World function

The value of the integral

$$\Omega(P',P)=\Omega(x',x)=\frac12(u_1-u_0)\int\limits_{u_0}^{u_1}g_{ij}U^iU^jdu,$$

taken along a geodesic $\Gamma$ joining two points $P'(x')$ and $P(x)$ in (geodesically-convex) space-time. Here $\Gamma$ is given by a parametrization $x^i=\xi^i(u)$, where $u$ is a canonical parameter and $U^i=d\xi^i/du$. The world function is equal, up to sign, to half the square measure of the geodesic joining $P'$ and $P$, and is a two-point invariant in the sense that its value does not change under coordinate transformations in a neighbourhood of $P'$ and $P$.

In flat space-time there is a system of coordinates such that

$$\Omega(x',x)=\frac12g_{ij}^0(x^{i\prime}-x^i)(x^{j\prime}-x^j),$$

where

$$g_{ij}^0=\operatorname{diag}(1,1,1,-1).$$

References

[1]	J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960) pp. Chapt. II