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 } . Suppose γ ( λ 0 ) = x ′ {\displaystyle \gamma (\lambda _{0})=x'} and γ ( λ 1 ) = x {\displaystyle \gamma (\lambda _{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 {\displaystyle \sigma (x,x')={\frac {1}{2}}(\lambda _{1}-\lambda _{0})\int _{\gamma }g_{\mu \nu }(z)t^{\mu }t^{\nu }d\lambda }
where t μ = d z μ d λ {\displaystyle t^{\mu }={\frac {dz^{\mu }}{d\lambda }}} is the tangent vector to the affinely parametrized geodesic γ ( λ ) {\displaystyle \gamma (\lambda )} . That is, σ ( x , x ′ ) {\displaystyle \sigma (x,x')} is half the square of the signed geodesic length from x {\displaystyle x} to x ′ {\displaystyle x'} computed along the unique geodesic segment, in U {\displaystyle U} , joining the two points. Synge's world function is well-defined, since the integral above is invariant under reparameterization. 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
σ ( x , x ′ ) = 1 2 η α β ( x − x ′ ) α ( x − x ′ ) β . {\displaystyle \sigma (x,x')={\frac {1}{2}}\eta _{\alpha \beta }(x-x')^{\alpha }(x-x')^{\beta }.}
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 M × M {\displaystyle M\times M} , though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of M × M {\displaystyle M\times M} ) 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.
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