Synge world function

In a manifold $ M $ with Riemannian metric $ g $, a natural object characterizing the geometry is the distance $ d ( P,Q ) $ between any two points $ P $, $ Q $ in $ M $, defined as the infimum of the length of all curves connecting the points. This does not itself generalize to pseudo-Riemannian manifolds (cf. Pseudo-Riemannian space; Pseudo-Riemannian geometry). In that case, however, for any $ P $ there is an open neighbourhood $ U _ {P} $ such that for $ Q \in U _ {P} $ the two points are joined by a unique geodesic $ \gamma _ {PQ } $ in $ U _ {P} $. If $ u $ is an affine parameter with $ \gamma _ {PQ } ( u _ {1} ) = P $, $ \gamma _ {PQ } ( u _ {2} ) = Q $, and $ X = d \gamma _ {PQ } /du $ is the corresponding tangent vector, then the quantity

$$ \Omega ( P,Q ) = { \frac{1}{2} } ( u _ {2} - u _ {1} ) ^ {2} g ( X,X ) $$

is independent of the choice of affine parameter and is well defined for both Riemannian and pseudo-Riemannian metrics $ g $. In the Riemannian case it reduces to $ d ( P,Q ) ^ {2} /2 $, while in the case of general relativity (pseudo-Riemannian metric of Lorentz signature) it evaluates to zero if $ P $, $ Q $ are null-separated and to plus or minus half the square of the space- or time-separation of the points otherwise. This function was introduced by H.S. Ruse [a3] and popularized by J.L. Synge [a4] as the world function for general relativity.

If $ g $ is differentiable of class $ C ^ {k} $( $ k \geq 2 $), then $ \Omega $ is of class $ C ^ {k - 2 } $ on the manifold $ D $ of pairs $ \{ {( P,Q ) } : {Q \in U _ {P} } \} $. On the diagonal $ \Delta = \{ {( P,P ) } : {P \in M } \} $ the first few partial derivatives with respect to the coordinates of the first argument are given, in index notation, by

$$ \Omega _ {,i } = 0, \Omega _ {,ij } = g _ {ij } $$

$$ \Omega _ {,ijk } = 0, \Omega _ {,ijkl } = - { \frac{1}{3} } ( R _ {ikjm } + R _ {imjk } ) , $$

where $ R $ is the Riemann tensor of $ g $( with sign convention $ R ^ {i} _ {jkl } = \partial _ {k} \Gamma ^ {i} _ {jl } - \dots $).

Since the world function is physically interpretable in terms of the squares of lengths and times, and is linked by the above formulas to the curvature, it could be used by Synge [a4] as a systematic tool in deriving the basic formulas of differential geometry together with their physical interpretation. In this spirit it was used by C.J.S. Clarke and F. de Felice [a1] to express the curvature corrections to radar-ranging measurements; further formulas of this kind have been presented in [a2].

References

[a1]	C.J.S. Clarke, F. de Felice, "Relativity on curved manifolds" , Cambridge Univ. Press (1990)
[a2]	J.M. Gambi, P. Romero, A. Sanmiguel, F. Vicente, "Fermi coordinate transformation under base-line change in relativistic celestial mechanics" Int. J. Theor. Phys. , 30 (1991) pp. 1097–1116
[a3]	H.S. Ruse, "Taylor's theorem in the tensor calculus" Proc. London Math. Soc. , 32 (1931) pp. 87
[a4]	J.L. Synge, "Relativity: the general theory" , North-Holland (1960)