I About the notion of non-standard inertial frame

[cianfa72]

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 ?

 

[Ibix]

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...

 

[cianfa72]

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.

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 $t$.

 

[PAllen]

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.

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.

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.

 

[Dale]

TL;DR: About the notion of nonstandard inertial frame in the context of flat spacetime

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 ?

I think that the term “inertial frame” means what the author using the term defines it to mean. There are variations in the literature.

 

[cianfa72]

No, as discussed in a recent thread, requiring Newton's laws to hold good requires Einstein synchronization because Newtons laws are isotropic.

Ok, since assuming isotropic laws includes the light propagation process. Isotropy forces Einstein synchronization, i.e. spacelike hyperplanes (slices) orthogonal to the timelike axis.

I think that the term “inertial frames” means what the author using the term defines it to mean. There are variations in the literature.

Yes, I'm aware of. In this context I mean basically inertial "coordinate system/chart".

 

[Dale]

Yes, I'm aware of. In this context I mean basically inertial "coordinate system/chart"

Same there. Different authors will have different meanings for what charts they consider to be inertial.

For example, some will consider stationary, non-rotating spherical coordinates to be non-inertial and others will consider them to be inertial. There is no hard and fast rule for what everyone calls "inertial".

 

[PeterDonis]

spatial planes (spacelike hyperplanes) not orthogonal in spacetime to the timelike axis.

You can do this, but you can't do it by only transforming the spatial coordinates from a standard inertial frame. In fact, the simplest way to do it transforms only the time coordinate--the spatial coordinates are left unchanged.

 

[Ibix]

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.

 

[Ibix]

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.

 

[cianfa72]

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.

 

[cianfa72]

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).

 

[PeterDonis]

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

Yes it does, because a tetrad by definition has four mutually orthogonal vectors.

 

[cianfa72]

Yes it does, because a tetrad by definition has four mutually orthogonal vectors.

Ah yes. The definition of frame field/tetrad/vierbein requires pointwise mutual orthogonality of frame's vectors fields evaluated at any point. Therefore, according to @Ibix, "all inertially moving objects have constant coordinate speed" doesn't require mutual orthogonality or the normalization condition.

 

[PeterDonis]

The definition of frame field/tetrad/vierbein requires pointwise mutual orthogonality of frame's vectors fields evaluated at any point.

Yes.

Therefore, according to @Ibix, "all inertially moving objects have constant coordinate speed" doesn't require mutual orthogonality or the normalization condition.

I'm not sure where your "therefore" here comes from; this statement is completely unrelated to the previous one. There's no logical connection between them.

 

[cianfa72]

I'm not sure where your "therefore" here comes from; this statement is completely unrelated to the previous one. There's no logical connection between them.

Right, the latter is unrelated to the former. I just restated the conditions @Ibix put forward in post #10.

 

[PeterDonis]

Right, the latter is unrelated to the former.

Ok, so your "therefore" should not have been there.

 

[cianfa72]

As far as I can tell, by picking a pointwise non-orthogonal basis vector fields (one timelike and three spacelike) with constant magnitude representing the 4 translation symmetries of Minkowski flat spacetime, we get the result "all inertially moving objects have constant coordinate speed" however, for instance, light propagation process is no longer isotropic in the associated coordinates/chart (note that in this specific case such basis vector fields actually define a coordinate basis, i.e. a chart).