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 convex normal 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:

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

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 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:

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

References

• Poisson, E.; Pound, A.; Vega, I. (2011). "The Motion of Point Particles in Curved Spacetime". Living Rev. Relativ. 14 (7). doi:10.12942/lrr-2011-7.