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

*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,

*σ*_{η}(*x*, *y*)
= \(1\over2\)(*x*^{m}
− *y*^{m})
*η*_{mn}
(*x*^{n}
−*y*^{n}) .

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

(∂*σ*(*x*,* y*)/∂*x*^{a})
*g*^{ab}(*x*)
(∂*σ*(*x*,* y*)/∂*x*^{b})
= −2 *σ*(*x*,*y*)
, det(∂^{2}*σ*(*x*, *y*)
∂*x*^{a}∂*y*^{b}) ≠ 0 ,

lim_{y → x} *σ*(*x*, *y*)
= 0 , lim_{y → x}
∂*σ*(*x*,* y*)/∂*x*^{a} = 0 ,

lim_{y → x}
∂^{2}*σ*(*x*, *y*)
/ ∂*x*^{a}∂*y*^{b}
= −*g*_{ab}(*x*) .

(These limit properties explain why *S*^{ 2}
is used rather than *S*.)

> __Online resources__:
see Wikipedia page.

**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].

@ __Related topics__:
in Ottewill & Wardell PRD(11)-a0906 [derivatives, transport equation approach].

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send feedback and suggestions to bombelli at olemiss.edu – modified 14 sep 2019