general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

, Synge's world function is a smooth locally defined function of pairs of points in a smooth

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...

M

with smooth

Lorentzian metric In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...

g

. Let

x, x'

be two points in spacetime, and suppose

x

belongs to a convex normal neighborhood

U

x, x'

(referred to the

Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...

associated to

g

) so that there exists a unique

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

\gamma(\lambda)

from

x

x'

included in

U

, up to the affine parameter

\lambda

. Suppose

\gamma(\lambda_0) = x'

and

\gamma(\lambda_1) = x

. Then Synge's world function is defined as: :

\sigma(x,x') = \frac (\lambda_-\lambda_) \int_ g_(z) t^t^ d\lambda

where

t^= \frac

is the tangent vector to the affinely parametrized geodesic

\gamma(\lambda)

. That is,

\sigma(x,x')

is half the square of the signed geodesic length from

x

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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...

, the Synge's world function simplifies to half the

spacetime interval In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differe ...

between the two points: it is globally defined and it takes the form :

\sigma(x,x') = \frac \eta_ (x-x')^ (x-x')^.

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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...

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\times M

, though this definition requires some arbitrary choice. Synge's world function (also its extension to a neighborhood of the diagonal of

M\times M

) appears in particular in a number of theoretical constructions of

quantum field theory in curved spacetime In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory treats spacetime as a fixed, classical background, while givin ...

. It is the crucial object used to construct a

parametrix In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator. A parametrix for a di ...

of Green’s functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a

globally hyperbolic manifold In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold ...

, and in the definition of Hadamard Gaussian states.

References

* * * * General relativity {{relativity-stub