Physics:Synge's world function
In general relativity, Synge's world function is a smooth locally defined function of pairs of points in a smooth spacetime [math]\displaystyle{ M }[/math] with smooth Lorentzian metric [math]\displaystyle{ g }[/math]. Let [math]\displaystyle{ x, x' }[/math] be two points in spacetime, and suppose [math]\displaystyle{ x }[/math] belongs to a convex normal neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x, x' }[/math] (referred to the Levi-Civita connection associated to [math]\displaystyle{ g }[/math]) so that there exists a unique geodesic [math]\displaystyle{ \gamma(\lambda) }[/math] from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ x' }[/math] included in [math]\displaystyle{ U }[/math], up to the affine parameter [math]\displaystyle{ \lambda }[/math]. Suppose [math]\displaystyle{ \gamma(\lambda_0) = x' }[/math] and [math]\displaystyle{ \gamma(\lambda_1) = x }[/math]. Then Synge's world function is defined as:
- [math]\displaystyle{ \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]\displaystyle{ t^{\mu}= \frac{dz^{\mu}}{d\lambda} }[/math] is the tangent vector to the affinely parametrized geodesic [math]\displaystyle{ \gamma(\lambda) }[/math]. That is, [math]\displaystyle{ \sigma(x,x') }[/math] is half the square of the signed geodesic length from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ x' }[/math] computed along the unique geodesic segment, in [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ 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
- Synge, J.L. (1960). Relativity: the general theory. North-Holland. ISBN 0-521-34400-X.
- Fulling, S. A. (1989). Aspects of quantum field theory in curved space-time. CUP. ISBN 0-521-34400-X.
- Poisson, E.; Pound, A.; Vega, I. (2011). "The Motion of Point Particles in Curved Spacetime". Living Rev. Relativ. 14 (7): 7. doi:10.12942/lrr-2011-7. PMID 28179832. Bibcode: 2011LRR....14....7P.
- Moretti, V. (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. doi:10.1007/s11005-021-01464-4. Bibcode: 2021LMaPh.111..130M.
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