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If I pick up a rock and set it on a ledge above my head, I do work in the process. The work I do is termed "potential energy". We know how to recover the energy (i.e. let it fall back to earth).

However, while resting above the surface the energy is said to be "stored in the gravitational field", presumably meaning the space between the elevated rock and earth. Suppose we perform this experiment on the moon where we know the space between the elevated rock and surface consists of an empty vacuum of space which contains absolutely nothing. How can empty space store energy? That is, how can "nothing" have any properties whatsoever?

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    $\begingroup$ Just because there is a vaccuum (lack or air) on the moon doesn#t mean there isn#t a gravitational field... $\endgroup$ – Polygnome Aug 6 '16 at 19:32
  • $\begingroup$ It's called "spacetime" and it is considered a physical object. The vacuum is simply not empty. Whether the vacuum energy is "stored" the same was as for other fields is not clear. Personally I don't think it is, but we don't fully understand what's going on, yet. $\endgroup$ – CuriousOne Aug 6 '16 at 19:35
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    $\begingroup$ For a standard spring, the force in the spring is $kx$. But the space-time fabric is a "strange spring" where the force is $\frac{k}{x^2}$. Space-time can be thought of as an invisible spring that can store energy. $\endgroup$ – Inquisitive Aug 6 '16 at 20:37
  • $\begingroup$ I have come to the same conclusion as "Curiousone." Space is not empty, rather has attributes such as compressibility. Space-time fabric provides a logical mechanism for the basic formation of mass (i.e., compressed space), explains why mass has energy, and provides a mechanism for the phenomenon of gravity. As a physicist who has studied gravity most of my life, I find it strange that modern physics has not embraced "space-time" as a tangible physical object that posses attributes of size, time, energy (compression), etc. $\endgroup$ – M. Pope Aug 7 '16 at 16:49
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You're right: potential energy, as taught in introductory physics courses, is a "cheat".

On a fundamental level, there's no such thing -- there's only energy stored in fields. Since field descriptions are more complicated, your course glossed over it by calling the field energy "potential energy". It's totally fine for solving problems, since the bookkeeping is the same, but it can be confusing if you ask where the energy actually is. This is a big logical hole in intro courses.

In your case, the energy is stored in the gravitational field between the two objects. It's totally analogous to how energy is stored in the electric field between two charges. Here, the gravitational field is the metric tensor $g_{\mu\nu}$ and the presence of the masses perturbs it from the flat metric $\eta_{\mu\nu}$.

This gravitational field is not "nothing", even if there are no matter particles there. Its value, energy density, momentum density, etc. can all be measured and observed. In fact, from a modern perspective, there are only fields. Even the rock in your example is just a complicated excitation of the electron, quark, gluon, etc. quantum fields.

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    $\begingroup$ Yours is a great answer. We engineers generally don't take modern physics classes because they're not needed for low-velocity situations like terrestrial structural analysis, etc. For example, a course in Special Relativity isn't required for a B.S. in Mechanical Engineering. And it wouldn't be covered on the EIT and PE exams. Your answer makes me wish I majored in Physics so that I could have a more complete understanding about how the universe works. Again, great answer! $\endgroup$ – Inquisitive Aug 6 '16 at 20:30
  • $\begingroup$ As you said, the energy is actually stored in the space between the two be it gravity or electric since it can be measured. And I agree "the gravitational field is not 'nothing'." And there are no particles there, else we measure those too. Whatever is present is just too small to identify, perhaps on the order of Planck's length (~10-35 m). SPACE the final frontier........ $\endgroup$ – M. Pope Aug 9 '16 at 16:02
  • $\begingroup$ @M.Pope "Whatever is present is just too small to identify" is probably a misunderstanding of this answer. The gravitational field is present. Fields are the fundamental objects in the universe, not particles; particles are just excitations of fields, and observers in different frames may not even agree if a particle is present or not. $\endgroup$ – probably_someone Apr 10 '18 at 17:11
  • $\begingroup$ "there are only fields" Yet these fields describe the probability to find a particle. $\endgroup$ – my2cts Nov 23 at 10:25
  • $\begingroup$ Here is something to scratch your head over: the gravitational field between two large sheets of matter is essentially zero. Change the separation between the sheets and you're changing the volume of that "zero" region. Potential energy, then, relates not to added stresses in the region between the sheets but to reduced stresses in the overall outside volume of space. $\endgroup$ – S. McGrew Nov 23 at 14:07
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Gravitational Potential Energy is the difference in the total energy of a mass between two points in space as a direct consequence of the difference in the length of a metre between the two locations.

In other words, it is a direct result of gravitational length contraction.

The total energy of a mass can be expressed as $$E=mc^2$$

But it is also Force × Distance I.e. it's the distance that a 1N force can be applied by the mass if all of its energy is released..

During free fall, the unit of distance decreases at a rate of $$g/c^2$$ wrt distance fallen.

Therefore the total energy of a mass reduces at a rate of $$mc^2g/c^2$$ Or $$mg$$ wrt distance fallen due to gravitational length contraction.

The change in total energy over a given change in height, mgh (or more generally over a given change in gravitational potential) is what we call Gravitational Potential Energy.

Gravitational length contraction is the result of a reduction in the wavelength of light between the two locations.

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