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Harmonic coordinates

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Coordinates in which the metric tensor satisfies the condition

where is the determinant defined by the components of the tensor . 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

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=33520
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article