Let us presume that I have a capacitor and suddenly charge it. This induces a (miniscule) spacetime curvature. Nearby objects would experience a gravitational attraction and by extension would have nonzero gravitational potential energy with respect to this localized energy. Where did that gravitational potential energy come from? Ignore for the moment that this gravity field would be so small as to be negligible in all but the most pedantic of circumstances.
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$\begingroup$ "The map is not the territory." Relativity may be the best model we have to explain/predict our observations of reality, but it is still a model. It is possible, however unlikely, that someday someone will come up with a simpler one that has as good or better predictive value. $\endgroup$– WGroleauCommented Feb 26, 2023 at 0:44
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$\begingroup$ different (and apparently not as catchy) but related: Gravitational field from virtual photons; does $\rho c^2 = \frac{1}{2} \epsilon_0 |E|^2$? $\endgroup$– uhohCommented Feb 26, 2023 at 10:06
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2 Answers
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Where did that gravitational potential energy come from?
The gravitational potential energy came from the battery. As the battery is drained to produce the energy in the capacitor it has less gravitational attraction and objects near it lose gravitational potential energy.
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The answer already provided is correct, but I'll point out that this is a common misunderstanding I see in the popular explanations of relativity. You see things like "as the bicyclist pedals faster and faster, he gets heavier and heavier" and the like. This is not really correct.
If you take mass energy equivalence to mean what it actually says, and you take its conservation to mean what it says, then the energy was already stored in the bicyclist before starting pedaling (in the form of fat, for example). From this perspective, as the bicyclist is pedaling, the energy just changes form from potential to kinetic, and the mass energy equivalence applies to both. This is a little bit off, because the bicyclist uses oxygen from the air, but a self contained battery, like your example, uses nothing external, and so could power a motor. Additionally, work is constantly provided to overcome air resistance, and so the bicyclist (or any other vehicle) is getting lighter as its energy stores are transferred to the air by this friction. However, the atmosphere is getting warmer from viscous friction, and so heavier.
Energy conservation means that the volume integral of energy density over all of space is a constant, and the mass energy equivalence means that there is the famous linear relation between the two, so that must be constant as well. The increase in mass of your capacitor accompanies a decrease in mass of the battery and just a rearrangement of gravitational fields.
answered Feb 24, 2023 at 20:30