Difference between revisions of "Harmonic coordinates"
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| − | Coordinates in which the [[Metric tensor|metric tensor]] | + | {{TEX|done}} |
| + | Coordinates in which the [[Metric tensor|metric tensor]] $g_{ik}$ satisfies the condition | ||
| − | + | $$\frac{\partial}{\partial x_k}(\sqrt{|g|}g^{ik})=0,$$ | |
| − | where | + | 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 | ||
| − | + | $$g^{ij}\Gamma_{ij}^k=0.$$ | |
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) |
Harmonic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_coordinates&oldid=16906