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In a [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103201.png" /> with [[Riemannian metric|Riemannian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103202.png" />, a natural object characterizing the geometry is the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103203.png" /> between any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103205.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103206.png" />, defined as the infimum of the length of all curves connecting the points. This does not itself generalize to pseudo-Riemannian manifolds (cf. [[Pseudo-Riemannian space|Pseudo-Riemannian space]]; [[Pseudo-Riemannian geometry|Pseudo-Riemannian geometry]]). In that case, however, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103207.png" /> there is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103208.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s1103209.png" /> the two points are joined by a unique geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032012.png" /> is an [[Affine parameter|affine parameter]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032014.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032015.png" /> is the corresponding tangent vector, then the quantity
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032016.png" /></td> </tr></table>
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is independent of the choice of affine parameter and is well defined for both Riemannian and pseudo-Riemannian metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032017.png" />. In the Riemannian case it reduces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032018.png" />, while in the case of general relativity (pseudo-Riemannian metric of Lorentz signature) it evaluates to zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032020.png" /> are null-separated and to plus or minus half the square of the space- or time-separation of the points otherwise. This function was introduced by H.S. Ruse [[#References|[a3]]] and popularized by J.L. Synge [[#References|[a4]]] as the world function for general relativity.
+
In a [[Manifold|manifold]]  $  M $
 +
with [[Riemannian metric|Riemannian metric]]  $  g $,
 +
a natural object characterizing the geometry is the distance  $  d ( P,Q ) $
 +
between any two points  $  P $,
 +
$  Q $
 +
in  $  M $,  
 +
defined as the infimum of the length of all curves connecting the points. This does not itself generalize to pseudo-Riemannian manifolds (cf. [[Pseudo-Riemannian space|Pseudo-Riemannian space]]; [[Pseudo-Riemannian geometry|Pseudo-Riemannian geometry]]). In that case, however, for any  $  P $
 +
there is an open neighbourhood  $  U _ {P} $
 +
such that for  $  Q \in U _ {P} $
 +
the two points are joined by a unique geodesic  $  \gamma _ {PQ }  $
 +
in  $  U _ {P} $.  
 +
If  $  u $
 +
is an [[Affine parameter|affine parameter]] with  $  \gamma _ {PQ }  ( u _ {1} ) = P $,
 +
$  \gamma _ {PQ }  ( u _ {2} ) = Q $,
 +
and  $  X = d \gamma _ {PQ }  /du $
 +
is the corresponding tangent vector, then the quantity
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032021.png" /> is differentiable of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032022.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032023.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032024.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032025.png" /> on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032026.png" /> of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032027.png" />. On the diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032028.png" /> the first few partial derivatives with respect to the coordinates of the first argument are given, in index notation, by
+
$$
 +
\Omega ( P,Q ) = {
 +
\frac{1}{2}
 +
} ( u _ {2} - u _ {1} ) ^ {2} g ( X,X )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032029.png" /></td> </tr></table>
+
is independent of the choice of affine parameter and is well defined for both Riemannian and pseudo-Riemannian metrics  $  g $.
 +
In the Riemannian case it reduces to  $  d ( P,Q )  ^ {2} /2 $,
 +
while in the case of general relativity (pseudo-Riemannian metric of Lorentz signature) it evaluates to zero if  $  P $,
 +
$  Q $
 +
are null-separated and to plus or minus half the square of the space- or time-separation of the points otherwise. This function was introduced by H.S. Ruse [[#References|[a3]]] and popularized by J.L. Synge [[#References|[a4]]] as the world function for general relativity.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032030.png" /></td> </tr></table>
+
If  $  g $
 +
is differentiable of class $  C  ^ {k} $(
 +
$  k \geq  2 $),
 +
then  $  \Omega $
 +
is of class  $  C ^ {k - 2 } $
 +
on the manifold  $  D $
 +
of pairs  $  \{ {( P,Q ) } : {Q \in U _ {P} } \} $.
 +
On the diagonal  $  \Delta = \{ {( P,P ) } : {P \in M } \} $
 +
the first few partial derivatives with respect to the coordinates of the first argument are given, in index notation, by
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032031.png" /> is the [[Riemann tensor|Riemann tensor]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032032.png" /> (with sign convention <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110320/s11032033.png" />).
+
$$
 +
\Omega _ {,i }  = 0,  \Omega _ {,ij }  = g _ {ij }
 +
$$
 +
 
 +
$$
 +
\Omega _ {,ijk }  = 0,  \Omega _ {,ijkl }  = - {
 +
\frac{1}{3}
 +
} ( R _ {ikjm }  + R _ {imjk }  ) ,
 +
$$
 +
 
 +
where  $  R $
 +
is the [[Riemann tensor|Riemann tensor]] of $  g $(
 +
with sign convention $  R  ^ {i} _ {jkl }  = \partial  _ {k} \Gamma  ^ {i} _ {jl }  - \dots $).
  
 
Since the world function is physically interpretable in terms of the squares of lengths and times, and is linked by the above formulas to the [[Curvature|curvature]], it could be used by Synge [[#References|[a4]]] as a systematic tool in deriving the basic formulas of [[Differential geometry|differential geometry]] together with their physical interpretation. In this spirit it was used by C.J.S. Clarke and F. de Felice [[#References|[a1]]] to express the curvature corrections to radar-ranging measurements; further formulas of this kind have been presented in [[#References|[a2]]].
 
Since the world function is physically interpretable in terms of the squares of lengths and times, and is linked by the above formulas to the [[Curvature|curvature]], it could be used by Synge [[#References|[a4]]] as a systematic tool in deriving the basic formulas of [[Differential geometry|differential geometry]] together with their physical interpretation. In this spirit it was used by C.J.S. Clarke and F. de Felice [[#References|[a1]]] to express the curvature corrections to radar-ranging measurements; further formulas of this kind have been presented in [[#References|[a2]]].

Latest revision as of 08:24, 6 June 2020


In a manifold $ M $ with Riemannian metric $ g $, a natural object characterizing the geometry is the distance $ d ( P,Q ) $ between any two points $ P $, $ Q $ in $ M $, defined as the infimum of the length of all curves connecting the points. This does not itself generalize to pseudo-Riemannian manifolds (cf. Pseudo-Riemannian space; Pseudo-Riemannian geometry). In that case, however, for any $ P $ there is an open neighbourhood $ U _ {P} $ such that for $ Q \in U _ {P} $ the two points are joined by a unique geodesic $ \gamma _ {PQ } $ in $ U _ {P} $. If $ u $ is an affine parameter with $ \gamma _ {PQ } ( u _ {1} ) = P $, $ \gamma _ {PQ } ( u _ {2} ) = Q $, and $ X = d \gamma _ {PQ } /du $ is the corresponding tangent vector, then the quantity

$$ \Omega ( P,Q ) = { \frac{1}{2} } ( u _ {2} - u _ {1} ) ^ {2} g ( X,X ) $$

is independent of the choice of affine parameter and is well defined for both Riemannian and pseudo-Riemannian metrics $ g $. In the Riemannian case it reduces to $ d ( P,Q ) ^ {2} /2 $, while in the case of general relativity (pseudo-Riemannian metric of Lorentz signature) it evaluates to zero if $ P $, $ Q $ are null-separated and to plus or minus half the square of the space- or time-separation of the points otherwise. This function was introduced by H.S. Ruse [a3] and popularized by J.L. Synge [a4] as the world function for general relativity.

If $ g $ is differentiable of class $ C ^ {k} $( $ k \geq 2 $), then $ \Omega $ is of class $ C ^ {k - 2 } $ on the manifold $ D $ of pairs $ \{ {( P,Q ) } : {Q \in U _ {P} } \} $. On the diagonal $ \Delta = \{ {( P,P ) } : {P \in M } \} $ the first few partial derivatives with respect to the coordinates of the first argument are given, in index notation, by

$$ \Omega _ {,i } = 0, \Omega _ {,ij } = g _ {ij } $$

$$ \Omega _ {,ijk } = 0, \Omega _ {,ijkl } = - { \frac{1}{3} } ( R _ {ikjm } + R _ {imjk } ) , $$

where $ R $ is the Riemann tensor of $ g $( with sign convention $ R ^ {i} _ {jkl } = \partial _ {k} \Gamma ^ {i} _ {jl } - \dots $).

Since the world function is physically interpretable in terms of the squares of lengths and times, and is linked by the above formulas to the curvature, it could be used by Synge [a4] as a systematic tool in deriving the basic formulas of differential geometry together with their physical interpretation. In this spirit it was used by C.J.S. Clarke and F. de Felice [a1] to express the curvature corrections to radar-ranging measurements; further formulas of this kind have been presented in [a2].

References

[a1] C.J.S. Clarke, F. de Felice, "Relativity on curved manifolds" , Cambridge Univ. Press (1990)
[a2] J.M. Gambi, P. Romero, A. Sanmiguel, F. Vicente, "Fermi coordinate transformation under base-line change in relativistic celestial mechanics" Int. J. Theor. Phys. , 30 (1991) pp. 1097–1116
[a3] H.S. Ruse, "Taylor's theorem in the tensor calculus" Proc. London Math. Soc. , 32 (1931) pp. 87
[a4] J.L. Synge, "Relativity: the general theory" , North-Holland (1960)
How to Cite This Entry:
Synge world function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Synge_world_function&oldid=48936
This article was adapted from an original article by C.J.S. Clarke (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article