Harmonic coordinates

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)