Revisiting the lesson of the Relativity of Simultaneity

[JesseM]

In a Michelson interferometer, since a split beam travels round-trip along two orthogonal arms before being compared, normally only differential accelerations between the two arms would be an issue.

But your definition of wavenumber isotropy only works if we include some assumption about the two arms having the same length, and if we want to do that without referring to coordinate distance, it seems like we'd have to talk about taking a single arm and rotating it. I suppose we might also define length in terms of some physical feature like multiples of the distance between atoms in a diamond placed alongside each arm.

No, but it's built into GR that it reduces to SR locally, meaning that as you zoom in on a smaller and smaller region of spacetime, the error in using the SR laws to make predictions in that region alone will go to zero.

It's the same for any flat coordinate system that you may choose though, not just for inertial coordinate systems.

What is the same? It's certainly not true that in any flat coordinate system in spacetime, the laws of physics in that coordinate system (written in non-tensor form) will be the same as those in inertial frames in SR.

Generally covariant laws of physics have all of the same physical features as any laws of physics which can be derived from them to describe spacetime using a family of inertial coordinate systems.

Sure, but I think some of these "physical features" may only definable be translating the equations into a non-tensor form in some physically-constructed coordinate system. For example, I don't think it's possible to explain what it means to say that GR is locally Lorentz-invariant without actually showing that in a local region, if you translate the tensor equations of GR into their non-tensor form in the family of local inertial coordinate systems defined by the Lorentz transform, the non-tensor form of the equations will be the same in all these coordinate systems.

So this is one way to test whether some feature of a coordinate-dependent law is really physical or partly mathematical.

Start with some generally covariant laws of physics, and then choose some arbitrary coordinate system to describe spacetime. Will the resulting coordinate-dependent law of physics always be Lorentz invariant?

And you think you can decide if it's Lorentz-invariant without transforming from your arbitrary coordinate system to the family of inertial coordinate systems given by the Lorentz transformation? How would you do that?

I agree that any feature that is inherited by all coordinate systems might be a real physical feature, but if Lorentz invariance is unique to inertial coordinate systems then it only represents one of many possible ways to cover a real physical feature with coordinates.

It isn't unique to inertial coordinate systems. If you are using some non-inertial coordinate system, you can still say that the laws of physics are Lorentz-invariant if it's true that, were you to transform the equations in your system to the family of inertial frames given by the Lorentz transform, the resulting equations would be identical for every member of this family. The statement "the laws of physics are Lorentz-invariant" describes a physical feature of the world which is true for everyone, it doesn't depend on whether you are moving inertially, or on what coordinate system you actually choose to use when approaching problems.