In general relativity, Synge's world function is a smooth locally defined function of pairs of points in a smooth spacetime M with smooth Lorentzian metric g . Let x , x ′ be two points in spacetime, and suppose x belongs to a convex normal neighborhood U of x , x ′ (referred to the Levi-Civita connection associated to g ) so that there exists a unique geodesic γ γ ( λ λ ) from x to x ′ included in U , up to the affine parameter λ λ . Suppose γ γ ( λ λ 0 ) = x ′ and γ γ ( λ λ 1 ) = x . Then Synge's world function is defined as: