I know a classical mechanics law points out the following (Newton's first law): material particles with constant velocity will continue to move uniformly in straight line. If material particles are in a state of rest they will continue to be at rest. This law is just valid for some kind of reference frames (RFs) (to be more accurate, it is just valid for RFs with certain singular states of motion).

The first question coming to my mind was:

Why are there valid and invalid reference-bodies in classical mechanics? I know Newton's laws are just valid at inertial reference frames, but the same question came to my mind: why?

Doing some research I read Einstein thought it was not possible to find the reason why bodies had different behaviours considered with respect to different reference systems (using classical mechanics). Also I read Newton tried to invalidate this preference but it was not possible.

  • $\begingroup$ That's an interesting question. I guess it has to do with the Universe were we live. We don't know why, but our physical laws include $F=ma$ if $m=const.$, it doesn't say anything about velocity, so RF's with difference velocities are indistinguishable, while accelerated ones are not... $\endgroup$ – FGSUZ Aug 29 '17 at 20:30
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    $\begingroup$ In General Relativity the objective distinction between accelerated and unaccelerated observers/reference frames is abandoned and there no longer is such a preference. $\endgroup$ – Adomas Baliuka Aug 29 '17 at 21:52
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    $\begingroup$ @AdomasBaliuka: Your statement is false. In GR, free-falling observers are inertial. $\endgroup$ – Ben Crowell Aug 30 '17 at 5:29
  • $\begingroup$ Your question is not clear to me. No frame of reference is "invalid". Are you asking why inertial frames have a special status in Newtonian Mechanics? ... It would be helpful if you provided the text of the quotes from Einstein and Newton, and gave references for those quotes. $\endgroup$ – sammy gerbil Aug 30 '17 at 8:12
  • $\begingroup$ Invalid reference frame is not the best way to explain it. A right word would be preference. Let's quote what Einsten wrote : " How does it come that certain reference-bodies (or their states of motion) are given priority over other reference-bodies (or their states of motion) ? What is the reason for this Preference?" You can find it on the chapter called "In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?" (Part II; The General Theory of Relativity) $\endgroup$ – JD_PM Aug 30 '17 at 20:37

Although for simplicity Newtonian Mechanics is usually introduced in the language of inertial reference frames, it can also be taught in a coordinate-free way using the tools of Differential Geometry, which is a subject based on the concept of a manifold (a generalization and formalization of the concept of a surface).

In this more sophisticated setting, spacetime is represented by a manifold, and Newton's 1st Law says that an object free of any forces will follow a straight line in this manifold. In this context a 'straight line' is defined in terms of the manifold, and not in terms of a preferred choice of coordinates.

As you can probably guess, this language follows over into relativity theory in which Newton's 1st Law remains true, with the main change being the geometrical structure of the manifold.

If this doesn't make much sense to you, then you'll understand why people usually introduce Newtonian mechanics in the language of a coordinate system (specifically, one in which a straight line has the simplest possible formula).

Of course, even in this coordinate-free representation one could also ask 'What's so special about straight lines?', and this is something which we take as an axiom, since you have to start somewhere, right?

Your question is important because coordinates are not intrinsic to nature, and shouldn't be essential to stating the fundamental laws. Indeed, that's why Differential Geometry is such an important subject in physics.

  • $\begingroup$ Okey, please correct me if I am wrong. A manifold is any region of space which locally looks like Euclidean geometry. Could we using that approximation relate mathematically bodies with reference frames no matter if reference frames are accelerated? I know GR avoids preference problem. I still do not know why, but I will work it soon. $\endgroup$ – JD_PM Aug 30 '17 at 1:33
  • $\begingroup$ But my question was more about why exists a difference between reference frames when we use classical mechanics. Shouldn't be something explaining why bodies behave different considered with respect to different reference frames (let's call them K and K')? I think using differential geometry guides me to GR, but I am trying to find the reason of that preference using just classical mechanics $\endgroup$ – JD_PM Aug 30 '17 at 1:46
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    $\begingroup$ @JD_PM You can express the laws of classical mechanics in general coordinates, but the equations take their simplest form in inertial coordinates. $\endgroup$ – Physics Footnotes Aug 30 '17 at 2:22

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