Synge's world function

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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 be two points in spacetime, and suppose belongs to a convex normal neighborhood of so that there exists a unique geodesic from to , up to the affine parameter . Suppose and . Then Synge's world function is defined as:

where is the tangent vector to the affinely parametrized geodesic . That is, is half the square of the geodesic length from to . 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: