# On general covariance

If newton's theory could be formulated in the language of differential geometry (symplectic manifolds), what do we really mean when we say that the theory is covariant under the Galilean group when it's really generally covariant. I've heard that any theory could be put in a generally covariant form (it's just a matter of mathematical ingenuity). I understand that general covariance doesn't imply that all frames are equivalent and newton's theory clearly distinguishes between inertial and non-inertial frames. What I am looking for is an algorithmic way to figure out which frames are "Physically equivalent" (For example, inertial frames in newton's theory related by Galilean transformations) given a covariant formulation of the theory.

As far as I understand, since the laws of GR are generally stated in a generally covariant form (and that's when everyone learns differential geometry), it propagates a false notion of it being the characteristic feature of GR. Of course, not all frames are physically equivalent in GR although it is what Einstein originally wanted to achieve. Am I correct in saying that general covariance is just a result of mathematical ingenuity and not a feature of any theory?

To answer your specific question, you look for some suitable invariant that identifies the frames you want. For example inertial frames have a zero proper acceleration. The invariant you need will depend on what you're looking for and what formulation of the theory you're using so it isn't possible to comment further.

Your general question is inevitably a matter of opinion, but we should note that General Relativity has a remarkably simple and elegant formulation that is generally covariant. Googling will find plenty of statements to the effect that it is the simplest such theory. While it is true that any theory can be formulated in a generally covariant way this seems like letting ingenuity get in the way of common sense.

• Can the statement "two frames are physically equivalent" be quantified in a generalized manner?
– Jay
May 23, 2022 at 13:48
• @Jay you need to define what physically equivalent means. I assume you mean they share some property e.g. being both inertial. They can't share all physical properties or they'd just be the same frame. May 23, 2022 at 16:16
• So the statement "laws of physics are the same in all inertial frames" is merely philosophical and has no mathematical basis?
– Jay
May 23, 2022 at 23:52
• Inertial frames are the same in the sense that no physical experiment can distinguish one from the other, which is different from just stating that they're a class of frames with zero proper acceleration.
– Jay
May 23, 2022 at 23:54
• @Jay a more precise statement would be to say Newton's Laws of motion are invariant under a Galilean transformation. I suspect this is what your statement two frames are physically equivalent is intended to mean. May 24, 2022 at 4:15

Newton's law can be formulated as a covariant law, i.e., valid for all Cartesian coordinate transformations inertial and non-inertial, if the ordinary time derivative is replaced by the so-called rotational time derivative. (see Zipfel "Modeling and Simulation of Aerospace Vehicle Dynamics", AIAA 2014)