# Gravitational potential energy and rest mass

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• e2m2a
In summary, the rest mass of an object increases when it acquires gravitational potential energy. This is largely a matter of conventions and definitions rather than physics, as so often happens in relativity.f

#### e2m2a

Does the rest mass of an object increase when it acquires gravitational potential energy, and if so, is this the reason why Einstein believed that the inertia of a mass increases in the presence of other masses?

This is largely a matter of conventions and definitions rather than physics, as so often happens in relativity.

A free-falling object in a static gravitational field in a conventional coordinate system has constant total energy, which can be considered to be due to the rest energy being decreased or increased by the potential energy and the change being converted to kinetic energy. Note that the rest energy decreases in a lower potential, closer to other massive objects.

I think that Einstein's original idea about inertia in this context was almost opposite to this, in that he initially noted that the coordinate speed of light slows down in a potential well (by twice the fractional amount that the rest energy decreases), which means that the coordinate acceleration induced by a fixed force also decreases in a potential well, which he attributed to an increase in inertia. That was however a very early way of looking at things, and I don't think it really matches current thinking in that respect.

In more complex situations, there is no really satisfactory conventional definition of either potential energy or inertia involving multiple masses in General Relativity, although there are useful approximations in practical situations.

An interesting aspect of inertia is that if nearby masses are accelerated, GR predicts that a test particle will in theory experience a "frame-dragging" linear acceleration in the same direction (much too small to be measurable with current technology) which means that it would need a force applied to keep it at rest. If this idea is pushed to the limit, then if the whole universe were accelerated relative to the test particle, it appears that the "sum for inertia" of the effects of every mass would mean that the mass would experience an acceleration which would be of the right magnitude to keep it exactly in step with the motion of the universe. It seems neat that this apparently explains inertia as a purely relative effect (conforming to "Mach's Principle") but unfortunately the details don't work out correctly with General Relativity, as this explanation of inertia appears to require a gravitational "constant" that actually varies with location and time instead of being a universal constant as assumed in GR (and supported by experiment).

This is largely a matter of conventions and definitions rather than physics, as so often happens in relativity.

A free-falling object in a static gravitational field in a conventional coordinate system has constant total energy, which can be considered to be due to the rest energy being decreased or increased by the potential energy and the change being converted to kinetic energy. Note that the rest energy decreases in a lower potential, closer to other massive objects.

I think that Einstein's original idea about inertia in this context was almost opposite to this, in that he initially noted that the coordinate speed of light slows down in a potential well (by twice the fractional amount that the rest energy decreases), which means that the coordinate acceleration induced by a fixed force also decreases in a potential well, which he attributed to an increase in inertia. That was however a very early way of looking at things, and I don't think it really matches current thinking in that respect.

In more complex situations, there is no really satisfactory conventional definition of either potential energy or inertia involving multiple masses in General Relativity, although there are useful approximations in practical situations.

An interesting aspect of inertia is that if nearby masses are accelerated, GR predicts that a test particle will in theory experience a "frame-dragging" linear acceleration in the same direction (much too small to be measurable with current technology) which means that it would need a force applied to keep it at rest. If this idea is pushed to the limit, then if the whole universe were accelerated relative to the test particle, it appears that the "sum for inertia" of the effects of every mass would mean that the mass would experience an acceleration which would be of the right magnitude to keep it exactly in step with the motion of the universe. It seems neat that this apparently explains inertia as a purely relative effect (conforming to "Mach's Principle") but unfortunately the details don't work out correctly with General Relativity, as this explanation of inertia appears to require a gravitational "constant" that actually varies with location and time instead of being a universal constant as assumed in GR (and supported by experiment).
I think your reply is interesting and raises more questions. I have searched the internet for an answer to this question, and surprisingly, I found very little information that gives a definitive answer. Pick any other subject in relativity and you will find tons of websites. But this particular question about gpe and rest mass seems to be unanswered or controversial. This surprises me because in view of all the recent confirming experiments of general relativity-- gravity probe b, gravity waves-- there are still questions in relativity that the answers are not clear.
I believe reading an excerpt from a paper Einstein wrote in a compilation by Stephen Hawking, where Einstein implied that the rest mass of an object is indeed affected by its gpe.

Note that even in Newtonian gravity, potential energy doesn't apply to a single mass, but rather to a configuration of masses, and there is no unambiguous way to assign it to specific masses in the general case; you can move A closer to B then take B away from A, and you can't say which one had what energy when. It is of course a complication that for GR we expect potential energy to contribute to the source strength and therefore we want to know where it is, but such Newtonian analogies tend to run into major problems.

There is a partial solution in Newtonian theory which is to assign energy density ##g^2/8 \pi G## to the field, in which case each source mass can lose the full potential energy due to all other masses and it is balanced by the same energy with a positive sign in the field. Something similar applies in GR, in that there are various pseudotensor quantities which can represent the relative gravitational energy density from a particular point of view, but this is a controversial and confusing area.

Assume this hypothetical situation. Suppose there is a test mass on the surface of a planet far removed from any appreciable gravitational affects of other planets, stars, etc. Just the one planet and the one test mass. In this hypothetical, does special relativity or gr predict that when you lift the test mass off the surface of the planet, because its gpe has increased, relative to an obeserver on the surface of the planet, does the rest mass of the test mass increase equal to the increase in gravitational potenial energy of the test mass divided by the speed of light squared?

Assume this hypothetical situation. Suppose there is a test mass on the surface of a planet far removed from any appreciable gravitational affects of other planets, stars, etc. Just the one planet and the one test mass. In this hypothetical, does special relativity or gr predict that when you lift the test mass off the surface of the planet, because its gpe has increased, relative to an obeserver on the surface of the planet, does the rest mass of the test mass increase equal to the increase in gravitational potenial energy of the test mass divided by the speed of light squared?
If you're not interested in what happens to the energy of the planet and you can assume the planet to be at rest, you can use that model within GR.

However, if you consider what happens to the energy of the planet when it is moved a little away from the test mass by the same action, you will get the same result for the change in potential energy of the planet, which doesn't add up, because the total energy imparted to the test mass plus planet system as a whole is only equal to the potential energy, so you then have to assume the loss of some other energy in the gravitational field to make up for it. But GR says there is no energy (in the sense of gravitational source energy) in the field. So at least one of these "energy" quantities is some sort of "effective" energy not real energy. And I'm puzzled about it too.

If you're not interested in what happens to the energy of the planet and you can assume the planet to be at rest, you can use that model within GR.

However, if you consider what happens to the energy of the planet when it is moved a little away from the test mass by the same action, you will get the same result for the change in potential energy of the planet, which doesn't add up, because the total energy imparted to the test mass plus planet system as a whole is only equal to the potential energy, so you then have to assume the loss of some other energy in the gravitational field to make up for it. But GR says there is no energy (in the sense of gravitational source energy) in the field. So at least one of these "energy" quantities is some sort of "effective" energy not real energy. And I'm puzzled about it too.[/But GR says there is not energy(in the sense of gravitational source energy) in the field]
Question about this. I thought gravitational waves do have energy in them? And does this imply that gravitational fields have energy? The reports from LIGO have indicated that the energy contained in the waves would be equivalent to about 2 or 3 solar masses.

Question about this. I thought gravitational waves do have energy in them? And does this imply that gravitational fields have energy? The reports from LIGO have indicated that the energy contained in the waves would be equivalent to about 2 or 3 solar masses.

Yes, gravitational waves carry energy, and one can calculate their effective energy density. That energy relates to rapid changes in the field, not the field itself.

Basically, there are at least two sorts of things we call "energy". One is local "stuff" as we would measure at that location and which acts as a gravitational source according to GR, and the other is some sort of global "stuff" within some sort of practical coordinate system which behaves more like Newtonian total energy. These can't both be the same thing, as the effective value of the local "stuff" changes according to potential. It is however mathematically possible for them both to be "conserved" at least to a good approximation.

Note also that in many ways "gravitational energy" is just an accounting term, not a physical thing. For example, if I lift an object to a greater height, it now has greater energy, but I had to supply that energy. I can't change the total energy of a system from within by reconfiguring it, in that any change in "gravitational energy" is actually simply moving energy from one place to another.

Going back to my original question, I think a paradox arises. My understanding is gravitational mass is only related to rest or proper mass. But in the hypothetical with one test mass and one planet if indeed the rest mass of both the test mass and the planet increased via the increase of their gpe, then their gravitational field would increase, which would mean their gpe would even be greater, and therefore, their rest mass would even be greater, and on and on ad infintum. Is this the case?

May I suggest a simple thought experiment:

A cannon firmly attached to the surface of a neutron star fires, cannonball flies one way, neutron star flies other way. The ratio of the two masses is the ratio of the two speeds.

This way we can find the mass of a cannonball on the surface of a neutron star.

Going back to my original question, I think a paradox arises. My understanding is gravitational mass is only related to rest or proper mass. But in the hypothetical with one test mass and one planet if indeed the rest mass of both the test mass and the planet increased via the increase of their gpe, then their gravitational field would increase, which would mean their gpe would even be greater, and therefore, their rest mass would even be greater, and on and on ad infintum. Is this the case?
Firstly, the relative increase due to gravitational potential energy is incredibly tiny, typically a few parts per trillion or billion, so there is no runaway regardless.

But as I already said, the trivial model of assuming that the rest energy of a particle is modified by gravitational potential to give it the gravitational energy simply does not work unless you also assume that there is a correcting change in energy elsewhere, for example in the field.

It is of course expected that the external gravitational field of any system should reflect the total source energy, taking any internal potential energy into account.

In General Relativity, there is an apparently consistent solution to this for any static system. The total gravitational effect of a system at rest includes an effect due to pressure. The total volume integral of the pressure over each of three perpendicular planes over the volume of a static system is equal (and opposite) to its internal potential energy in Newtonian gravity. The same effect in GR (as part of what is called the "Komar mass") appears to compensate exactly for the fact that potential energy would otherwise be applied twice, with each constituent component particle of an object both being reduced in energy by its potential energy and causing the rest of the object to be reduced by the same energy.

However, the pressure term does not provide an explanation in a dynamic system. The physicist Richard Tolman pointed out that if the pressure suddenly changed inside a star, for example as a result of some sort of collapse, then this would apparently change its gravitational field, even if no energy had been added or removed externally. This is now known as one of Tolman's paradoxes. Similarly, it does not provide an explanation for the potential energy of a system involving two bodies orbiting around one another.

As far as I know, this is still a generally unresolved area. As GR is difficult to handle analytically, especially in dynamic situations, the tiny additional effects due to potential energy modifying the source strength are not very relevant to current experimental modelling, but they are of interest on the theoretical side. If current GR is inaccurate at that level, this will not make much difference except in extremely strong gravitational situations, but it could for example affect whether black holes actually form. However, that sort of speculation is outside the scope of these forums.

May I suggest a simple thought experiment:

A cannon firmly attached to the surface of a neutron star fires, cannonball flies one way, neutron star flies other way. The ratio of the two masses is the ratio of the two speeds.

This way we can find the mass of a cannonball on the surface of a neutron star.
Well, for a start the cannon would have turned into a thin film on the surface of the star...

But ignoring that, there are problems with what time and space coordinates to use, and what you mean by the "mass" and "speed" of an extended object (especially a neutron star).

Firstly, the relative increase due to gravitational potential energy is incredibly tiny, typically a few parts per trillion or billion, so there is no runaway regardless.

But as I already said, the trivial model of assuming that the rest energy of a particle is modified by gravitational potential to give it the gravitational energy simply does not work unless you also assume that there is a correcting change in energy elsewhere, for example in the field.

It is of course expected that the external gravitational field of any system should reflect the total source energy, taking any internal potential energy into account.

In General Relativity, there is an apparently consistent solution to this for any static system. The total gravitational effect of a system at rest includes an effect due to pressure. The total volume integral of the pressure over each of three perpendicular planes over the volume of a static system is equal (and opposite) to its internal potential energy in Newtonian gravity. The same effect in GR (as part of what is called the "Komar mass") appears to compensate exactly for the fact that potential energy would otherwise be applied twice, with each constituent component particle of an object both being reduced in energy by its potential energy and causing the rest of the object to be reduced by the same energy.

However, the pressure term does not provide an explanation in a dynamic system. The physicist Richard Tolman pointed out that if the pressure suddenly changed inside a star, for example as a result of some sort of collapse, then this would apparently change its gravitational field, even if no energy had been added or removed externally. This is now known as one of Tolman's paradoxes. Similarly, it does not provide an explanation for the potential energy of a system involving two bodies orbiting around one another.

As far as I know, this is still a generally unresolved area. As GR is difficult to handle analytically, especially in dynamic situations, the tiny additional effects due to potential energy modifying the source strength are not very relevant to current experimental modelling, but they are of interest on the theoretical side. If current GR is inaccurate at that level, this will not make much difference except in extremely strong gravitational situations, but it could for example affect whether black holes actually form. However, that sort of speculation is outside the scope of these forums.

Is it possible that the correcting change in energy would be due to a tiny gravitaional wave generated when the 2 masses are momentarily accelerated and deaccelerated when they are separated?

Is it possible that the correcting change in energy would be due to a tiny gravitaional wave generated when the 2 masses are momentarily accelerated and deaccelerated when they are separated?
Sorry, no, because the amount of energy depends only on the static configuration but the energy in a gravitational wave is proportional to some power of the rate of change.

One could for example imagine two small but heavy masses held apart by a light rod. In that case, the integral of the pressure in the rod would simply be the force times the length of the rod, so it would as expected be equal to the potential energy. However, if the rod is pushed aside so the masses begin to accelerate, in the Newtonian picture this does not change the potential energy at all, at least until the masses have moved enough to make a difference, but the "pressure" term will have become zero long before that, which suggests that according to GR the effective total gravitational source effect should suddenly change by the potential energy, which doesn't seem plausible, as in the Tolman paradox.

This is one reason why I personally suspect that GR is not completely correct, but it's the best theory we have so far and seems consistent with the majority of the experimental evidence.

ok. thanks for your replies. much to ponder.

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Well, for a start the cannon would have turned into a thin film on the surface of the star...

But ignoring that, there are problems with what time and space coordinates to use, and what you mean by the "mass" and "speed" of an extended object (especially a neutron star).

Let's say we have 100 separated balls, each has mass 1 kg. We drop 99 of those balls on each other, the result is a heap of warm balls, the heap has mass 99 kg. Then we slowly and carefully lower the last ball on the heap, this ball stays cool. Let's say the heap has now mass 99.9 kg.

Now the cool ball is the lightest of 100 balls, so its mass is less than 1% of the total mass of all balls, so its mass is less than 1% of 99.9 kg, in other words its mass is less than 0.999 kg.

What are you trying to show? If you lower something carefully, you are extracting energy from the system involving both the thing you are lowering and whatever you are lowering it onto. We would normally assume that the extracted energy is associated with the thing being lowered, and that what it is being lowered onto is at rest, and this works in most simple cases.

However, it is not a general solution. If you try to look at it from the point of view of the planet or whatever onto which things are being lowered, the energy of the planet is affected by the same amount. You can try to make up rules to resolve the ambiguity (for example specifically associating the energy with force moving through a distance, so in the simple case the planet didn't move enough to make any difference to the energy) but they don't work in all cases (for example if you move object A towards object B then move object B away by the same distance, where according to that rule the energy would apparently be removed from A but added to B).

What are you trying to show? If you lower something carefully, you are extracting energy from the system involving both the thing you are lowering and whatever you are lowering it onto. We would normally assume that the extracted energy is associated with the thing being lowered, and that what it is being lowered onto is at rest, and this works in most simple cases.

However, it is not a general solution. If you try to look at it from the point of view of the planet or whatever onto which things are being lowered, the energy of the planet is affected by the same amount. You can try to make up rules to resolve the ambiguity (for example specifically associating the energy with force moving through a distance, so in the simple case the planet didn't move enough to make any difference to the energy) but they don't work in all cases (for example if you move object A towards object B then move object B away by the same distance, where according to that rule the energy would apparently be removed from A but added to B).

I tried to cleverly avoid some of that ambiguity. An observer on that "planet" made of 100 balls says that the cool ball is lighter than the others, so we trust the observer and deduce that the cool ball's mass was reduced in the lowering process.

Let's say the thing onto which an object is being lowered is a black hole. At the final stages of the lowering process the object is moving towards the black hole, and the energy being extracted is moving away from the black hole, energy of the black hole is affected by the object moving closer, and the extracted energy moving away. The black hole is not time dilated at all when the object has reached the event horizon, while the lowered object is very time dilated.

When one mass is much larger than the other, there is very little ambiguity anyway. However, it also seems reasonable to assume that in general the change in energy of both objects should be equal to the change in potential due to the other times the rest energy. But that's where the problem comes in. Where is the missing energy? Does the missing energy affect the gravitational source strength?

A similar factor-of-two problem occurs even in the electromagnetic field case, in that two charges near each other would both effectively experience a change in energy due to the potential created by the other, but that accounts for double the energy involved in placing them at those locations.

In both cases, one possibility is to assume that all of the potential energy is in the field, and none of it in the source objects. Another is to assume that each of the source objects has the full potential energy but the field has opposite energy equal to one copy of the potential energy. Another solution which is mathematically consistent but physically implausible is that every object possesses exactly half of its conventional potential energy and there is nothing in the field!

The problem overall is that "potential energy" does not have a physical location, and it's very difficult to find a consistent way of assigning one even by convention.