Future null geodesic completeness and incompleteness

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$. A future-directed affinely parametrized null geodesic is a smooth nonconstant curve $\gamma :J\to M$, where $J\subseteq \mathbb{R}$ is an interval with nonempty interior, such that its tangent $\dot{\gamma }$ is future-directed and null at every interior parameter and $D_{\lambda }\dot{\gamma }=0$. Here “null” means $g(\dot{\gamma },\dot{\gamma })=0$ and $\dot{\gamma }\mathrm{\ne }0$, and future-directed means $g(\dot{\gamma },N)<0$. The geodesic equation and covariant derivative are those of the metric-selected connection.

The maximal future affine interval of such a geodesic is the future endpoint interval of its maximal extension as an affinely parametrized geodesic in $M$. It is future complete when that interval has no finite upper endpoint, equivalently when its affine parameter tends to $+\mathrm{\infty }$. The spacetime is future null geodesically complete when every future-directed null geodesic is future complete, and future null geodesically incomplete when at least one is not.

An affine change of parameter $\tilde{\lambda }=a\lambda +b$ with $a>0$ preserves whether the maximal future interval is unbounded above, so these properties do not depend on the chosen positive affine normalization.

Geodesic of an affine connection

Causal curves, chronological and causal reachability, and inextendibility

Metric compatibility, torsion freedom and the metric-selected connection class

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

This definition distinguishes an incomplete geodesic in $M$ from a curvature singularity or a point added to an extension. No equation for matter, Einstein equation, or claim about the physical adequacy of $(M,g,N)$ is included.