Null congruences and Raychaudhuri focusing

Maintain signature (−,+,+,+) and G=c=1. Let k be the future null tangent with affine parameter lambda, so its covariant derivative along itself vanishes. A two-dimensional spacelike screen transverse to the ray defines a small area A. Expansion is the logarithmic derivative of that area. Its units depend on the affine normalization: if lambda is a length, theta is an inverse length. Rescaling an affine parameter rescales numerical values without changing which rays meet.

The screen deformation separates into an isotropic trace, a trace-free symmetric shear, and an antisymmetric twist. Squared shear and twist are nonnegative when contracted with the positive screen metric. A null family orthogonal to a hypersurface has zero twist, which is the case needed for horizon generators and the null normals used in the focusing argument. A Killing generator is generally nonaffinely parameterized, so inserting it directly into the affine formula would require an additional nonaffinity term.

For a hypersurface-orthogonal null family, removing twist leaves only nonpositive geometric terms if the null Ricci contraction is nonnegative. In Einstein gravity, contracting the field equation with k twice removes the trace and cosmological-constant terms because k is null. The null energy condition then supplies this sign. This condition concerns every null projection of the stress tensor, not merely a positive energy measured by one chosen observer.

With initially negative expansion, the inequality makes the expansion more negative. Integrating its reciprocal shows that focusing occurs no later than twice the inverse magnitude of the initial expansion, provided the geodesic extends that far. Shear or positive null Ricci curvature can make focusing earlier. The bound does not require the congruence to be shear-free, but an exact simple area solution does. Quantum expectation values of the stress tensor can violate the classical null energy condition, so the same conclusion cannot be imported unchanged into evaporation.

For zero shear, zero twist, and zero null Ricci contraction, the inequality is saturated. The solution has a denominator linear in affine parameter, and the transverse area shrinks as its square until it reaches zero. At that point the chosen ray family ceases to provide a regular local coordinate description. Individual geodesics can continue through a caustic in a completely smooth spacetime, as ordinary converging light rays illustrate.

A singularity theorem needs additional global causal and topological assumptions to turn focusing into geodesic incompleteness. Likewise, an area theorem needs the special global role of event-horizon generators to rule out certain future focal behavior. Keeping those steps distinct prevents an everyday lens focus from becoming a false proof of a spacetime singularity. Raychaudhuri's equation provides a geometric evolution law; the global interpretation comes from the hypotheses of the theorem in which it is used.