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In the literature I have seen the following definition of an inertial frame:

A frame is called inertial if any material point interacting with no other bodies or fields moves with constant velocity in a straight line with respect to this frame.

It is claimed that if another frame moves uniformly with respect to an inertial one, then it is also inertial.

In Newtonian mechanics that can be easily proved using the Galileo transformations.

Is there a more direct general way to see that without computations so that it would work simultaneously both in Newtonian mechanics and in special relativity?

Of course the above conclusion can be obtained similarly using the Lorentz transformations. However, usually in courses in special relativity the Lorentz transformations are deduced the other way around. The argument is based on the following two assumptions (among others): (1) in any inertial frame the speed of light is the same; (2) if a frame moves uniformly with respect to an inertial frame then it is inertial.

The second assumption is the focus of my question.

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Is there a more direct general way to see that (if another frame moves uniformly with respect to an inertial one, then it is also inertial) without computations so that it would work simultaneously both in classical mechanics and in special relativity?

Yes, but it requires a bit of setup and vocabulary. First, for both classical mechanics and relativity set up a spacetime using three dimensions of space and one dimension of time. Note that an inertial frame is then one in which force-free objects form straight lines in spacetime.

Now, if we have two inertial frames then we have the requirement that all straight lines in one frame must be mapped to straight lines in the other. The class of transformations that does this is called affine transforms.

Finally, note that if a frame moves uniformly with respect to another then the time axis of one frame is tilted with respect to the other. This can happen with a shear transform, which maps squares to rhombuses. You can visualize it like taking a deck of cards and sliding them so that the cards remain flat but the stack of cards as a whole is tilted. Shear transforms are affine transforms.

So combining the above if you start with an inertial frame then a frame moving uniformly relative to it will also be inertial because uniform motion is a shear transform which is an affine transform which preserves straight lines which (for free objects) defines an inertial frame.

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  • $\begingroup$ Why straight lines in one inertial frame are mapped into straight lines in another one? $\endgroup$
    – MKO
    Sep 10, 2020 at 10:43
  • $\begingroup$ @MKO in an inertial frame any material point interacting with no other bodies or fields moves with constant velocity in a straight line with respect to this frame, so it is a straight line in spacetime. If the first frame is inertial and a straight line in the first frame maps to a curved line in the second frame then the second frame is not inertial because it has a non-interacting body which does not go in a straight line. So a transform between inertial frames must map straight lines to straight lines $\endgroup$
    – Dale
    Sep 10, 2020 at 11:44
  • $\begingroup$ You have shown in your last comment that transform between two inertial frames is affine. However the question was why the transform from an inertial frame to a frame moving uniformly with respect to it is also affine. Since we do not know that the second frame is inertial (we want to prove it eventually!) we do know how a free body moves with respect to it. $\endgroup$
    – MKO
    Sep 10, 2020 at 13:16
  • $\begingroup$ @MKO “why the transform from an inertial frame to a frame moving uniformly with respect to it is also affine”: because a frame moving uniformly is a shear transform and shear transforms are affine, as explained in the answer. Eg $(t,x)\rightarrow (t,x+vt)$ is a shear which is affine $\endgroup$
    – Dale
    Sep 10, 2020 at 14:41
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    $\begingroup$ @MKO this is more effort than it is worth for 0 reputation. After this I am done. “Moving uniformly” means that every second everything shifts by the same amount in the same direction. That is shear. Yes, the transforms are different because in the Galilean transform it is pure shear and in the Lorentz transform it is shear plus rotation (relativity of simultaneity) but the “uniform motion” refers to the shear part. I am done. Upvote or not as you wish. I gave a good answer to a tough question. I am sorry that the terminology is unfamiliar to you but I did my best. Good luck with your studies! $\endgroup$
    – Dale
    Sep 11, 2020 at 11:31

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