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Possibly a very amateur misunderstanding, but if it were possible for an object to appear out of nowhere and remained and rest, then its position would not have moved at all. Since the equation for calculation of GPE relies on change in position rather than the position itself, if this was zero then surely the gravitational potential energy would be zero, too? However, if it were on Earth and you dropped it, it would fall. Wouldn't that suggest that the object does in fact have GPE? Or perhaps is this just an impossible theoretical situation?

Thanks in advance for any help!

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    $\begingroup$ You can't spontaneously make something from nothing. The rest energy (mass) of the new thing had to come from somewhere. If you carefully account for the energy budget, there won't be a problem $\endgroup$
    – Paul T.
    Jun 3 at 12:46

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Yes.

Gravitational potential does not require a change in position, it requires a hypothetical change in position. Wikipedia defines gravitational potential as the energy "that would be needed to move an object to that location from a fixed reference location" (emphasis mine). Two masses always have gravitational potential with respect to each other. It's possible to calculate the potential energy of a mass with respect to some point even in total absence of any information of how it got there, whether it was positioned through normal methods, or was magically put into place by some physics-defying method. A mass in a gravitational field will fall when released - to suggest that a mass spontaneously appearing in a gravitational field might not have potential energy implies that the mass can defy gravity. Assuming the only thing "special" about the mass is its spontaneous appearance, it will be affected by gravity just like every other bit of matter.

Furthermore, gravitational energy is always computed with respect to some particular point. It's not mathematically possible for a mass to have zero potential energy with respect to all points, as that implies there is zero potential energy difference between any of those points, which is clearly not the case. A mass cannot have 0 PE with respect to the earth and 0 PE with respect to the sun, as that implies there is 0 PE between the earth and the sun, which is obviously untrue. Note that we can calculate the earth's PE with respect to the sun regardless of whether the earth was positioned by hard physics, or placed by a divine being. Potential energy describes a state, not the process that led to it. Once the object magically appears, the state is indistinguishable from one where the object was placed normally. The object is there, so it has potential energy - how it got there is irrelevant.

Of course, a mass "spontaneously appearing" may be physics-defying in itself, but if we take that at face value and assume all other physics operate as usual, the mass must have potential energy.

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  • $\begingroup$ To remove the objections to the idea of a mass "spontaneously appearing," can't you simply consider that the mass moved into position from very far away? $\endgroup$
    – D. Halsey
    Jun 3 at 18:11
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In reality, we don't know with complete certainty. We've never seen this happen. While we have excellent mathematical models you cannot consistently ask a mathematical model for a prediction of an effect when it also asserts that the cause is impossible. Conservation of energy forbids this. So we strongly doubt this could happen.

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    $\begingroup$ I disagree, it's perfectly reasonable to analyze a physics problem computing the potential energy of an object relative to some point, even with no information about how it got there. You can calculate the PE of a rollercoaster at the top of a hill without knowing or caring whether it climbed the hill normally or somehow popped into existence in that position. Regardless of how the object gets there, it will fall when released, so it must have potential energy. To say "we don't know" implies that the object could possibly just hang there, defying the law of gravity. $\endgroup$ Jun 3 at 14:34
  • $\begingroup$ @NuclearHoagie How would you test your hypothesis? $\endgroup$
    – John Doty
    Jun 3 at 15:23
  • $\begingroup$ It's straightforward to prove that a mass cannot have 0 PE with respect to all points. If the object has zero PE with respect to any arbitrary point, then all points have zero PE with respect to each other, which contradicts physical reality. It's mathematically impossible for a mass to have zero PE with respect to all points, it's like claiming there's a number which has zero difference from all other numbers. Potential energy analyses only care about states, not how those states came to be - the notion of "cause" is irrelevant. $\endgroup$ Jun 3 at 15:38
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    $\begingroup$ I do not understand the downvotes to this answer. In my opinion, this is the only answer compatible with the known Physics. I doubt that the original question can be considered within mainstream Physics. It is well known that from a false hypothesis, it is possible to prove everything (ex falso quodlibet). $\endgroup$
    – GiorgioP
    Jun 3 at 15:53
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    $\begingroup$ @NuclearHoagie In your example of two protons, you are forgetting that just before the collision in the same space-time region there is the same energy as after. $\endgroup$
    – GiorgioP
    Jun 3 at 15:55
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Let's imagine that you used a Star Trek type transporter to make a big rock suddenly appear in space. Then yes certainly this would warp the space-time surrounding the Rock and gravity would occur and spread at the speed of light across space. But of course as others have said above, not quite possible yet and it would take a heck of a lot of energy to make that matter appear in empty space.

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