Coordinate Conditions - Indeterminacy in General Relativity

Indeterminacy in General Relativity

The Einstein field equations do not determine the metric uniquely, even if one knows what the metric tensor equals everywhere at an initial time. This situation is analogous to the failure of the Maxwell equations to determine the potentials uniquely. In both cases, the ambiguity can be removed by gauge fixing. Thus, coordinate conditions are a type of gauge condition. No coordinate condition is generally covariant, but many coordinate conditions are Lorentz covariant or rotationally covariant.

Naively, one might think that coordinate conditions would take the form of equations for the evolution of the four coordinates, and indeed in some cases (e.g. the harmonic coordinate condition) they can be put in that form. However, it is more usual for them to appear as four additional equations (beyond the Einstein field equations) for the evolution of the metric tensor. The Einstein field equations alone do not fully determine the evolution of the metric relative to the coordinate system. It might seem that they would since there are ten equations to determine the ten components of the metric. However, due to the second Bianchi identity of the Riemann curvature tensor, the divergence of the Einstein tensor is zero which means that four of the ten equations are redundant, leaving four degrees of freedom which can be associated with the choice of the four coordinates. It should be noted that the same result can be derived from a Kramers-Moyal-van-Kampen expansion of the Master equation (using the Clebsch-Gordon coefficients for decomposing tensor products).