The singularity theorems of General Relativity and their low regularity extensions

Abstract

On the occasion of Sir Roger Penrose's 2020 Nobel Prize in Physics, we review the singularity theorems of General Relativity, as well as their recent extension to Lorentzian metrics of low regularity. The latter is motivated by the quest to explore the nature of the singularities predicted by the classical theorems. Aiming at the more mathematically minded reader, we give a pedagogical introduction to the classical theorems with an emphasis on the analytical side of the arguments. We especially concentrate on focusing results for causal geodesics under appropriate geometric and initial conditions, in the smooth and in the low regularity case. The latter comprise the main technical advance that leads to the proofs of C 1 -singularity theorems via a regularisation approach that allows to deal with the distributional curvature. We close with an overview on related lines of research and a future outlook.

Raychaudhuri equation and the dynamics of cosmic evolution

“…The general form of the Raychuadhuri equation (RE) for a congruence of time-like geodesics in an (n + 1)dimensional space-time takes the form [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26],…”

“…For greater details regarding how the divergence of Θ is implied by the presence of conjugate point and incomplete geodesic using the geometrical RE is shown in appendix B for an exact and comprehensive idea to the general readership. Readers may also refer to [10,14] in this regard.…”

“…2 takes the form (13) in Einstein gravity and (14) in Modified gravity as follows : for both the cases. Now for convergence ˜> R 0 so one may conclude that convergence will occur during the evolution of the Universe if q > 0 i.e, CC occurs only in decelerating phase.…”

“…If a space-time manifold has a congruence of closed geodesic which are not complete then the space-time has singularity. The notion of geodesic incompleteness facilitated proof of certain detailed theorems [11][12][13][14]. A space-time singularity marks not only the breakdown of Einstein Gravity in particular but also physics in general-' End of Everything'.…”

Timelike Ricci bounds for low regularity spacetimes by optimal transport

“…From a geometric perspective, C 1 [9] and C 1,1 [21] are the lowest regularities under which the classical Hawking singularity theorem [12,13] has been proven under distributional timelike Ricci bounds. (See [9,19] for C 1 -versions of the Hawking-Penrose singularity theorem, and [36] for an overview over singularity theorems in general relativity.) Incidentally, our results build a first bridge between [9,21] and the synthetic Hawking singularity theorem for timelike nonbranching low regularity spacetimes from [3,Thm.…”