# Singularities and the geometry of spacetime

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## 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. A* **294**: 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.

## Keywords

Point Conjugate Null Geodesic Geodesic Curve Cauchy Surface Timelike Curve## References

- 1.Bondi H. 1952.
*Cosmology*. Cambridge University Press, Cambridge.Google Scholar - 2.Bondi H. and T. Gold. 1948. The steady-state theory of the expanding universe.
*Mon. Not. Roy. Ast. Soc.***108**: 252–270. ADSCrossRefzbMATHGoogle Scholar - 3.Boyer R.H. and R.W. Lindquist. 1967. Maximal analytic extension of the Kerr solution.
*J. Math. Phys.***8**: 265–281.ADSCrossRefzbMATHMathSciNetGoogle Scholar - 4.Calabi E. and L. Markus. 1962. Relativistic space forms.
*Ann. Math.***75**: 63–76.ADSCrossRefzbMATHMathSciNetGoogle Scholar - 5.Carter B. 1966. The complete analytic extension of the Reissner-Nordström metric in the special case
*e*2 =*m*2.*Phys. Lett.***21**: 423–424.ADSCrossRefGoogle Scholar - 6.Coxeter H.S.M. and G.J. Whitrow. 1950. World-structure and non-Euclidean honeycombs.
*Proc. Roy. Soc. A***201**: 417–437. ADSCrossRefzbMATHMathSciNetGoogle Scholar - 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.Ellis G.F.R. 2014. Stephen Hawking’s 1966 Adams Prize Essay.
*Eur. Phys. J. H*, DOI: 10.1140/epjh/e2014-50014-x - 9.Geroch R.P. 1966. Singularities in closed universes.
*Phys. Rev. Lett.***17**: 445–447.ADSCrossRefzbMATHGoogle Scholar - 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. ADSCrossRefzbMATHMathSciNetGoogle Scholar - 11.Hawking S.W. 1965a. Properties of expanding universes. Ph.D. Thesis. Cambridge.Google Scholar
- 12.Hawking S.W. 1965b. Occurrence of singularities in open universes.
*Phys. Rev. Lett.***15**: 689–690.ADSCrossRefMathSciNetGoogle Scholar - 13.Hawking S.W. and G.F.R. Ellis. 1965c. Singularities in homogeneous world models.
*Phys. Rev. Lett.***17**: 246–247.CrossRefGoogle Scholar - 14.Hawking S.W. 1966a. Singularities in the universe.
*Phys. Rev. Lett.***17**: 444–445.ADSCrossRefGoogle Scholar - 15.Hawking S.W. 1966b. Perturbations of an expanding universe.
*ApJ.***145**: 544–554. ADSCrossRefGoogle Scholar - 16.Hawking S.W. 1966c. The occurrence of singularities in cosmology.
*Proc. Roy. Soc. Lond. A***294**: 511–521. ADSCrossRefzbMATHMathSciNetGoogle Scholar - 17.Hawking S.W. and R.J. Tayler. 1966d. Helium production in anisotropic big bang universes.
*Nature***209**: 1278–1279. ADSCrossRefGoogle Scholar - 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.Hill E.L. 1955. Relativistic theory of discrete momentum space and discrete space-time.
*Phys. Rev.***100**: 1780–1783 ADSCrossRefzbMATHMathSciNetGoogle Scholar - 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.Hoyle F. 1948. A new model for the expanding universe.
*Mon. Not. Roy. Ast. Soc.***108**: 372–382. ADSCrossRefzbMATHGoogle Scholar - 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 ADSCrossRefzbMATHMathSciNetGoogle Scholar - 23.Hoyle, F. and J.V. Narlikar. 1964b. On the avoidance of singularities in C-field cosmology.
*Proc. Roy. Soc. A***278**: 465–478.ADSCrossRefzbMATHGoogle Scholar - 24.Kantowski R. 1966. Some relativistic cosmological models. Ph.D. Thesis. University of Texas.Google Scholar
- 25.Kobayashi S. and K. Nomizu. 1963.
*Foundations of Differential Geometry*. Wiley Interscience, New York, Vol. I.Google Scholar - 26.Kronheimer E.H. and R. Penrose. 1967. On the structure of causal spaces.
*Proc. Camb. Phil. Soc.***63**: 481–501.ADSCrossRefzbMATHMathSciNetGoogle Scholar - 27.Kruskal M.D. 1960. Maximal extension of Schwarzschild metric.
*Phys. Rev.***119**: 1743–1745. ADSCrossRefzbMATHMathSciNetGoogle Scholar - 28.Kundt W. 1963. Note on the completeness of spacetimes.
*Z. f. Physik***172**: 488–489. ADSCrossRefzbMATHMathSciNetGoogle Scholar - 29.Kundt W. and M. Trümper. 1962. Beiträge zur Theorie der Gravitations-Strahlungsfelder.
*Akad. Wiss. Mainz***12**: 967–1000. Google Scholar - 30.Lichnerowicz A. 1955.
*Global Theory of Connection and Holonomy Groups*. Noordhoff, Leyden.Google Scholar - 31.Lifshitz E.M. and I.M. Khalatnikov. 1963. Investigations in relativistic cosmology.
*Adv. Phys.***12**: 185–249.ADSCrossRefMathSciNetGoogle Scholar - 32.Markus L. 1955. Line element fields and Lorentz structures on differentiable manifolds.
*Ann. Math.***62**: 411–417.CrossRefzbMATHMathSciNetGoogle Scholar - 33.Milnor J. 1963.
*Morse Theory*. Ann. of Math. Studies no. 31, Princeton.Google Scholar - 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.Penrose R. 1963. In:
*Relativity, Groups, and Topology*. Gordon and Breach, New York.Google Scholar - 36.Penrose R. 1965a. Zero rest-mass fields including gravitation: asymptotic behaviour.
*Proc. Roy. Soc. A***284**: 159–203. ADSCrossRefzbMATHMathSciNetGoogle Scholar - 37.Penrose R. 1965b. Gravitational collapse and space-time singularities.
*Phys. Rev. Lett.***14**: 57–59.ADSCrossRefzbMATHMathSciNetGoogle Scholar - 38.Rindler W. 1956. Visual horizons in world models.
*Mon. Not. Roy. Ast. Soc.***116**: 662–677. ADSCrossRefMathSciNetGoogle Scholar - 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.Trümper M. 1964. Contributions to actual problems in general relativity (preprint).Google Scholar
- 41.Walker A.G. 1944. Completely symmetric spaces.
*J. Lond. Math. Soc.***19**: 219–226.CrossRefzbMATHGoogle Scholar - 42.Yano Y. and S. Bochner. 1953.
*Curvature and Betti Numbers*. Ann. of Math. Studies no. 32, Princeton.Google Scholar - 43.Zeeman E.C. 1964. Causality implies the Lorentz group.
*J. Math. Phys.***5**: 490–493.ADSCrossRefzbMATHMathSciNetGoogle Scholar