Synge's world function

{{Short description|Locally defined function in general relativity}}

In general relativity, Synge's world function is a smooth locally defined function of pairs of points in a smooth spacetime $M$ with smooth Lorentzian metric $g$. Let $x,\; x\text{'}$ be two points in spacetime, and suppose $x$ belongs to a convex normal neighborhood $U$ of $x,\; x\text{'}$ (referred to the Levi-Civita connection associated to $g$) so that there exists a unique geodesic $\backslash gamma(\backslash lambda)$ from $x$ to $x\text{'}$ included in $U$, up to the affine parameter $\backslash lambda$. Suppose $\backslash gamma(\backslash lambda\_0)\; =\; x\text{'}$ and $\backslash gamma(\backslash lambda\_1)\; =\; x$. Then Synge's world function is defined as:

:$\backslash sigma(x,x\text{'})\; =\; \backslash frac\{1\}\{2\}\; (\backslash lambda\_\{1\}-\backslash lambda\_\{0\})\; \backslash int\_\{\backslash gamma\}\; g\_\{\backslash mu\backslash nu\}(z)\; t^\{\backslash mu\}t^\{\backslash nu\}\; d\backslash lambda$

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

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

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 $M\backslash times\; M$, though this definition requires some arbitrary choice.

Synge's world function (also its extension to a neighborhood of the diagonal of $M\backslash times\; M$ ) 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.