In general relativity, Synge's world function is an example of a bitensor, i.e. a tensorial function of pairs of points in the spacetime. Let x , x ′ be two points in spacetime, and suppose x belongs to a convex normal neighborhood of x so that there exists a unique geodesic γ ( λ ) from x to x ′ , up to the affine parameter λ . Suppose γ ( λ 0 ) = x ′ and γ ( λ 1 ) = x . Then Synge's world function is defined as:
σ ( x , x ′ ) = 1 2 ( λ 1 − λ 0 ) ∫ γ g μ ν ( z ) t μ t ν d λ where t μ = d z μ d λ is the tangent vector to the affinely parametrized geodesic γ ( λ ) . That is, σ ( x , x ′ ) is half the square of the geodesic length from x to x ′ . Synge's world function is well-defined, since the integral above is invariant under reparametrization. In particular, for Minkowski spacetime, the Synge's world function simplifies to half the spacetime interval between the two points:
σ ( x , x ′ ) = 1 2 η α β ( x − x ′ ) α ( x − x ′ ) β 