Singularity

A spacetime singularity is a breakdown of general relativity's geometric description of the universe. Naively it is pictured as a point of infinite density and infinite curvature, as at the centre of a black hole or at the Big Bang. But because a singularity is, by construction, not part of the spacetime manifold — you cannot stand at it and measure the curvature — physicists define it indirectly, through geodesic incompleteness: a spacetime is singular if some freely-falling observer (a timelike or null geodesic) reaches the edge of the manifold in finite proper time or affine parameter, with no possible continuation. The observer's history simply stops, and nothing lies downstream.

This definition is what makes the Penrose–Hawking singularity theorems possible. Rather than solving Einstein's equations to find where the curvature blows up — which requires symmetry assumptions — the theorems prove that under three mild premises (a trapped surface, an energy condition, and global hyperbolicity) at least one geodesic must terminate. The existence of a singularity is established without any description of what it looks like.

Crucially, a singularity is best read not as 'infinite density physically exists' but as a signpost: it marks the boundary of general relativity's own validity. Where curvature grows without bound, the classical theory is announcing that a deeper, quantum theory of gravity must take over. The singularities at r = 0 inside black holes and at the beginning of the cosmos are therefore the sharpest classical pointers toward the unfinished problem of quantum gravity.

Schwarzschild's 1916 solution showed two apparent infinities; the horizon one was shown to be a removable coordinate artifact by Eddington, Lemaître and Finkelstein, while the r = 0 curvature singularity was long dismissed as an artifact of perfect spherical symmetry until Penrose's 1965 theorem proved it generic.