In relativity momentum is associated with flux of energy (not just mass), e.g. for a single particle $\textbf{p} = (E/c^2)\textbf{v}$

Imagine a rod which initially is hotter on one end. If the rod is thermally isolated, heat will start to propagate through the rod until the temperature equalizes.

Does this mean that rod will start moving in the direction of the hotter end to compensate for non-zero momentum associated with the heat flow? It should be moving until the heat flow stops and should preserve the center of energy point (not center of mass).

Are there any known physical effects that are related to this? It looks kinda cool. I realize that in the "real-world" it should be small, but maybe in high-energy or plasma physics..

  • $\begingroup$ Sounds feasible, but the effect would be tiny in a solid rod, and difficult to disentangle from the classical thermal expansion & contraction. $\endgroup$ – PM 2Ring Dec 23 '18 at 20:42
  • $\begingroup$ @PM2Ring sure, but maybe it'll be more pronounced in some other physical systems? The rod example was just for illustration. $\endgroup$ – xaxa Dec 23 '18 at 21:14

Yes, there will be such an effect. Rather than trying to use relativistic mass to solve problems like this (which, e.g., doesn't work in the transverse directions), you're better off using the identity $m^2=E^2-p^2$ (in units with $c=1$) and/or the stress-energy tensor.

This is probably simplest to analyze if you imagine the case where the energy transfer occurs by radiation rather than conduction. The mechanism can't make any difference to the outcome of the conservation law. The light traveling from the hot end to the cool end has zero mass, so we have $0=E^2-p^2$, or $p=E$. Therefore the light brings not just energy with it, but also momentum. In SI units, the amount of momentum transported is $E/c$. This is similar to examples like the Nichols radiometer.

So the hot end experiences a reaction from emitting the radiation, while the cold end gets an impulse from absorbing it. This then becomes like one of those artificial freshman mechanics problems where coal is being tossed from one car in a train to another. The center of mass stays fixed while the cars move.

  • $\begingroup$ Do you know of any situations where this effect would be significant? Radiation is kind of extreme case, because, as you said, it has no mass. I assume this effect should be taken into account in relativistic hydrodynamics, so maybe it plays a role somewhere in nuclear physics or astrophysics? $\endgroup$ – xaxa Dec 23 '18 at 21:22
  • $\begingroup$ Relativistic electric car takes energy from the battery, then accelerates it to the velocity of the road. That creates most of the force propelling the car. $\endgroup$ – stuffu Dec 24 '18 at 13:43
  • $\begingroup$ @stuffu by relativistic electric car do you mean cars using lead-acid battery? (journals.aps.org/prl/abstract/10.1103/PhysRevLett.106.018301) $\endgroup$ – xaxa Dec 24 '18 at 19:17
  • $\begingroup$ What does the battery technology matter?? Okay it was lead-acid. But how about this: There is a inward flow of fuel in the sun. That speeds up the spinning of the sun, but the heat energy that is being conducted outwards cancels out that. $\endgroup$ – stuffu Dec 25 '18 at 8:51

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