Transfer of heat in relativity and conservation of momentum 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..
 A: 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.
