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The value of the integral
 
The value of the integral
  
<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/w/w098/w098140/w0981401.png" /></td> </tr></table>
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$$\Omega(P',P)=\Omega(x',x)=\frac12(u_1-u_0)\int\limits_{u_0}^{u_1}g_{ij}U^iU^jdu,$$
  
taken along a geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981402.png" /> joining two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981404.png" /> in (geodesically-convex) [[Space-time|space-time]]. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981405.png" /> is given by a parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981406.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981407.png" /> is a canonical parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981408.png" />. The world function is equal, up to sign, to half the square measure of the geodesic joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w0981409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w09814010.png" />, and is a two-point invariant in the sense that its value does not change under coordinate transformations in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w09814011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098140/w09814012.png" />.
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taken along a geodesic $\Gamma$ joining two points $P'(x')$ and $P(x)$ in (geodesically-convex) [[Space-time|space-time]]. Here $\Gamma$ is given by a parametrization $x^i=\xi^i(u)$, where $u$ is a canonical parameter and $U^i=d\xi^i/du$. The world function is equal, up to sign, to half the square measure of the geodesic joining $P'$ and $P$, and is a two-point invariant in the sense that its value does not change under coordinate transformations in a neighbourhood of $P'$ and $P$.
  
 
In flat space-time there is a system of coordinates such that
 
In flat space-time there is a system of coordinates such that
  
<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/w/w098/w098140/w09814013.png" /></td> </tr></table>
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$$\Omega(x',x)=\frac12g_{ij}^0(x^{i\prime}-x^i)(x^{j\prime}-x^j),$$
  
 
where
 
where
  
<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/w/w098/w098140/w09814014.png" /></td> </tr></table>
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$$g_{ij}^0=\operatorname{diag}(1,1,1,-1).$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Synge,  "Relativity: the general theory" , North-Holland &amp; Interscience  (1960)  pp. Chapt. II</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Synge,  "Relativity: the general theory" , North-Holland &amp; Interscience  (1960)  pp. Chapt. II</TD></TR></table>

Latest revision as of 10:30, 27 November 2018

The value of the integral

$$\Omega(P',P)=\Omega(x',x)=\frac12(u_1-u_0)\int\limits_{u_0}^{u_1}g_{ij}U^iU^jdu,$$

taken along a geodesic $\Gamma$ joining two points $P'(x')$ and $P(x)$ in (geodesically-convex) space-time. Here $\Gamma$ is given by a parametrization $x^i=\xi^i(u)$, where $u$ is a canonical parameter and $U^i=d\xi^i/du$. The world function is equal, up to sign, to half the square measure of the geodesic joining $P'$ and $P$, and is a two-point invariant in the sense that its value does not change under coordinate transformations in a neighbourhood of $P'$ and $P$.

In flat space-time there is a system of coordinates such that

$$\Omega(x',x)=\frac12g_{ij}^0(x^{i\prime}-x^i)(x^{j\prime}-x^j),$$

where

$$g_{ij}^0=\operatorname{diag}(1,1,1,-1).$$

References

[1] J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960) pp. Chapt. II
How to Cite This Entry:
World function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=World_function&oldid=43496
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article