World Function

In General
* Idea: The spacetime (squared) interval between two points, which conceptually encodes all the information in the metric, but does not mention a differentiable structure in its definition and is therefore appealing for generalizations of spacetimes (e.g., discrete ones), at least at the kinematical level.
$ Def: Given two points x, y M, the world function is defined as

(x, y):=   S ²(x, y) ,

where S(x, y) is the geodesic distance between x and y if it is defined, and the sign depends on whether x and y are or are not, respectively, causally related.
* Example: In Minkowski space,

_eta(x, y) = (x^m–y^m) _mn (xⁿ–yⁿ) .

* Properties: It is symmetric, non-negative, and satisfies (in any dimension, with signature (–, +, ..., +), and under the appropriate differentiability assumptions)

((x, y)/x^a) g^ab(x) ((x, y)/x^b) = –2 (x,y) ,   det(²(x, y)/x^ay^b) 0 ,

lim_{y to x} (x, y) = 0 ,   lim_{y to x} (x, y)/x^a = 0 ,

lim_{y to x} ²(x, y)/^ay^b = –g_ab(x) .

(These limit properties explain why S ² is used rather than S.)

And Gravitation
* Idea: All curvature tensors can be written as coincidence limits of derivatives of the world function, and Einstein's equation becomes a set of fourth order pde's for .

References
@ General: in Synge 60; Rylov AdP(63).
@ Special cases: Roberts ALC(93)gq/99 [in FRW spacetime].
@ Applications: Bahder AJP(01)gq [spacetime navigation]; Le Poncin-Lafitte et al CQG(04) [and light deflection]; > s.a. tests of general relativity with light.
@ And quantum gravity: Álvarez PLB(88) [quantum spacetime]; Rylov JMP(90) [discrete spacetime].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 21 jun 2008