The classical singularity theorems of general relativity due to Hawking and Penrose hold for \(C^2\)-Lorentzian metrics. The theorems basically state that in general relativity Lorentzian spacetimes (manifolds with a Lorentzian metric), satisfying certain reasonable physical conditions, are geodesically incomplete, i.e., singular. In other words such spacetimes cannot be extended in a \(C^2\)-Lorentzian way.
However, in general relativity, some lower regularity metrics are interesting. In particular, \(C^{1,1}\)-metrics, i.e., metrics that are differentiable, with all derivatives locally Lipshitz, do arise; for example, those that are used to match interior and exterior relativistic spacetimes in physical models. It is natural, therefore to ask whether an extension could be made with a lower regularity metric. Previous investigations [the third author et al., Classical Quantum Gravity 32, No. 15, Article ID 155010, 12 p. (2015; Zbl 1327.83199); Classical Quantum Gravity 32, No. 7, Article ID 075012, 19 p. (2015; Zbl 1328.83123)] show that the Penrose singularity theorem [R. Penrose, Phys. Rev. Lett. 14, 57–59 (1965; Zbl 0125.21206)] and the Hawking singularity theorem [S. W. Hawking, Proc. R. Soc. Lond., Ser. A 300, 187–201 (1967; Zbl 0163.23903)] hold for metrics that are \(C^{1,1}\). This paper states and proves a \(C^{1,1}\)-version of the Hawking-Penrose singularity theorem [S. W. Hawking and R. Penrose, Proc. R. Soc. Lond., Ser. A 314, 529–548 (1970; Zbl 0954.83012)] and its generalization by G. J. Galloway and J. M. M. Senovilla [Classical Quantum Gravity 27, No. 15, Article ID 152002, 10 p. (2010; Zbl 1195.83065)].
The usual assumptions of the Hawking-Penrose singularity theorem are modified with an eye towards a lower regularity proof. Weak versions of the strong energy condition and genericity condition for \(C^{1,1}\)-metrics and for the \(C^0\)-trapped submanifolds are given.
The proof is extensive. It proceeds using regularization showing that under these weak conditions, causal geodesics become non-maximizing. In the process it gives new insight into the matrix Riccati equation for certain approximating metrics.
The overall proof of the singularity theorem for \(C^{1,1}\)-metrics is significantly complicated by the fact that with the lower regularity the curvature tensor is only defined almost everywhere. The standard proof of the singularity theorems relies on the existence of conjugate points (or focal points) along suitable classes of geodesics in the Lorentzian manifold, and these points are shown to exist by studying Jacobi fields (or, equivalently, Riccati equations) along the geodesics. But, with the lower regularity, the Jacobi fields aren’t well defined along all the geodesics. This greatly complicates the necessary analysis of the maximizing properties of causal geodesics, and it necessitates the detailed analysis of the matrix Riccati equation for certain approximating metrics.
In summary, this is a very important addition to understanding the global structure of general relativistic spacetimes. It is of interest to both mathematicians and physicists in gravity and geometric analysis. In addition, there may be independent interest by mathematicians studying the matrix Riccati equation.