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 formula_1 with smooth Lorentzian metric formula_2. Let formula_3 be two points in spacetime, and suppose formula_4 belongs to a convex normal neighborhood formula_5 of formula_3 (referred to the Levi-Civita connection associated to formula_2) so that there exists a unique geodesic formula_8 from formula_4 to formula_10 included in formula_5, up to the affine parameter formula_12. Suppose formula_13 and formula_14. Then Synge's world function is defined as: where formula_16 is the tangent vector to the  affinely parametrized geodesic formula_8. That is, formula_18 is half the square of the signed geodesic length from formula_4 to formula_10 computed along the unique geodesic segment, in formula_5, joining the two point . 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: it is globally defined and it takes the form Obviously Synge's function can be defined also in Riemannian manifolds and in that case it has non-negative sign. Generally speaking, Synge’s  function is only locally defined and an attempt to define an extension to domains larger than convex normal neighborhoods generally leads to a multivalued function since there may be several geodesic segments joining a pair of points in the spacetime. It is however possible to define it in a neighborhood of the diagonal of formula_23, though this definition requires some arbitrare choice. Synge's world function (also its extension to a neighborhood of the diagonal of formula_23 ) appears in particular in a number of theoretical constructions of quantum field theory in curved spacetime. It is the crucial object used to  construct a parametrix of Green’s functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a globally hyperbolic manifold, and in the definition of Hadamard Gaussian states. References.