Synge's world function
In general relativity, Synge's world function is a smooth locally defined function of pairs of points in a smooth spacetime M {\displaystyle M} with smooth Lorentzian metric g {\displaystyle g} . Let x , x ′ {\displaystyle x,x'} be two points in spacetime, and suppose x {\displaystyle x} belongs to a convex normal neighborhood U {\displaystyle U} of x , x ′ {\displaystyle x,x'} (referred to the Levi-Civita connection associated to g {\displaystyle g} ) so that there exists a unique geodesic γ ( λ ) {\displaystyle \gamma (\lambda )} from x {\displaystyle x} to x ′ {\displaystyle x'} included in U {\displaystyle U} , up to the affine parameter λ {\displaystyle \lambda } .