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
be two points in spacetime, and suppose
belongs to a convex normal neighborhood
(referred to the Levi-Civita connection associated to
Then Synge's world function is defined as: where
is the tangent vector to the affinely parametrized geodesic
is half the square of the signed geodesic length from
computed along the unique geodesic segment, in
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.
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