# 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 ${\displaystyle x,x'}$ be two points in spacetime, and suppose ${\displaystyle x}$ belongs to a convex normal neighborhood of ${\displaystyle x}$ so that there exists a unique geodesic ${\displaystyle \gamma (\lambda )}$ from ${\displaystyle x}$ to ${\displaystyle x'}$, 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 geodesic length from ${\displaystyle x}$ to ${\displaystyle x'}$. 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:
${\displaystyle \sigma (x,x')={\frac {1}{2}}\eta _{\alpha \beta }(x-x')^{\alpha }(x-x')^{\beta }}$