Synge's world function

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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}