Causal curve

Definition

Let $(M,g)$ be a Lorentzian manifold with a time orientation. A piecewise $C^1$ curve $\gamma:I\to M$ is future-directed causal if, wherever $\dot\gamma$ exists and is nonzero,

and $\dot\gamma$ lies in the chosen future cone. A past-directed causal curve is defined using the past cone. Either is called a causal curve.

A causal curve is timelike if $g(\dot\gamma,\dot\gamma)<0$ wherever its tangent is nonzero, and null if $g(\dot\gamma,\dot\gamma)=0$. Definitions using locally Lipschitz curves impose the same condition almost everywhere and include the piecewise-smooth curves used here.

Future-directed causal and timelike curves define the causal and chronological relations on spacetime.