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.