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I've seen it said before that we often ignore potential energy in relativity because it can be included in the mass term. It is commonly said that a hydrogen atom has less mass than the sum of its parts, due to the negative binding potential energy, for example. But I haven't seen an intuitive argument for why this would be the case. In fact, it seems counterintuitive.

Suppose you have two equal masses, separated by a distance $d$. If they are both stationary, and noninteracting, then their total mass is $2m$. If you were to apply some force $F$ for an instant $dt$ to the first mass, and the same force to the second mass, the system's center of mass would gain a velocity $dv=Fdt/m$. While center of mass is not talked about often in relativity, I understand that it does have a definition, though obscure.

Now suppose one of these masses has a positive charge, and the other a negative charge, of equal magnitudes. There is now a potential energy of interaction which differs from the potential energy the masses would have if they were infinitely far apart. While it is true that they will attract each other, their changes in motion oppose each other. Applying the same force to both charges, the center of mass seems that it should still gain the same $dv$; in fact, the deviations in the two individual masses from this velocity due to their mutual attraction should exactly cancel. It seems that mass 1 should have a velocity $dv+dv_{0}$, while mass 2 should have a velocity $dv-dv_{0}$, where $dv_{0}$ is the change due to their attraction.

So if the center of mass gains the same change in velocity for the same force in both situations, how does the system with the electrostatic potential have less mass? Is mass not related to how hard something is to accelerate? Where is my misunderstanding?

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    $\begingroup$ Does this answer your question? Why does binding energy of particles, which constitutes most of macroscopic mass, make them harder to accelerate? $\endgroup$
    – naturallyInconsistent
    Commented Jun 20 at 5:50
  • $\begingroup$ For even more details, see physics.stackexchange.com/a/804393/364064 $\endgroup$
    – naturallyInconsistent
    Commented Jun 20 at 5:53
  • $\begingroup$ I'm not sure. I guess I'm more looking to understand why potential energy can be treated the same as, say, thermal energy. Kinetic/thermal energy seem very different in character from potential energy, even though they are both forms of energy, and while I can visualize why maybe increasing thermal energy would make acceleration harder, I can't do the same with potential energy. $\endgroup$
    – Aidan Beecher
    Commented Jun 20 at 5:59
  • $\begingroup$ I'm trying to understand this more by visualizing what's happening to the individual particles experiencing the potential that makes the system harder to push around. I don't know if this is possible. $\endgroup$
    – Aidan Beecher
    Commented Jun 20 at 6:00
  • $\begingroup$ That is why I gave the first comment before the 2nd. You have to stop thinking of inertia as a property of mass. It is a property of energy. When an object has more energy, it will have more inertia, i.e. by definition, harder to push around. When negative potential energy robs that energy, then it becomes easier to push around. There is no visualisation possible. This is purely a mathematical connection, consequence, and not visualisable. $\endgroup$
    – naturallyInconsistent
    Commented Jun 20 at 6:07

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