TOPICS
Search

Harmonic Coordinates


Harmonic coordinates satisfy the condition

 Gamma^lambda=g^(munu)Gamma_(munu)^lambda=0,
(1)

or equivalently,

 partial/(partialx^kappa)(sqrt(g)g^(lambdakappa))=0.
(2)

It is always possible to choose such a system. Using the d'Alembertian,

  square ^2phi=(g^(lambdakappa)phi_(;lambda))_(;kappa)=g^(lambdakappa)(partial^2phi)/(partialx^lambdapartialx^kappa)-Gamma^lambda(partialphi)/(partialx^lambda).
(3)

But since Gamma^lambda=0 for harmonic coordinates, the result is a generalization of the harmonic equation

 del ^2x=0
(4)

to

  square ^2x^mu=0.
(5)

See also

d'Alembertian

Explore with Wolfram|Alpha

References

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

Referenced on Wolfram|Alpha

Harmonic Coordinates

Cite this as:

Weisstein, Eric W. "Harmonic Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicCoordinates.html

Subject classifications