In other words, the description of the world as given by the laws of physics does not depend on our choice of coordinate systems.
This situation is analogous to the failure of the Maxwell equations to determine the potentials uniquely.
The Einstein field equations alone do not fully determine the evolution of the metric relative to the coordinate system.
The same result can be derived from a Kramers-Moyal-van-Kampen expansion of the Master equation (using the Clebsch–Gordan coefficients for decomposing tensor products)[citation needed].
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.
[5] This Kerr-Schild condition goes well beyond removing coordinate ambiguity, and thus also prescribes a type of physical space-time structure.
[4][6] When choosing coordinate conditions, it is important to beware of illusions or artifacts that can be created by that choice.
Similarly, for numerical methods one needs to avoid caustics (coordinate singularities).
If one combines a coordinate condition which is Lorentz covariant, such as the harmonic coordinate condition mentioned above, with the Einstein field equations, then one gets a theory which is in some sense consistent with both special and general relativity.