Synge's world function

In general relativity, Synge's world function is an example of a smooth locally defined bitensor, i.e. a tensorial function of pairs of points in a smooth spacetime $M$ with smooth 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 $\gamma (\lambda )$ from $x$ to $x'$ included in $U$ , up to the affine parameter $\lambda$ . Suppose $\gamma (\lambda _{0})=x'$ and $\gamma (\lambda _{1})=x$ . Then Synge's world function is defined as:

\sigma (x,x')={\frac {1}{2}}(\lambda _{1}-\lambda _{0})\int _{\gamma }g_{\mu \nu }(z)t^{\mu }t^{\nu }d\lambda

where $t^{\mu }={\frac {dz^{\mu }}{d\lambda }}$ is the tangent vector to the affinely parametrized geodesic $\gamma (\lambda )$ . That is, $\sigma (x,x')$ is half the square of the signed geodesic length from $x$ to $x'$ computed along the unique geodesic segment, in $U$ , 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

\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, this bitensor 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 $M\times M$ , though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of $M\times M$ ) 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 functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a Globally hyperbolic manifold, and in the definition of Hadamard Gaussian states.