TL;DR

About the notion of nonstandard inertial frame in the context of flat spacetime

I had a DM with @PAllen about the notion of non-standard inertial frame.

Let's consider a (global) inertial frame in the context of flat spacetime. Apply now to it a general transformation of spatial coordinates alone.

Such a transformation yields a frame that is no longer inertial. Although one might say something like “we no longer have a standard inertial frame”. The point being that inertially moving bodies may have a nonlinear coordinate description.

This means that, using the coordinate time ##t## as parameter, the functions ##x(t), y(t), z(t)## describing in that frame a body moving inertially (i.e. zero proper acceleration) might be nonlinear. Hence the sentence "we no longer have a standard inertial frame".

What do you think about, does the definition of inertial frame prescribe that the description of a body moving intertially must be linear in it ?

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I think the usual interpretation of "inertial frame" requires that all objects exhibit inertia - i.e. objects that have constant coordinate speed experience no proper acceleration. That means Cartesian coordinates only. (As an aside, it still allows your spatial planes not to be orthogonal to your timelike axis, which is one view on why Einstein had to specify his synchronisation convention as well as specifying that Newton's laws hold good).

I would add that if you actually physically implemented some spatial coordinate system as a frame of rods then put it somewhere in space, the frame itself would be inertial, regardless of its form. A literal inertial frame, and the only implied requirement is that objects at rest in the system will be inertial. So I think it's possible to argue this view, but my understanding is that it isn't the customary usage.

I would not be confused by something like "an inertial frame in spherical polars", but it should probably be understood as shorthand for "take a global inertial frame and apply a coordinate transform to spherical polars".

Now let's see what everyone else says...

Ibix said:

I think the usual interpretation of "inertial frame" requires that all objects exhibit inertia - i.e. objects that have constant coordinate speed experience no proper acceleration. That means Cartesian coordinates only. (As an aside, it still allows your spatial planes not to be orthogonal to your timelike axis, which is one view on why Einstein had to specify his synchronisation convention as well as specifying that Newton's laws hold good).

Ok, you mean that the requirement that zero proper acceleration has constant coordinate speed allows for spatial planes (spacelike hyperplanes) not orthogonal in spacetime to the timelike axis.

Ibix said:

I would add that if you actually physically implemented some spatial coordinate system as a frame of rods then put it somewhere in space, the frame itself would be inertial, regardless of its form. A literal inertial frame, and the only implied requirement is that objects at rest in the system will be inertial.

What about any object moving inertially ? I believe there is also the implied requirement that their coordinate speed must be constant as well.

Note that coordinate speed involves derivatives of spatial/spacelike coordinates w.r.t. coordinate time ##t##.

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Ibix said:

I think the usual interpretation of "inertial frame" requires that all objects exhibit inertia - i.e. objects that have constant coordinate speed experience no proper acceleration.

Ok. That would rule out e.g. the coordinate transform ##x'=log(x)## because an object with no proper acceleration moving in the x' direction would not have constant coordinate speed. On the other hand, any combination of translation, rotation, or boost would, of course be ok.

Ibix said:

That means Cartesian coordinates only. (As an aside, it still allows your spatial planes not to be orthogonal to your timelike axis, which is one view on why Einstein had to specify his synchronisation convention as well as specifying that Newton's laws hold good).

No, as discussed in a recent thread, requiring Newton's laws to hold good requires Einstein synchronization because Newtons laws are isotropic. For example, consider an idealized gun and ideal calorimeter at mutual rest. Invariant is the statement that the calorimeter will have the same reading for any orientation of the set up. However, if you also require that kinetic energy depends only on speed (definitely part of Newton's laws), then isotropic synchronization is forced.

Ibix said:

I would add that if you actually physically implemented some spatial coordinate system as a frame of rods then put it somewhere in space, the frame itself would be inertial, regardless of its form. A literal inertial frame, and the only implied requirement is that objects at rest in the system will be inertial. So I think it's possible to argue this view, but my understanding is that it isn't the customary usage.

Right, if you only require that objects at rest with respect to an inertial reference object (defined, e.g. by constant radar reflection times) have a constant coordinate position, then you could have an 'non standard' inertial frame such that an object with zero proper acceleration may have non-constant coordinate velocity.

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cianfa72 said:

What about any object moving inertially ? I believe there is also the implied requirement that their coordinate speed must be constant as well.

As I said, I don't think that represents the usual meaning of an inertial frame. It's a defensible definition, I think, just not the usual one.

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PAllen said:

Ok. That would rule out e.g. the coordinate transform ##x'=log(x)## because an object with no proper acceleration moving in the x' direction would not have constant coordinate speed. On the other hand, any combination of translation, rotation, or boost would, of course be ok.

I think also scaling, even anisotropically as long as it is globally the same, would be fine. But yes, I think that definition restricts "inertial frame" to uniformly scaled axes.

Perhaps more precisely, I think it restricts your coordinate basis vectors to being parallel to a timelike Killing field and three of the spacelike Killing fields that correspond to translational symmetry. If the only requirement is that all inertial objects have constant coordinate speed, I don't think it requires the basis vectors to be mutually orthogonal or normalised, merely that their magnitude is constant. I see your point about "Newton's laws hold good" in a more general sense being a tighter restriction, one that does require orthonormality.

Ibix said:

Perhaps more precisely, I think it restricts your coordinate basis vectors to being parallel to a timelike Killing field and three of the spacelike Killing fields that correspond to translational symmetry. If the only requirement is that all inertial objects have constant coordinate speed, I don't think it requires the basis vectors to be mutually orthogonal or normalised, merely that their magnitude is constant.

So for instance in Minkowski spacetime that has 10 independent symmetries, pick 4 independent translational KVFs (they define a frame field/tetrad) and coordinates such that their (coordinate) basis vectors are parallel to those frame field's vectors at any point. Then "all inertially moving objects have constant coordinate speed" doesn't require mutual orthogonality but just constant magnitude for coordinate basis vectors.

Can we extend the construction given in #11 to any spacetime, e.g. even to non maximally symmetric spacetimes ?

I'm not positive about that, since a frame field may not define a coordinate basis (i.e. there is no guarantee that coordinate lines exist such that the associated basis coincides with the frame field's vectors at any point).

Another point related to post #18. For a given spacetime suppose there is a set of 4 (algebraically) independent vector fields of constant magnitude (one timelike and three spacelike) with vanishing Lie brackets (i.e. they commute).Then there exist local charts with the property that their coordinate basis's vectors are the same as those 4 vector fields evaluated at any point.

Suppose now the spacetime global topology allows for a global chart. From what said in this thread, this implies the chart being defined is classified as inertial and since extends globally, it follows the underlying spacetime is flat Minkowski.

On a second thought, I believe my post #19 isn't entirely correct.

For a given spacetime, assume there exists a Lorentzian frame field (one timelike and three spacelike mutually orthonormal vector fields) with the additional property that its vector fields commute (i.e. zero Lie bracket). The 4d manifold is locally Minkowskian and the frame field defines locally a coordinate (holonomic) basis.

Suppose now the topology of the spacetime allows for a global chart (i.e. the local charts implied by Frobenius' theorem happen to extend globally). Does this imply the underlying 4d manifold is Minkowski flat ?

PeterDonis said:

For your second assumption, that the topology allows for a global inertial chart, why do you think that doesn't require the spacetime to be globally Minkowski?

Ah yes, you're right (I edited my post #20).

Coming back to what @Ibix said in post #10, I've not a clear understanding why a non orthonormal frame field (one timelike and three spacelike) with constant magnitude basis vectors should result in "all inertially moving objects have constant coordinate speed" claim.

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cianfa72 said:

I've not a clear understanding why a non orthonormal frame field (one timelike and three spacelike) with constant magnitude basis vectors should result in "all inertially moving objects have constant coordinate speed" claim.

Because the inner product of an inertial world line's four velocity and any of the Killing fields is constant.