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Difference between revisions of "Harmonic coordinates"

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Coordinates in which the [[Metric tensor|metric tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046450/h0464501.png" /> satisfies the condition
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Coordinates in which the [[Metric tensor|metric tensor]] $g_{ik}$ satisfies the condition
  
<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/h/h046/h046450/h0464502.png" /></td> </tr></table>
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$$\frac{\partial}{\partial x_k}(\sqrt{|g|}g^{ik})=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046450/h0464503.png" /> is the determinant defined by the components of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046450/h0464504.png" />. In several cases use of harmonic coordinates leads to a considerable simplification of the calculations: an example is the derivation of the equations of motion in general relativity.
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where $g$ is the determinant defined by the components of the tensor $g_{ik}$. In several cases use of harmonic coordinates leads to a considerable simplification of the calculations: an example is the derivation of the equations of motion in general relativity.
  
 
====References====
 
====References====
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Harmonic coordinates also are introduced by the property that the coordinate functions are harmonic [[#References|[a1]]]. This is equivalent to the equation
 
Harmonic coordinates also are introduced by the property that the coordinate functions are harmonic [[#References|[a1]]]. This is equivalent to the equation
  
<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/h/h046/h046450/h0464505.png" /></td> </tr></table>
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$$g^{ij}\Gamma_{ij}^k=0.$$
  
 
They simplify the formula for the [[Ricci curvature|Ricci curvature]]. For two-dimensional manifolds [[Isothermal coordinates|isothermal coordinates]] are harmonic.
 
They simplify the formula for the [[Ricci curvature|Ricci curvature]]. For two-dimensional manifolds [[Isothermal coordinates|isothermal coordinates]] are harmonic.

Latest revision as of 19:16, 9 October 2014

Coordinates in which the metric tensor $g_{ik}$ satisfies the condition

$$\frac{\partial}{\partial x_k}(\sqrt{|g|}g^{ik})=0,$$

where $g$ is the determinant defined by the components of the tensor $g_{ik}$. In several cases use of harmonic coordinates leads to a considerable simplification of the calculations: an example is the derivation of the equations of motion in general relativity.

References

[1] V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1954) (Translated from Russian)


Comments

Harmonic coordinates also are introduced by the property that the coordinate functions are harmonic [a1]. This is equivalent to the equation

$$g^{ij}\Gamma_{ij}^k=0.$$

They simplify the formula for the Ricci curvature. For two-dimensional manifolds isothermal coordinates are harmonic.

Note that the best coordinates are neither the normal nor the harmonic ones, but the Jost–Karcher coordinates, cf. [a2], pp. 59-60.

References

[a1] A.L. Besse, "Einstein manifolds" , Springer (1987)
[a2] "Elie Cartan et les mathématiques d'aujourd'hui" Astérisque (1985)
How to Cite This Entry:
Harmonic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_coordinates&oldid=16906
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article