{{Short description|Locally defined function in general relativity}} In general relativity, '''Synge's world function''' is a smooth locally defined function of pairs of points in a smooth spacetime <math>M</math> with smooth Lorentzian metric <math>g </math>. Let <math>x, x'</math> be two points in spacetime, and suppose <math>x</math> belongs to a convex normal neighborhood <math>U</math> of <math>x, x'</math> (referred to the Levi-Civita connection associated to <math>g </math>) so that there exists a unique geodesic <math>\gamma(\lambda)</math> from <math>x</math> to <math>x'</math> included in <math>U</math>, up to the affine parameter <math>\lambda</math>. Suppose <math>\gamma(\lambda_0) = x'</math> and <math>\gamma(\lambda_1) = x</math>. Then Synge's world function is defined as: :<math>\sigma(x,x') = \frac{1}{2} (\lambda_{1}-\lambda_{0}) \int_{\gamma} g_{\mu\nu}(z) t^{\mu}t^{\nu} d\lambda</math> where <math>t^{\mu}= \frac{dz^{\mu}}{d\lambda}</math> is the tangent vector to the affinely parametrized geodesic <math>\gamma(\lambda)</math>. That is, <math>\sigma(x,x')</math> is half the square of the signed geodesic length from <math>x</math> to <math>x'</math> computed along the unique geodesic segment, in <math>U</math>, 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 :<math>\sigma(x,x') = \frac{1}{2} \eta_{\alpha \beta} (x-x')^{\alpha} (x-x')^{\beta}.</math> 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 <math>M\times M </math>, though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of <math>M\times M </math> ) 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 ==
* {{cite book |first=John, L. |last=Synge |authorlink=John Lighton Synge |title=Relativity: the general theory |publisher=North-Holland |year=1960 |isbn=978-0-720-40066-3}}
* {{cite book |first=Stephen, A. |last=Fulling |title=Aspects of quantum field theory in curved space-time |publisher=CUP |year=1989 |isbn=0-521-34400-X}}
* {{cite journal |last1=Poisson |first1=E. |last2=Pound |first2=A. |last3=Vega |first3=I. |title=The Motion of Point Particles in Curved Spacetime |journal=Living Rev. Relativ. |volume=14 |issue=7 |year=2011 |page=7 |doi=10.12942/lrr-2011-7 |pmid=28179832 |pmc=5255936 |arxiv=1102.0529 |bibcode=2011LRR....14....7P |doi-access=free}}
* {{cite journal |last1=Moretti |first1=Valter |title=On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods |journal=Letters in Mathematical Physics |volume=111 |issue=5 |year=2021 |page=130 |doi=10.1007/s11005-021-01464-4 |arxiv=2107.04903 |bibcode=2021LMaPh.111..130M |doi-access=free}} * Moretti, Valter (2024) <small>[https://moretti.maths.unitn.it/manifolds.pdf Geometric Methods in Mathematical Physics II: Tensor Analysis on Manifolds and General Relativity], Chapter 7. Lecture Notes Trento University (2024)</small> {{Reflist}} Category:General relativity
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