Curvature invariant (general relativity) in the context of Lorentzian manifold

A gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by definition no longer part of the regular spacetime and cannot be determined by "where" or "when". Gravitational singularities exist at a junction between general relativity and quantum mechanics; therefore, the properties of the singularity cannot be described without an established theory of quantum gravity. Trying to find a complete and precise definition of singularities in the theory of general relativity, the best theory of gravity available, remains a difficult problem. A singularity in general relativity can be defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete.

General relativity predicts that any object collapsing beyond its Schwarzschild radius would form a black hole, inside which a singularity will form. A black hole singularity is, however, covered by an event horizon, so it is never in the causal past of any outside observer, and at no time can it be objectively said to have formed. General relativity also predicts that the initial state of the universe, at the beginning of the Big Bang, was a singularity of infinite density and temperature. However, classical gravitational theories are not expected to be accurate under these conditions, and a quantum description is likely needed. For example, quantum mechanics does not permit particles to inhabit a space smaller than their Compton wavelengths.