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The singularities of gravitational collapse and cosmology

1970/01/27 by S. W. Hawking, Stephen William Hawking, Roger Penrose · 2,101 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Classical mechanics #Conjugate points #Cosmic censorship hypothesis #Cosmological constant #Cosmology and Gravitation Theories #Energy condition #General relativity #Geodesic #Geodesics in general relativity #Gravitational collapse #Gravitational singularity #Hypersurface #Mathematical analysis #Mathematical physics #Mathematics #Null (SQL) #Physics #Quantum mechanics

paper · doi:10.1098/rspa.1970.0021

published in Proceedings of the Royal Society of London A Mathematical and Physical Sciences 314(1519), 529-548 (Royal Society)

openalex publication_date 1970/01/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Abstract A new theorem on space-time singularities is presented which largely incorporates and generalizes the previously known results. The theorem implies that space-time singularities are to be expected if either the universe is spatially closed or there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface) or there is a point p whose past null cone encounters sufficient matter that the divergence of the null rays through p changes sign somewhere to the past of p (i. e. there is a minimum apparent solid angle, as viewed from p for small objects of given size). The theorem applies if the following four physical assumptions are made: (i) Einstein’s equations hold (with zero or negative cosmological constant), (ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’), (iii) there are no closed timelike curves, (iv) every timelike or null geodesic enters a region where the curvature is not specially alined with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.) In common with earlier results, timelike or null geodesic incompleteness is used here as the indication of the presence of space-time singularities. No assumption concerning existence of a global Cauchy hypersurface is required for the present theorem.

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