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The Penrose-Hawking singularity theorems characterize spacetimes in the theory of Einstein-gravity (general relativity) which have “singularities”, points where the Riemann curvature is undefined (or would be undefined if these points were included in the spacetime manifold) such as appears notably in black hole spacetimes.
Hellman suggest this theorem as an example of noncomputable physics. See Frank for a response.
A related problem is that of the maximal Cauchy development for the Einstein equations. In this case, at least Zorn's lemma can be avoided.
The original texts:
Roger Penrose: Gravitational Collapse and Space-Time Singularities, Phys. Rev. Lett. 14 (1965) 57 [arXiv:10.1103/PhysRevLett.14.57]
Stephen W. Hawking, George F. R. Ellis: The large scale structure of space-time, Cambridge Univ. Press (1973, 2010) [doi:10.1017/CBO9780511524646, Wikipedia entry]
Exposition and discussion:
Review:
Klaas Landsman, Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose (arXiv:2101.02687)
José M. M. Senovilla, A critical appraisal of the singularity theorems (arXiv:2108.07296)
See also:
For further developments see
In view of constructive mathematics:
Geoffrey Hellman, Mathematical constructivism in spacetime, British Journal for the Philosophy of Science 49 (3):425-450 (1998) PDF
Matthew Frank, Axioms and aesthetics in constructive mathematics and differential geometry. PhD-thesis, Chicago, 2004.
Jan Sbierski, On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification PDF
Cautioning that the singularity theorem makes statements about non-extendible curves but not explicitly about “singularities” in the sense of points of diverging curvature:
Informed commentary is made in reply to Physics.SE:q/790724, especially Physics.SE:a/796154:
Penrose’s theorem guarantees, under certain hypotheses, that spacetime is null geodesically incomplete. […] The primary point of Kerr’s paper is that this theorem has nothing to do with the central ring singularity of his namesake spacetime. The theorem does not tell us that the ring singularity is there: the singularity could, in principle, be excised from the spacetime and replaced with a self-supporting stationary matter distribution without contradicting the theorem in any way.
Last revised on July 30, 2024 at 09:14:10. See the history of this page for a list of all contributions to it.