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 ${\displaystyle M}$ with smooth Lorentzian metric ${\displaystyle g}$. Let ${\displaystyle x,x'}$ be two points in spacetime, and suppose ${\displaystyle x}$ belongs to a convex normal neighborhood ${\displaystyle U}$ of ${\displaystyle x,x'}$ (referred to the Levi-Civita connection associated to ${\displaystyle g}$) so that there exists a unique geodesic ${\displaystyle \gamma (\lambda )}$ from ${\displaystyle x}$ to ${\displaystyle x'}$ included in ${\displaystyle U}$, up to the affine parameter ${\displaystyle \lambda }$. Suppose ${\displaystyle \gamma (\lambda _{0})=x'}$ and ${\displaystyle \gamma (\lambda _{1})=x}$. Then Synge's world function is defined as:

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

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

${\displaystyle \sigma (x,x')={\frac {1}{2}}\eta _{\alpha \beta }(x-x')^{\alpha }(x-x')^{\beta }.}$

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 ${\displaystyle M\times M}$, though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of ${\displaystyle M\times M}$ ) 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.