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I guess absolute time is associated to classical mechanics because people like Newton believed in that concept, but are there actually any statements whose derivation is based on this assumption?

I've looked into the Noether's theorem and some proofs regarding the Lagrange equations and it didn't seem like this assumption was necessary.

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The concept of absolute time was perhaps more of a postulate, rather than an axiom.

If Newton's physics were based on any axioms, it would have to be the principle of relativity, which is perhaps the most profound concept, fundamental to his laws of motion. This principle was first stated by Galileo$^1$. It states that the laws of physics should be the same in all inertial reference frames.

To this principle, Newton added his laws of motion, universal gravitation, and an assertion of an absolute time. He stated

"Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time.

So it is perhaps more accurate to not think of this absolute time as being based on an axiom, but rather a postulate, based on his (Newton's) understandings of space and time and other scientists understandings during that era.

$^1$ Newton's laws are invariant under Galilean transformations, described by the equations (for motion along the x-axis) $$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$\boxed{ t'=t }$$ Note that it becomes apparent that both Galileo and Newton (and others in that era) thought of time as absolute, encompassed in this last equation.

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    $\begingroup$ Thank you for your answer! In other words, absolute time was just a common missbelieve at that time, but the mathematical formulation of the theory is not based on this assumption? $\endgroup$
    – Filippo
    Jun 25 at 9:56
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    $\begingroup$ Yes, it was believed to be true and the formulation of his laws of motion were based on this. $\endgroup$
    – joseph h
    Jun 25 at 9:59
  • $\begingroup$ @josephh: I don't know why you keep making comments that suggest you wish absolute time is not needed in Newtonian mechanics. The additivity of velocity fails completely in Einstein's theory of special relativity, which has been empirically verified so far. Since Newtonian mechanics proves the additivity of velocity, obviously it is due to the assumption of absolute time, since other assumptions used in that proof are okay. $\endgroup$
    – user21820
    Jun 26 at 6:07
  • $\begingroup$ @user21820 "you keep making comments that suggest you wish absolute time is not needed in Newtonian mechanics" What? I actually suggest the complete opposite. Perhaps you meant this for someone else? $\endgroup$
    – joseph h
    Jun 26 at 6:28
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    $\begingroup$ You're right and your quite welcome. Cheers. $\endgroup$
    – joseph h
    Jun 26 at 7:27
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Newton and others did not think necessary to state that the time was absolute (at least not as a postalate$^1$), because they considered this self-evident. Indeed, if they doubted this, they would have come with something similar to (special) relativity much earlier.

I think the assumption about absolute nature of time is most obvious when deriving Galilean transformations as opposed to Lorentz transformations (I mean the pedestrian derivations where one adds velocities). You get the former one, assuming that the time is absolute, and the latter, assuming that the speed of light is constant.

Example: addition of velocities
As an example of a statement based on the absolute time we can take the addition of velocities. The position of an object in reference frame B is given by $$ x_B(t) = x_A(t) + X(t), $$ where $X(t)$ is the position of the origin of reference frame A. Assuming that the time is the same in the two reference frames, we can differentiate this equation, obtaining: $$ v_B = v_A + V $$


1 But see the Newton's quote in @josephh answer.

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  • $\begingroup$ Thank you for your answer! I was expecting someone to come up with Galilean transformations - could you please name a reference where I can find the type of derivation you have in mind? I'd look to look go through the proof and understand exatly on what assumptions it is based. $\endgroup$
    – Filippo
    Jun 25 at 9:41
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    $\begingroup$ It is usually covered in basic mechanics courses, as addition of velocities - perhaps even Holliday&Reznik have it, but on elementary level. Wikipedia suggests Arnold's Mathematical methods of classical mechanics. Essentially, the positions in different reference Frames are related by $x_B(t)=X(t)+x_A(t)$ and one differentiates it in respect to time to obtain velocity addition - thsi already implies that time is absolute. Lorentz transformations are discussed in a similar way the introductory texts on special relativity (in more advanced texts they may be derived from symmetries). $\endgroup$ Jun 25 at 9:49
  • $\begingroup$ Thank you, I'll check it out! $\endgroup$
    – Filippo
    Jun 25 at 9:58
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    $\begingroup$ "Newton and others did not think necessary to state that the time was absolute" — except Newton did bother to state this in Principia, as mentioned in joseph h's answer. $\endgroup$ Jun 26 at 4:05
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    $\begingroup$ @Arthur thanks, I made it clearer. $\endgroup$ Jun 26 at 5:11
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Absolute time means there is unique procedure to determine which events are simultaneous and this is present everywhere in Newtonian mechanics.

Take for example third Newton law. It states, that if object A acts on object B with some force, so does the object B on object A with force equal in magnitude and opposite in direction. But if the force is changing as system evolves, how do we know when exactly is the reaction force to be applied? The answer is, at the same time. I.e. there is an assumption that "at the same time" has absolute meaning, otherwise two different observers would assign reaction force at different events.

I've looked into the Noether's theorem and some proofs regarding the Lagrange equations and it didn't seem like this assumption was necessary.

How so? In newtonian mechanics, one demands Lagrangian to be invariant under Galilean transformations. These transformations do not transform time coordinate, so absoluteness of time is present right there in the symmetries of Lagrangian. I.e. if you start with list of all possible Lagrangians of Newtonian mechanics and find out their symmetries, you will notice that there is a class of coordinate transformations that keep all of them invariant and that these transformation can be used to define special kind of time. So it is there.

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  • $\begingroup$ "otherwise two different observers would assign reaction force at different events" I don't understand, can you elaborate, please? $\endgroup$
    – Filippo
    Jun 25 at 12:03
  • $\begingroup$ @Filippo Imagine both objects A and B have their own clocks. Then if object A acts on object B by some force at certain time of B's clocks, two different observers would not agree at which time of object's A clocks should they assign the reaction force. $\endgroup$
    – Umaxo
    Jun 25 at 12:10
  • $\begingroup$ What do you mean by "assigning" the reaction force? $\endgroup$
    – Filippo
    Jun 25 at 12:30
  • $\begingroup$ @Filippo at which time the reaction force acts. $\endgroup$
    – Umaxo
    Jun 25 at 12:31

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