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I just asked this question on Can you answer it for me?

When chemical energy is released, mass is reduced ( -- if only by a negligible amount. Presumably that's true for all energy. And presumably that works in reverse as well: storing energy involves an increase in mass. It seems to follow that moving some object against a gravitational gradient, increases the mass of the object -- potential energy is being stored. Somehow that's difficult to understand. Is it really true?
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You increase the mass of a body if you increase its internal energy. But potential energy isn't internal to the body, it's a feature of the relationship between the body and the environment. So mass doesn't depend on the gravitational field you are in.
+Sean Carroll: Thanks! That makes sense. Is it all problematical to say that by moving an object against a gravitational gradient one has transferred energy from whatever provides the energy to move the object to <something>, The energy must be preserved, but it's not visible anywhere as mass or as energy. So the energy must be just in the changed configuration of objects.

I guess that works. A collection of objects (where the collection itself is considered as a single entity) may have different internal (gravitational potential) energies depending on its (internal) configuration. To go from one configuration to another may require the addition of energy from outside. This can be analogized to how chemical energy is stored by looking at where electrons are with respect to a nucleus. Is that a reasonable way to think about it?
+Sean Carroll But it now occurs to me that saying that the energy is stored in the configuration shouldn't be the entire answer. Doesn't it have to be the case that the collection of objects has (negligibly) more mass when in one configuration than another. (This should apply both to energy stored as chemical potential and as gravitational potential.) Otherwise aren't we saying that energy can be a property of nothing other than (abstract) geometry? Is that really possible? Mustn't stored energy manifest itself as mass (or additional internal motion) somehow or other?

In other words, if one has an isolated system consisting of a collection of masses and one adds energy to the system by pulling those masses apart (and then somehow locking them in that new configuration), doesn't the system have to have more mass?
I got additional answers on ( As I now understand it, the following is the case.

If one had an isolated system of masses, enclosed, say, in a rigid container, and modified it by separating the internal masses, then the system as a whole would have more mass than before. The added energy is stored in the curvature of space -- some of which would occur outside the container. That could be measured, say, by applying a force to the container and checking its acceleration. The before and after results would be different.

That all seems very strange, but I guess that's the way it is. The individual masses aren't changed but the entire system has more mass. That additional mass is stored in the curvature of space -- whatever it means to say that!
That sounds right. It doesn't really matter that the energy is stored in "the curvature of space"; it would equally well work in a Newtonian language, where you said the energy was stored in the gravitational potential field. (But it wouldn't work perfectly, because in Newtonian gravity the gravitational field itself is not a source for gravity.)
It still seems mysterious. Mass is stored as either space curvature or an energy field depending on whether one is speaking relativistic or not. Either way, it's hard for me to conceptualize mass in that way. The field is somehow "thicker," which suggests that all energy fields have mass.
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