The European Physical Journal H

, Volume 39, Issue 4, pp 413–503 | Cite as

Singularities and the geometry of spacetime

  • Stephen Hawking
Historical Document

Abstract

The aim of this essay is to investigate certain aspects of the geometry of the spacetime manifold in the General Theory of Relativity with particular reference to the occurrence of singularities in cosmological solutions and their relation with other global properties. Section 2 gives a brief outline of Riemannian geometry. In Section 3, the General Theory of Relativity is presented in the form of two postulates and two requirements which are common to it and to the Special Theory of Relativity, and a third requirement, the Einstein field equations, which distinguish it from the Special Theory. There does not seem to be any alternative set of field equations which would not have some undeseriable features. Some exact solutions are described. In Section 4, the physical significance of curvature is investigated using the deviation equation for timelike and null curves. The Riemann tensor is decomposed into the Ricci tensor which represents the gravitational effect at a point of matter at that point and the Welyl tensor which represents the effect at a point of gravitational radiation and matter at other points. The two tensors are related by the Bianchi identities which are presented in a form analogous to the Maxwell equations. Some lemmas are given for the occurrence of conjugate points on timelike and null geodesics and their relation with the variation of timelike and null curves is established. Section 5 is concerned with properties of causal relations between points of spacetime. It is shown that these could be used to determine physically the manifold structure of spacetime if the strong causality assumption held. The concepts of a null horizon and a partial Cauchy surface are introduced and are used to prove a number of lemmas relating to the existence of a timelike curve of maximum length between two sets. In Section 6, the definition of a singularity of spacetime is given in terms of geodesic incompleteness. The various energy assumptions needed to prove the occurrence of singularities are discussed and then a number of theorems are presented which prove the occurrence of singularities in most cosmological solutions. A procedure is given which could be used to describe and classify the singularites and their expected nature is discussed. Sections 2 and 3 are reviews of standard work. In Section 4, the deviation equation is standard but the matrix method used to analyse it is the author’s own as is the decomposition given of the Bianchi identities (this was also obtained independently by Trümper). Variation of curves and conjugate points are standard in a positive-definite metric but this seems to be the first full account for timelike and null curves in a Lorentz metric. Except where otherwise indicated in the text, Sections 5 and 6 are the work of the author who, however, apologises if through ignorance or inadvertance he has failed to make acknowledgements where due. Some of this work has been described in [Hawking S.W. 1965b. Occurrence of singularities in open universes. Phys. Rev. Lett. 15: 689–690; Hawking S.W. and G.F.R. Ellis. 1965c. Singularities in homogeneous world models. Phys. Rev. Lett.17: 246–247; Hawking S.W. 1966a. Singularities in the universe. Phys. Rev. Lett. 17: 444–445; Hawking S.W. 1966c. The occurrence of singularities in cosmology. Proc. Roy. Soc. Lond. A294: 511–521]. Undoubtedly, the most important results are the theorems in Section 6 on the occurrence of singularities. These seem to imply either that the General Theory of Relativity breaks down or that there could be particles whose histories did not exist before (or after) a certain time. The author’s own opinion is that the theory probably does break down, but only when quantum gravitational effects become important. This would not be expected to happen until the radius of curvature of spacetime became about 10-14 cm.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bondi H. 1952. Cosmology. Cambridge University Press, Cambridge.Google Scholar
  2. 2.
    Bondi H. and T. Gold. 1948. The steady-state theory of the expanding universe. Mon. Not. Roy. Ast. Soc. 108: 252–270. ADSCrossRefMATHGoogle Scholar
  3. 3.
    Boyer R.H. and R.W. Lindquist. 1967. Maximal analytic extension of the Kerr solution. J. Math. Phys. 8: 265–281.ADSCrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Calabi E. and L. Markus. 1962. Relativistic space forms. Ann. Math. 75: 63–76.ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Carter B. 1966. The complete analytic extension of the Reissner-Nordström metric in the special case e2 = m2. Phys. Lett. 21: 423–424.ADSCrossRefGoogle Scholar
  6. 6.
    Coxeter H.S.M. and G.J. Whitrow. 1950. World-structure and non-Euclidean honeycombs. Proc. Roy. Soc. A 201: 417–437. ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Doroshkevich A.G., Ya.B. Zeldovich, and I.D. Novikov. 1966. Gravitational collapse of nonsymmetric and rotating masses. J. Exp. Theor. Phys. 22: 122–130.ADSGoogle Scholar
  8. 8.
    Ellis G.F.R. 2014. Stephen Hawking’s 1966 Adams Prize Essay. Eur. Phys. J. H, DOI:10.1140/epjh/e2014-50014-x
  9. 9.
    Geroch R.P. 1966. Singularities in closed universes. Phys. Rev. Lett. 17: 445–447.ADSCrossRefMATHGoogle Scholar
  10. 10.
    Graves J.L. and D.R. Brill. 1960. Oscillatory character of the Reissner-Nordström metric for an ideal charged wormhole. Phys. Rev. 120: 1507–1513. ADSCrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hawking S.W. 1965a. Properties of expanding universes. Ph.D. Thesis. Cambridge.Google Scholar
  12. 12.
    Hawking S.W. 1965b. Occurrence of singularities in open universes. Phys. Rev. Lett. 15: 689–690.ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hawking S.W. and G.F.R. Ellis. 1965c. Singularities in homogeneous world models. Phys. Rev. Lett. 17: 246–247.CrossRefGoogle Scholar
  14. 14.
    Hawking S.W. 1966a. Singularities in the universe. Phys. Rev. Lett. 17: 444–445.ADSCrossRefGoogle Scholar
  15. 15.
    Hawking S.W. 1966b. Perturbations of an expanding universe. ApJ. 145: 544–554. ADSCrossRefGoogle Scholar
  16. 16.
    Hawking S.W. 1966c. The occurrence of singularities in cosmology. Proc. Roy. Soc. Lond. A 294: 511–521. ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Hawking S.W. and R.J. Tayler. 1966d. Helium production in anisotropic big bang universes. Nature 209: 1278–1279. ADSCrossRefGoogle Scholar
  18. 18.
    Heckmann O. and E. Schücking. 1962. Relativistic cosmology. In: Gravitation. An Introduction to Current Research, ed. by L. Witten. Wiley, New York, pp. 438–469.Google Scholar
  19. 19.
    Hill E.L. 1955. Relativistic theory of discrete momentum space and discrete space-time. Phys. Rev. 100: 1780–1783 ADSCrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Hocking J.G. and G.S. Young. 1961. Topology. Addison-Wesley Publishing Co. Inc., Reading, MA London. Reprinted, Dover Publications Inc., New York, 1988.Google Scholar
  21. 21.
    Hoyle F. 1948. A new model for the expanding universe. Mon. Not. Roy. Ast. Soc. 108: 372–382. ADSCrossRefMATHGoogle Scholar
  22. 22.
    Hoyle F. and J.V. Narlikar. 1964a. Time-symmetric electrodynamics and the arrow of time in cosmology. Proc. Roy. Soc. A 277: 1–23 ADSCrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Hoyle, F. and J.V. Narlikar. 1964b. On the avoidance of singularities in C-field cosmology. Proc. Roy. Soc. A 278: 465–478.ADSCrossRefMATHGoogle Scholar
  24. 24.
    Kantowski R. 1966. Some relativistic cosmological models. Ph.D. Thesis. University of Texas.Google Scholar
  25. 25.
    Kobayashi S. and K. Nomizu. 1963. Foundations of Differential Geometry. Wiley Interscience, New York, Vol. I.Google Scholar
  26. 26.
    Kronheimer E.H. and R. Penrose. 1967. On the structure of causal spaces. Proc. Camb. Phil. Soc. 63: 481–501.ADSCrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Kruskal M.D. 1960. Maximal extension of Schwarzschild metric. Phys. Rev. 119: 1743–1745. ADSCrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kundt W. 1963. Note on the completeness of spacetimes. Z. f. Physik 172: 488–489. ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Kundt W. and M. Trümper. 1962. Beiträge zur Theorie der Gravitations-Strahlungsfelder. Akad. Wiss. Mainz 12: 967–1000. Google Scholar
  30. 30.
    Lichnerowicz A. 1955. Global Theory of Connection and Holonomy Groups. Noordhoff, Leyden.Google Scholar
  31. 31.
    Lifshitz E.M. and I.M. Khalatnikov. 1963. Investigations in relativistic cosmology. Adv. Phys. 12: 185–249.ADSCrossRefMathSciNetGoogle Scholar
  32. 32.
    Markus L. 1955. Line element fields and Lorentz structures on differentiable manifolds. Ann. Math. 62: 411–417.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Milnor J. 1963. Morse Theory. Ann. of Math. Studies no. 31, Princeton.Google Scholar
  34. 34.
    Misner C.W. 1965. Taub-NUT space as a counterexample to almost anything (preprint). In: Relativity Theory and Astrophysics, Vol. 1: Relativity and Cosmology. Lectures in Applied Mathematics, Vol. 8, edited by J. Ehlers. American Mathematical Society (Providence, Rhode Island, 1967), p. 160.Google Scholar
  35. 35.
    Penrose R. 1963. In: Relativity, Groups, and Topology. Gordon and Breach, New York.Google Scholar
  36. 36.
    Penrose R. 1965a. Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. Roy. Soc. A 284: 159–203. ADSCrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Penrose R. 1965b. Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14: 57–59.ADSCrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Rindler W. 1956. Visual horizons in world models. Mon. Not. Roy. Ast. Soc. 116: 662–677. ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Sandage A.R. 1961. The light travel time and the evolutionary correction to magnitudes of distant galaxies. ApJ. 134: 916–926. ADSCrossRefMathSciNetGoogle Scholar
  40. 40.
    Trümper M. 1964. Contributions to actual problems in general relativity (preprint).Google Scholar
  41. 41.
    Walker A.G. 1944. Completely symmetric spaces. J. Lond. Math. Soc. 19: 219–226.CrossRefMATHGoogle Scholar
  42. 42.
    Yano Y. and S. Bochner. 1953. Curvature and Betti Numbers. Ann. of Math. Studies no. 32, Princeton.Google Scholar
  43. 43.
    Zeeman E.C. 1964. Causality implies the Lorentz group. J. Math. Phys. 5: 490–493.ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© EDP Sciences and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Stephen Hawking
    • 1
  1. 1.Gonville and Caius CollegeCambridgeUK

Personalised recommendations