I am a little confused: I read that there are general covariant formulations of Newtonian mechanics (e.g. here). I always thought:
1) A theory is covariant with respect to a group of transformations if the form of those equations is conserved.
2) general covariance means not only transformations defined by arbitrary velocities between different systems, but also transformations defined by arbitrary accelerations conserve the form of such equations.
But in that case the principle of general relativity (the form of all physical laws must be conserved under arbitrary coordinate transformations) would not be unique to general relativity. Where is my error in reasoning, or stated differently which term do I misunderstand?
Regards and thanks in advance!
Answer
The development of general relativity has led to a lot of misconceptions about the significance of general covariance. It turns out that general covariance is a manifestation of a choice to represent a theory in terms of an underlying differentiable manifold.
Basically, if you define a theory in terms of the geometric structures native to a differentiable manifold (i.e. tangent spaces, tensor fields, connections, Lie derivatives, and all that jazz), the resulting theory will automatically be generally-covariant when expressed in coordinates (guaranteed by the manifold's atlas).
It turns out that most physical theories can be expressed in this language (e.g. symplectic manifolds in the case of Hamiltonian Mechanics) and can therefore be presented in a generally covariant form.
What turns out to be special(?) about the general theory of relativity is that space and time combine to form a (particular type of) Lorentzian manifold and that the metric tensor field on the manifold is correlated with the stuff occupying the manifold.
In other words, general covariance was not the central message of general relativity; it just seemed like it was because it was a novelty at the time, and a poorly understood one at that.
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