Synge's world function explained

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( λ ) from x to x' included in U , up to the affine parameter λ . Suppose \gamma( λ _{0)} = x' and \gamma( λ _{1)} = x . Then Synge's world function is defined as:\sigma(x,x') = (1)/(2) ( λ _{1} -λ _{0} ) \int _{\gamma} g _{\mu\nu} (z) t ^{\mu} t ^{\nu} d λ where t ^{\mu} = (dz ^{\mu})/(d λ) is the tangent vector to the affinely parametrized geodesic \gamma( λ ) . 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') = (1)/(2) η _{\alpha} (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 x M , though this definition requires some arbitrary choice.Synge's world function (also its extension to a neighborhood of the diagonal of M x 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

• Book: Synge, John, L. . Relativity: the general theory . North-Holland . 1960 . 0-521-34400-X.
• Book: Fulling, Stephen, A. . Aspects of quantum field theory in curved space-time . CUP . 1989 . 0-521-34400-X.
• Poisson . E. . Pound . A. . Vega . I. . The Motion of Point Particles in Curved Spacetime . Living Rev. Relativ. . 14 . 7 . 2011 . 7 . 10.12942/lrr-2011-7 . 28179832 . 5255936 . 1102.0529 . 2011LRR....14....7P . free.
• Moretti . Valter . 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 . 2021 . 130 . 10.1007/s11005-021-01464-4 . 2107.04903 . 2021LMaPh.111..130M . free.
• Moretti, Valter (2024) Geometric Methods in Mathematical Physics II: Tensor Analysis on Manifolds and General Relativity, Chapter 7. Lecture Notes Trento University (2024)