Gravitational singularity

Remarks

The definition locates the singularity by its effect, the geodesics that end, and not as a set of points of $M$. Clause 1 rules out spacetimes that are incomplete only because part of a larger spacetime has been cut away. Smoothness classes matter: one usually asks for extensions of class $C^2$, and other classes give other notions.

Example: the maximal analytic extension of the Schwarzschild metric, in Kruskal–Szekeres coordinates, is inextendible. A timelike geodesic falling radially inward reaches $r = 0$ in finite proper time. So it is singular. The Kretschmann scalar $48M^2/r^6$ blows up along such a geodesic, which shows that no $C^2$ extension exists.

Non-example: the exterior region $r > 2M$ of the Schwarzschild metric, with its usual coordinates, also has incomplete radial geodesics. But it embeds as a proper open subset of the Kruskal–Szekeres spacetime, so it is extendible and fails clause 1. The coordinate singularity at $r = 2M$ is not a gravitational singularity. Minkowski space is geodesically complete and fails clause 2.