Inductive reasoning in small space-time scale

Hi All,

I would like to hear opinions about an explanation I gave some time ago to provide physical fundamentation to the concept of instantaneous velocity.

I started showing a typical situation where one calculates average velocity between events separated in time by one second. Along the trajectory I discussed to what extent that number (the calculated average velocity) was able to specify the physical attribute of the body's velocity at the middle point. Obviously, some examples were taken to show that this number may have or not a close relation with the real velocity at that moment.

Now, if you take a pair of events closer to this middle point (point P from now on) one is intuitivelly conduced to the belief that the average velocity may have a closer relation to that the body experiences at P. But we are still capable of showing examples where the avg. vel. does have nothing to do with the way the body moves when it is in the point P. Then I came with the following allegation: "It just so happens that, according to our observations, we live in a world that, for small times cales, the universe seems to really behave with increasing calm. Note that the universe didn't have to behave like this. We could so well live in a world where in small time scales, the dynamics still maintain itself violent and even unpredictable. Supose you make a movie where we see the flight of a butterfly. Slowing it down we still see its wings going up and down, slowing it down even more, the up and down movement is still perceptible, although in a smaller spatial scale, and as go further and further on this process, we always have the up and down movement visible. I didn't say that there but here I would say that this hypothesis represents some sort of fractal in nature (what, to the present moment, is not oberved in this world).

In this strange but, in principle, possible world, even taking smaller and smaller time scales, the average velocity between points which are at the vicinity of P will never have a close relation to the state of movement of the body at the point P. We seem to be lucky to live in a calmer micro-time world."

Best regards,



Gold Member
In classical physics this is a result of the fact that no force is infinite. Since [itex]\vec{F}=m\vec{a}[/tex] this means that the second derivative of position always exists, implying that velocity is always differentiable, which of corse implies that it is continuous. However, if you keep going smaller and smaller, eventually quantum effects become significant, and this "calmer micro-time world" is not observed. The wavefunctions undergo finite changes in infintesemal, or possibly even nonexistant periods of time.
I agree with you. Have you already heard the noum Zitterbewegung in physics ? As my old-man memory tells me this term apllies when calculating path "Feynmann" integrals in QM. My teacher said that the less well behaved is a path near the classical one, the more it contributes to the action S.

Possibly, our world is not so "calmer micro-time" as classical analysis say.

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