Coordinate Conditions - Other Coordinates

Other Coordinates

Many other coordinate conditions have been employed by physicists, though none as pervasively as those described above. Almost all coordinate conditions used by physicists, including the harmonic and synchronous coordinate conditions, would be satisfied by a metric tensor that equals the Minkowski tensor everywhere. (However, since the Riemann and hence the Ricci tensor for Minkowski coordinates is identically zero, the Einstein equations give zero energy/matter for Minkowski coordinates; so Minkowski coordinates cannot be an acceptable final answer.) Unlike the harmonic and synchronous coordinate conditions, some commonly used coordinate conditions may be either under-determinative or over-determinative.

An example of an under-determinative condition is the algebraic statement that the determinant of the metric tensor is −1, which still leaves considerable gauge freedom. This condition would have to be supplemented by other conditions in order to remove the ambiguity in the metric tensor.

An example of an over-determinative condition is the algebraic statement that the difference between the metric tensor and the Minkowski tensor is simply a null four-vector times itself, which is known as a Kerr-Schild form of the metric. This Kerr-Schild condition goes well beyond removing coordinate ambiguity, and thus also prescribes a type of physical space-time structure. The determinant of the metric tensor in a Kerr-Schild metric is negative one, which by itself is an under-determinative coordinate condition.

When choosing coordinate conditions, it is important to beware of illusions or artifacts that can be created by that choice. For example, the Schwarzschild metric may include an apparent singularity at a surface that is separate from the point-source, but that singularity is merely an artifact of the choice of coordinate conditions, rather than arising from actual physical reality.

If one is going to solve the Einstein field equations using approximate methods such as the Post-Newtonian expansion, then one should try to choose a coordinate condition which will make the expansion converge as quickly as possible (or at least prevent it from diverging). Similarly, for numerical methods one needs to avoid caustics (coordinate singularities).