, 391 (2000)

    ADS  Google Scholar 

  42. The blowing-up of the expansion of the Ricci time-like eigendirection can take place due to many different reasons: a caustic, the failure of hypersurface orthogonality or of local coordinates, the failure of the vector field to be a Ricci eigendirection, or a change in the algebraic type of the Ricci tensor, among others. There are well-known examples of hypersurface-orthogonal time-like congruences showing this behaviour even in flat spacetime

  43. L Fernández-Jambrina and L M González-Romero, Phys. Rev. D66, 024027 (2002)

    Google Scholar 

  44. L Fernández-Jambrina and L M González-Romero, J. Math. Phys. 45, 2113 (2004)

    Article  MATH  ADS  Google Scholar 

  45. A K Raychaudhuri, http://xxx.lanl.gov/abs/gr-qc/0202056

  46. L Fernández-Jambrina, http://xxx.lanl.gov/abs/gr-qc/0202088

  47. A K Raychaudhuri, Gen. Relativ. Gravit. 36, 343 (2004)

    Article  ADS  Google Scholar 

  48. L Fernández-Jambrina, Gen. Relativ. Gravit. 37, 421 (2005)

    Article  ADS  Google Scholar 

  49. A K Raychaudhuri, Gen. Relativ. Gravit. 37, 425 (2005)

    Article  ADS  Google Scholar 

  50. A Komar, Phys. Rev. 104, 544 (1956)

    Article  MATH  ADS  Google Scholar 

  51. R M Wald, General relativity (The University of Chicago Press, Chicago, 1984)

    MATH  Google Scholar 

  52. see http://lambda.gsfc.nasa.gov/product/map/dr2/parameters.cfm

  53. J M M Senovilla, Class. Quant. Grav. 17, 2799 (2000)

    Article  MATH  ADS  Google Scholar 

  54. R Penrose, Techniques of differential topology in relativity, Regional Conference Series in Applied Math. 7 (SIAM, Philadelphia, 1972)

    MATH  Google Scholar 

  55. J M M Senovilla, Singularity-free Spacetimes, in Rotating objects and relativistic physics edited by F J Chinea and L M González-Romero, Lect. Notes in Phys. 423 (Springer, Berlin, 1993), pp. 185–193

    Google Scholar 

  56. F J Tipler, C J S Clarke and G F R Ellis, in General relativity and gravitation: One hundred years after the birth of Albert Einstein edited by A Held (Plenum Press, New York, 1980)

    Google Scholar 

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Authors and Affiliations

  1. Física Teórica, Universidad del País Vasco, Apartado 644, 48080, Bilbao, Spain

    J. M. M. Senovilla

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  1. J. M. M. Senovilla
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The first instance of a non-singular cosmological solution, compatible with the energy and causality conditions and the Raychaudhuri equation, was given by Senovilla. This generated an intense interest in AKR on the nature and behaviour of such solutions. His initial proposal of the vanishing of the space-time average of the energy density (and other physical and kinematical quantities) as a condition for their existence got refined through many discussions with Dadhich and Senovilla. It evolved finally to the realization that the vanishing of the spatial average of the energy density is what distinguishes all these non-singular solutions, as stated in this theorem proved by Senovilla. In our opinion, this theorem, to which both Raychaudhuri and Senovilla have contributed, provides a succinct and complete characterization of the nature of all non-singular cosmological models — Editors.

In memory of Amal Kumar Raychaudhuri (1924–2005).

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Senovilla, J.M.M. A singularity theorem based on spatial averages. Pramana - J Phys 69, 31–47 (2007). https://doi.org/10.1007/s12043-007-0109-2

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