How does potential energy fit into mass-energy equivalence in SR? As with all forms of energy, a potential energy of ##E## added to static system ought to increase the system's mass by ##E/c^2##. This is often illustrated by saying that a compressed spring has slightly more mass than an uncompressed spring. But how does this square with the fact that potential energy is only specified up to an additive constant?(adsbygoogle = window.adsbygoogle || []).push({});

For instance, suppose we have a system of charged particles separated by some fixed distance. Each particle, with charge ##q_i## (say, all positive charges) and produces a potential of ##\phi_i##. Then the potential energy of the system is ##U = \frac{1}{2} \sum_i q_i \phi_i## (with the 1/2 factor to avoid double counting). If each particle has a rest mass of ##m## and say there are ##N## particles, what is the mass of this system? That's an unambiguous, measurable quantity—we could, for instance, use the equivalence principle to do this by measuring the strength of the gravitational field induced by the system—and it must be greater than just ##Nm## since work was done to assemble the system. Naïvely (at least, I assume it's naïvely since it seems to be wrong), the rest energy seems like it ought to be ##Nm + U/c^2##. This doesn't make sense, though, because the ##\phi_i##'s are not uniquely specified. I can shift each of them by some constant amount, corresponding to picking a different reference point for the potential. So how is the effect of potential energy on the system's mass unambiguously determined?

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# Potential energy and mass/energy equivalence

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