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 article:

• Stephen Hawking?, Roger Penrose, The Nature of Space and Time Princeton: Princeton University Press. ISBN 0-691-03791-4. (1996)

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 theorem

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