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In general relativity, **Synge's world function** is a smooth locally defined function of pairs of points in a smooth spacetime with smooth Lorentzian metric . Let be two points in spacetime, and suppose belongs to a convex normal neighborhood of (referred to the Levi-Civita connection associated to ) so that there exists a unique geodesic from to included in , up to the affine parameter . Suppose and . Then Synge's world function is defined as:

where is the tangent vector to the affinely parametrized geodesic . That is, is half the square of the signed geodesic length from to computed along the unique geodesic segment, in , 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

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 , though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of ) 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.

- Synge, John, L. (1960).
*Relativity: the general theory*. North-Holland. ISBN 0-521-34400-X.`{{cite book}}`

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- Fulling, Stephen, A. (1989).
*Aspects of quantum field theory in curved space-time*. CUP. ISBN 0-521-34400-X.`{{cite book}}`

: CS1 maint: multiple names: authors list (link)

- 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, Valter (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. - Moretti, Valter (2024) Geometric Methods in Mathematical Physics II: Tensor Analysis on Manifolds and General Relativity, Chapter 7. Lecture Notes Trento University (2024)