Harmonic coordinate condition

From Wikipedia, the free encyclopedia

In general relativity, a harmonic coordinate x^α is one which satisfies the d'Alembert's equation when regarded as a scalar field. Solutions of Laplace's equation are called harmonic functions. In space-time, d'Alembert's equation is the generalization of Laplace's equation, so its solutions are also called "harmonic".

1 Motivation
2 Derivation
3 Alternative form
4 Effect on the wave equation
5 See also
6 References
7 External links

[edit] Motivation

The laws of physics can be expressed in a generally invariant form. In other words, the real world does not care about our coordinate systems. However, for us to be able to solve the equations, we must fix upon a particular coordinate system. A coordinate condition selects one (or a smaller set of) such coordinate system(s). The Cartesian coordinates used in special relativity satisfy d'Alembert's equation, so a harmonic coordinate system is the closest approximation available in general relativity to an inertial frame of reference in special relativity.

[edit] Derivation

In general relativity, we have to use the covariant derivative instead of the partial derivative in d'Alembert's equation, so we get:

$0 = (x^\alpha)_{; \beta ; \gamma} g^{\beta \gamma} = ((x^\alpha)_{, \beta , \gamma} - (x^\alpha)_{, \sigma} \Gamma^{\sigma}_{\beta \gamma}) g^{\beta \gamma} \!.$

Since the coordinate x^α is not actually a scalar, this is not a tensor equation. That is, it is not generally invariant. But coordinate conditions must not be generally invariant because they are supposed to pick out (only work for) certain coordinate systems and not others. Since the partial derivative of a coordinate is the Kronecker delta, we get:

$0 = (\delta^\alpha_{\beta , \gamma} - \delta^\alpha_{\sigma} \Gamma^{\sigma}_{\beta \gamma}) g^{\beta \gamma} = (0 - \Gamma^{\alpha}_{\beta \gamma}) g^{\beta \gamma} = - \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} \!.$

And thus, dropping the minus sign, we get the harmonic coordinate condition:

$0 = \Gamma^{\alpha}_{\beta \gamma} g^{\beta \gamma} \!.$

This condition is especially useful when working with gravitational waves.

[edit] Alternative form

Consider the covariant derivative of the density of the reciprocal of the metric tensor:

$0 = (g^{\mu \nu} \sqrt {-g})_{; \rho} = (g^{\mu \nu} \sqrt {-g})_{, \rho} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \rho} \sqrt {-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \rho} \sqrt {-g} - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt {-g} \!.$

Contracting ν with ρ and applying the harmonic coordinate condition to the second term, we get:

$0 = (g^{\mu \nu} \sqrt {-g})_{, \nu} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \nu} \sqrt {-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \nu} \sqrt {-g} - g^{\mu \nu} \Gamma^{\sigma}_{\sigma \nu} \sqrt {-g} \!$

$= (g^{\mu \nu} \sqrt {-g})_{, \nu} + 0 + g^{\mu \alpha} \Gamma^{\beta}_{\alpha \beta} \sqrt {-g} - g^{\mu \alpha} \Gamma^{\beta}_{\beta \alpha} \sqrt {-g} \!$

Thus, we get that an alternative way of expressing the harmonic coordinate condition is:

$0 = (g^{\mu \nu} \sqrt {-g})_{, \nu} \!.$

[edit] Effect on the wave equation

For example, consider the wave equation applied to the electromagnetic vector potential:

$0 = A_{\alpha ; \beta ; \gamma} g^{\beta \gamma}.\!$

Let us evaluate the right hand side:

$A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} = A_{\alpha ; \beta , \gamma} g^{\beta \gamma} - A_{\sigma ; \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma} - A_{\alpha ; \sigma} \Gamma^{\sigma}_{\beta \gamma} g^{\beta \gamma}.$

Using the harmonic coordinate condition we can eliminate the right-most term and then continue evaluation as follows:

$A_{\alpha ; \beta ; \gamma} g^{\beta \gamma} = A_{\alpha ; \beta , \gamma} g^{\beta \gamma} - A_{\sigma ; \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma}$

$= A_{\alpha , \beta , \gamma} g^{\beta \gamma} - A_{\rho , \gamma} \Gamma^{\rho}_{\alpha \beta} g^{\beta \gamma} - A_{\rho} \Gamma^{\rho}_{\alpha \beta , \gamma} g^{\beta \gamma} - A_{\sigma , \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma} - A_{\rho} \Gamma^{\rho}_{\sigma \beta} \Gamma^{\sigma}_{\alpha \gamma} g^{\beta \gamma}$