nLab Penrose singularity theorem

Idea

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.

Related concepts

• conformal compactification

• cosmic censorship hypothesis

References

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:

• Stephen Hawking, Roger Penrose, The Nature of Space and Time, Princeton University Press (1996) [ISBN 0-691-03791-4, arXiv:hep-th/9409195, Wikipedia entry]

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:

• Wikipedia, Penrose-Hawking singularity theorems

For further developments see

• Alan Coley, Mathematical General Relativity (arXiv:1807.08628)

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:

• Roy Kerr, Do Black Holes have Singularities? [arXiv:2312.00841]

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.