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In Galilean dynamics, we do not have just one Euclidean 3-space $E_3$, as an arena for the actions of the physical world evolving with time, we have a different $E_3$ for each moment in time, with no natural identification between these various $E_3$s.

Sect 17.2, Chapter Space Time of Road to Reality

This paragraph confuses me, because if we have no natural identification between the different $E_3$s, then how can we talk about motion at all?

As an example, consider an apple on a tree which later falls in some frame and consider snap shots of it's fall. If we are to say that these snap shots have no natural correspondances with each other, how could we say the apple has even changed position?

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    $\begingroup$ Galilean spacetime is fully relative. We only talk about motion as relative motion between any two objects. Absolute "Newtonian" space is an irrelevant concept. Historically Galileo was way ahead of Newton with regards to this fundamental concept. $\endgroup$ Apr 12, 2023 at 21:56
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    $\begingroup$ Even if it is relative, the space at 2 seconds later has nothign to do with the space now (apparently). So, even relative motion doesn't make sense @FlatterMann $\endgroup$ Apr 12, 2023 at 22:02
  • $\begingroup$ That is exactly why absolute space is an irrelevant concept. What is always defined is the distance between two objects. I am not sure that Penrose does a very good job explaining this, though. The language certainly suggests that there is some relevance to an absolute, sliced space. That is just not so. $\endgroup$ Apr 12, 2023 at 22:23
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    $\begingroup$ @FlatterMann I think all things considered, Penrose actually did a good job explaining these things (I certainly gained atleast $2\epsilon$ worth of understanding after reading :). I think OP just needs to keep reading/re-read the chapter to understand what Penrose is saying. Also, OP, you certainly can compare the different $\Bbb{R}^3$'s, just there is no canonical isomorphism between them. See my remarks at the end of your question. The lack of canonical isomorphism encodes exactly that no inertial observer is preferred over any other. $\endgroup$
    – peek-a-boo
    Apr 13, 2023 at 0:09
  • $\begingroup$ @peek-a-boo I have only seen the short excerpt. I certainly hope that the entire chapter is clearer on the facts than this part. There is, of course, the case of a reference frame that is moving arbitrarily, which will lead to arbitrary (and unphysical) fictitious forces... I think the OP question is one of the central problems in Newtonian physics: What is a "good" reference frame? $\endgroup$ Apr 13, 2023 at 0:25

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I guess some of the confusion might stem from the use of mathematical lingo that you are possibly misinterpreting. In particular, the term "natural" means something like "obvious in a given context", and "no natural identification" doesn't mean that there is no possible identification.

You have to consider what he's saying here within the context of the previously discussed absolute Aristotelian spacetime. There was this notion that there is an absolute reference frame that you can refer to in order to say that a particular point at one time is the "same point" at some other time. That is, the assumption was that there's a way for everyone to agree on this - that's the "natural" identification (e.g. if you're given two arbitrary coordinate grids in two different times, if you can relate them to the absolute frame, you can unambiguously identify the corresponding points).

But, while you were reading the previous paragraph, have you stayed in the same place? You might say yes, but I might say that you haven't, since the Earth moved some 450 km on its orbit during that time. You and I named two different points in space as "the same spot".

Penrose is not saying that there is no way to connect points at different times, he's saying there's no single unambiguous "natural" way to identify points from these different $\mathbb E^3$s that everyone will agree on, which was taken for granted in the Aristotelian conception.

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  • $\begingroup$ I don't have Penrose's book, so I wonder what he means by "natural"? Newton's laws spell out the "natural" identification: it's a body moving freely without any forces acting on it (let's blink gravity out of the picture for a moment). There is, of course, not just one such body, but there are entire equivalence classes of such bodies. There is certainly no mathematical principle that lets us pick a frame... but nature provides plenty of them. $\endgroup$ Apr 13, 2023 at 0:31
  • $\begingroup$ I'd just like to mention that I now understand as lack of naturality as the physical understanding of motion being dependent on reference frmaes $\endgroup$ Apr 29, 2023 at 4:24
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The key phrase is "dynamics"...in "Galilean dynamics". If there is no absolute rest frame....pick a point: the origin: $p_0=(0, 0, 0)$ at $t=0$. Then at $t=\epsilon$, different rest frames associate that point with a new point in someone else space...something like $p_{\epsilon}=(-\epsilon v_x, -\epsilon v_y, -\epsilon v_y)$, where $\vec v$ is velocity difference in the two frames.

Note that since there is no upper bound on $||v||$, $p_{\epsilon}$ can be anywhere in the other frame's definition of $E_3$. Hence: no natural connection.

A similar, but inverse, cognitive problem happens in Special Relativity, where an event $E$ has an exactly defined location in Minkowski space for all reference frames, but new Gendankeneers tend to associate its location for latter times with the 3 + 1 space in which it was produced, say a flashbulb in a train station. The event doesn't remain on the train platform, any more than it remains on the passing train that also observed the flash: it is but a point in space-time.

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