View source for World function

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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Synge,   "Relativity: the general theory" , North-Holland &amp; Interscience  (1960)  pp. Chapt. II</TD></TR></table>