# Synge's world function

In general relativity, Synge's world function is an example of a bitensor, i.e. a tensorial function of pairs of points in the spacetime. Let $x, x'$ be two points in spacetime, and suppose $x$ belongs to a normal convex neighborhood of $x$ so that there exists a unique geodesic $\gamma(\lambda)$ from $x$ to $x'$, up to the affine parameter $\lambda$. Suppose $\gamma(\lambda_0) = x'$ and $\gamma(\lambda_1) = x$. Then Synge's world function is defined as:
$\sigma(x,x') = \frac{1}{2} (\lambda_{1}-\lambda_{0}) \int_{\lambda_{0}}^{\lambda_{1}} g_{\mu\nu}(z) t^{\mu}t^{\nu} d\lambda$
where the integral is evaluated along the geodesic connecting the two points. That is, $\sigma(x,x')$ is half the square of the geodesic length from $x$ to $x'$. 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:
$\sigma(x,x') = \frac{1}{2} \eta_{\alpha \beta} (x-x')^{\alpha} (x-x')^{\beta}$