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Consider a rod with length $L_0$ in its rest frame (call is $\mathcal S'$). Suppose this is the distance between the ends of the rod in the $\hat x$ direction.

As well known from special relativity, this rod will appear to have length $L_0/\gamma$ with respect to a different observer $\mathcal S$ moving at velocity $v$ in the $\hat x$ direction with respect to $\mathcal S'$.

This is all fine and well, but we also know that the rod isn't really an indivisible monolithic object: it is a bunch of atoms held together via various forces. How does the relativistic contraction look like if we think of the rod from a microscopic point of view? How is the contraction described then? Do the electromagnetic forces holding the atoms together appear stronger, thus pulling the atoms closer together? Or are the atoms themselves contracted (as in, I guess, the wavefunctions get "squished" in the velocity direction)? Or something else? Is there a good way to describe this?

The answer might depend on the specific structure we assume for the rod, but for the sake of argument, I suppose one can assume a simple standard crystalline structure for it. Or something else, if there is a better way to look at the problem.

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Your question is posed as a long list of "or" options, but it's not or, it's and. The electric and magnetic fields in the original frame transform to a new set of fields in the new frame. The probability densities get length-contracted (and there is also a more complicated change in the phases of the wavefunctions). The motion is time-dilated.

As a simple example, consider a classical model of an isolated hydrogen atom. The electron is in a circular orbit in the yz plane, and the atom is moving in the x direction at velocity $v$. To first order in $v$, the only real change we observe is that there is a magnetic field from the proton, but this magnetic field makes zero force on the electron. To second order in $v$, the electric field in the orbital plane increases by a factor of $\gamma$, and this is what is required in order to impart the observed centripetal acceleration because of the increase in the transverse inertia of the electron.

You can do this in the xy plane instead, and there would be a length contraction. The analysis gets more complicated, but the list of effects is the same.

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The solid objects are held together with electromagnetic forces which will transform entirely in line with relativistic predictions, as noted by @user281656.

I would pose it that if you pursue analysis along this direction you will arrive at confusing results because you will be comparing the object that is at rest and moving at some speed. These are two different reference frames. Drawing comparisons between them is a rich source of the 'paradoxes' of SR.

A more consistent approach is to ask what happens to an object when it is accelerated from zero velocity to some large velocity, from the point of view of the observer in the lab-frame. Given acceleration that would eventually lead to the length-contraction in the lab-frame, one can show that some of the energy spend on accelerating the object will go towards actually contracting it (http://iopscience.iop.org/article/10.1070/PU1975v018n08ABEH004917/meta). I can probably dig out the actual PDF, if you don't have access.

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    $\begingroup$ Cryo: "There will be a length contraction".....This of course either is or is not true depending on the details of how you accelerate the rod. If you apply forces at different parts of the rod that accelerate each part at the same rate (as measured in the lab frame) then the rod (obviously) does not contract in the lab frame --- instead it expands in its own new (post-acceleration) frame. In my experience, we risk confusing students when we make blanket statements like "after acceleration, the rod is contracted". $\endgroup$
    – WillO
    Dec 6, 2020 at 0:46
  • $\begingroup$ @WillO, agree. Will correct $\endgroup$
    – Cryo
    Dec 6, 2020 at 1:14
  • $\begingroup$ I don't see why the part about acceleration is relevant here. Sure, acceleration will lead to all sorts of physical effects affecting length etc, but that's different that the length contraction that is purely due to different inertial frames, isn't it? The latter is not due to physical effects contracting the rod, but rather must be explainable by the EM fields etc appearing different in different inertial frames. In other words, you can imagine that it's not the rod that is moving at different $v$s, but just that we are looking at it from different reference frames moving at different $v$s $\endgroup$
    – glS
    Dec 6, 2020 at 11:17
  • $\begingroup$ @glS, to compare length you need a way of measuring them, and what you are suggesting is to measure them in different reference frames, despite knowing that spatial dimension of the two reference frames will not be aligned, in 4d space-time. You can do it in principle, but it is not trivial. A simpler thing to do is to have a single way of measuring, in one reference frame. Then you need to start with object at small velocity, measure, accelerate it, and measure again, using the same procedure, in the same non-inertial lab-frame $\endgroup$
    – Cryo
    Dec 6, 2020 at 15:54
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    $\begingroup$ I don't care about acceleration, I care about properties of the object, as measured from the lab-frame, before and after the acceleration $\endgroup$
    – Cryo
    Dec 6, 2020 at 15:54

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