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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. (en)
- Em relatividade geral, a função mundo de Synge é um exemplo de um bitensor, isto é, uma função tensorial de pares de pontos no espaço-tempo. (pt)
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- 3677 (xsd:nonNegativeInteger)
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- Em relatividade geral, a função mundo de Synge é um exemplo de um bitensor, isto é, uma função tensorial de pares de pontos no espaço-tempo. (pt)
- 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: (en)
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- Função mundo de Synge (pt)
- Synge's world function (en)
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