Synge's world function

In general relativity, Synge's world function is a smooth locally defined function of pairs of points in a smooth spacetime $M$ with smooth Lorentzian metric $g$. Let $x,x'$ be two points in spacetime, and suppose $x$ belongs to a convex normal neighborhood $U$ of $x,x'$ (referred to the Levi-Civita connection associated to $g$) so that there exists a unique geodesic $\gamma (\lambda )$ from $x$ to $x'$ included in $U$, 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 signed geodesic length from $x$ to $x'$ computed along the unique geodesic segment, in $U$, joining the two points. Synge's world function is well-defined, since the integral above is invariant under reparameterization. In particular, for Minkowski spacetime, the Synge's world function simplifies to half the spacetime interval between the two points: it is globally defined and it takes the form

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

Obviously Synge's function can be defined also in Riemannian manifolds and in that case it has non-negative sign. Generally speaking, Synge’s function is only locally defined and an attempt to define an extension to domains larger than convex normal neighborhoods generally leads to a multivalued function since there may be several geodesic segments joining a pair of points in the spacetime. It is however possible to define it in a neighborhood of the diagonal of $M\times M$, though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of $M\times M$ ) appears in particular in a number of theoretical constructions of quantum field theory in curved spacetime. It is the crucial object used to construct a parametrix of Green’s functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a globally hyperbolic manifold, and in the definition of Hadamard Gaussian states.

References

• Synge, J.L. (1960). Relativity: the general theory. North-Holland. ISBN 0-521-34400-X.

• Fulling, S. A. (1989). Aspects of quantum field theory in curved space-time. CUP. ISBN 0-521-34400-X.

• Poisson, E.; Pound, A.; Vega, I. (2011). "The Motion of Point Particles in Curved Spacetime". Living Rev. Relativ. 14 (7): 7. arXiv:1102.0529. Bibcode:2011LRR....14....7P. doi:10.12942/lrr-2011-7. PMC 5255936. PMID 28179832.

• Moretti, V. (2021). "On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods". Letters in Mathematical Physics. 111 (5): 130. arXiv:2107.04903. Bibcode:2021LMaPh.111..130M. doi:10.1007/s11005-021-01464-4.