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].

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

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jan 2016