 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, yM, the world function is defined as

σ(x, y):= ± $$1\over2$$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,

ση(x, y) = $$1\over2$$(xmym) ηmn (xnyn) .

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

(∂σ(x, y)/∂xa) gab(x) (∂σ(x, y)/∂xb) = −2 σ(x,y) ,   det(∂2σ(x, y) / ∂xayb) ≠ 0 ,

limyx σ(x, y) = 0 ,   limyxσ(x, y) / ∂xa = 0 ,

limyx2σ(x, y) / ∂xayb = −gab(x) .

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

And Gravitation > s.a. spacetime structure.
* 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 partial differential equations for σ.

References
@ General: in Synge 60; Rylov AdP(63).
@ Special cases: Roberts ALC(93)gq/99 [in FLRW 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]; Kothawala PRD(13)-a1307 [minimal length]; Jia a1909 [quantum causal structure, including matter]; Padmanabhan MPLA(20)-a1911 [and correlator for density of spacetime events]; > s.a. types of approaches.
@ Related topics: in Ottewill & Wardell PRD(11)-a0906 [derivatives, transport equation approach].