# Effect of gravitational potential on energy

Gold Member
In a previous thread, I described a "double potential energy paradox". I allowed the discussions to stray off topic (resulting in locking of the thread), and I also thought someone had found a promising way to resolve the paradox. However, the main point was actually left unresolved, and I'm still trying to understand whether I've made a mistake (which is quite possible, but I can't find it myself) or whether I've found a different way of looking at the location of gravitational energy in a Newtonian approximation.

Consider a system containing multiple massive objects, where the field is weak everywhere and velocities are non-relativistic. In a linearized GR approximation (as for example described in Wald "General Relativity" section 4.4) described relative to a flat background coordinate system, we find that the time component of the metric causes a fractional decrease in clock rate (that is, time dilation) by the sum of the Newtonian potentials due to each source.

When an object (not necessarily of negligible mass) is within a gravitational potential, all time rates and rest energy values relating to local measurements are decreased by the time dilation when observed in the background coordinate system. This for example explains the red-shift of photons produced at a lower potential when compared with photons produced by a similar reaction at a higher potential, as in the Pound-Rebka experiment.

When the potential is assumed to be created by a fixed central mass, the difference in the time-dilated rest energy at different potentials is exactly the same as the difference in potential energy (and for a free-falling body is the negative of the difference in kinetic energy, preserving the total energy). This suggests that these two quantities could be identified with one another.

However, this model only takes one side of the picture into account. If we consider for example a two-body Newtonian case, where two masses m_1 and m_2 are orbiting about their center of mass with at distances r_1 and r_2, then in the time dilation model, both of these masses would experience a decrease in rest energy due to the potential of the other one.

In the pure Newtonian model, the total potential energy, relative to when the masses are at infinite separation, is

-G m_1 m_2/(r_1 + r_2).

In this case the potential energy can be unambiguously shared between the two masses proportionally to their radial distances from the center of mass (or equivalently inversely proportionally to the magnitudes of the masses).
The potential energy of each mass is then given by:

-G m_1 m_2/(r_1+r_2) * r_1/(r_1+r_2)

-G m_1 m_2/(r_1+r_2) * r_2/(r_1+r_2)

or using the mass ratio instead as

-G m_1 m_2/(r_1+r_2) * m_2/(m_1+m_2)

-G m_1 m_2/(r_1+r_2) * m_1/(m_1+m_2).

However, in the time dilation model, the potential due to each of the two masses still gives rise to the full time dilation at the location of the other. For example, the potential due to the second mass at the location of the first is

-G m_2/(r_1+r_2)

and the energy loss is therefore given by multiplying this by m_1, which gives the same as the total potential energy. For the other mass, the terms m_1 and m_2 are switched round, but the result is the same. This says that the total energy decrease due to the time dilation effect is exactly twice the change in potential energy.

In fact, the same factor of two appears in all cases where both sides of the time dilation effect are considered, including the case of a single central mass built up from smaller masses. This means that the change to the energy due to the time dilation cannot after all be identified with the location of the potential energy, as there is always a factor of two mismatch.

So what is the explanation for this mismatch?

1. Could there be a self-consistent way that the time dilation could be "shared out" (so that for example two equal masses would each affect each other's time rate by only half the usual amount)? I couldn't find one, and this idea seems to conflict directly with the linearized GR expression for the scalar potential.

2. Could it be that although the locally energy is simply multiplied by the time dilation in the context of the Pound-Rebka experiment, there is some additional first-order factor which applies when considering a system of multiple masses, which for example would decrease the effect by as much as a half when two equal masses were involved?

3. Could there be some other positive internal form of energy created when a mass is in a potential which has the effect of compensating for the double decrease due to time dilation?

For example, in a static central mass situation, the GR "Komar mass" expression adds to the energy another term which consists of the integral of three times the pressure over the volume of the central mass. This gives the right mathematical result for the central mass case, but doesn't extend to the two-body dynamic case. This pressure term can be loosely likened to twice the thermal kinetic energy of a monatomic ideal gas, as in the virial theorem. However, it doesn't seem to make a lot of sense as a real physical quantity unless one adds a restriction that the "Komar mass" expression only applies to a monatomic idea gas.

In the case of two bodies in free fall in each other's potential, it seems that any pressure induced in the other body could only be a tidal effect, and would therefore be of the wrong order to have a first order effect on the energy change due to time dilation.

4. Finally, it could perhaps be that the above factor of two is actually correct, and that the missing energy (equal in magnitude to the potential energy) has gone somewhere else in the system. In that case, the mathematically obvious solution is that the missing energy has gone into the gravitational field, and is identical in form to the Maxwell energy density of an electrostatic field, as this gives exactly the correct amount of energy for any combination of sources (at least in the weak-field non-relativistic approximation). This solution would also provide a specific explanation as to where gravitational energy is located in the linear approximation.

I've previously been offered various other suggestions which all essentially say that it's obviously not valid to apply the time dilation to the local rest energy to get the effective rest energy in the background coordinates, because, as I've shown, it gives the wrong answer. This doesn't seem to me to be a very scientific approach. If there's a problem extending the GR time dilation effect from the Pound-Rebka situation to the Newtonian two-body system, I'd like to know specifically why the problem arises and what the corrected solution should be.

If there's a problem extending the GR time dilation effect from the Pound-Rebka situation to the Newtonian two-body system, I'd like to know specifically why the problem arises and what the corrected solution should be.

Hmm. It feels sort of unsafe to mix a Newtonian concept like gravitational potential energy with a very relativistic effect - time dilation. This could lead to some double accounting.

If we consider for example a two-body Newtonian case, where two masses m_1 and m_2 are orbiting about their center of mass with at distances r_1 and r_2, then in the time dilation model, both of these masses would experience a decrease in rest energy due to the potential of the other one.

Do you calculate the potentials using the dilated or undilated masses ( so to speak ) ?

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Gold Member
Hmm. It feels sort of unsafe to mix a Newtonian concept like gravitational potential energy with a very relativistic effect - time dilation. This could lead to some double accounting.

Do you calculate the potentials using the dilated or undilated masses ( so to speak ) ?

For purposes of the linearized approximation, I'm assuming the original masses; this choice only has a second-order effect, which is far too small to be relevant to the original paradox, and is disregarded anyway in the linearized approximation. The potential and the related time dilation are of the order of Gm/rc^2, and this is where the factor of two arises. Any correction to the potential due to the effect of m being itself in a potential is of the order of (Gm/rc^2)^2, which is negligible in comparison.

Gold Member
Please can some GR expert help with this?

I don't consider "you are mixing Newtonian concepts with GR" a helpful answer in this case. After all, the PPN formalism is in a sense based on Newtonian principles and can be used to analyze GR to at least one higher order of accuracy than I need to resolve this paradox. I really want to know what SPECIFICALLY is wrong, and I've even given a list of options to help narrow it down, the majority of which involve me having made a mistake.

I've read bits of MTW, Wald, Rindler and Dirac books on GR without finding any clear discussion of this area. Both MTW and Wald discuss energy within GR, and their arguments as to why there cannot be energy in the field seem perfectly valid to me. I've covered bits of paper with expansions of tensors to check whether I might have missed some term which was significant at that level. I can't use GR to describe multiple moving masses exactly (and I'm not sure whether anyone else can) but I can use linearized GR approximations, and those seem to be well within their zone of validity in this case.

pervect
Staff Emeritus
In a previous thread, I described a "double potential energy paradox". I allowed the discussions to stray off topic (resulting in locking of the thread), and I also thought someone had found a promising way to resolve the paradox. However, the main point was actually left unresolved, and I'm still trying to understand whether I've made a mistake (which is quite possible, but I can't find it myself) or whether I've found a different way of looking at the location of gravitational energy in a Newtonian approximation.

Consider a system containing multiple massive objects, where the field is weak everywhere and velocities are non-relativistic. In a linearized GR approximation (as for example described in Wald "General Relativity" section 4.4) described relative to a flat background coordinate system, we find that the time component of the metric causes a fractional decrease in clock rate (that is, time dilation) by the sum of the Newtonian potentials due to each source.

When an object (not necessarily of negligible mass) is within a gravitational potential, all time rates and rest energy values relating to local measurements are decreased by the time dilation when observed in the background coordinate system. This for example explains the red-shift of photons produced at a lower potential when compared with photons produced by a similar reaction at a higher potential, as in the Pound-Rebka experiment.

When the potential is assumed to be created by a fixed central mass, the difference in the time-dilated rest energy at different potentials is exactly the same as the difference in potential energy (and for a free-falling body is the negative of the difference in kinetic energy, preserving the total energy). This suggests that these two quantities could be identified with one another.

However, this model only takes one side of the picture into account. If we consider for example a two-body Newtonian case, where two masses m_1 and m_2 are orbiting about their center of mass with at distances r_1 and r_2, then in the time dilation model, both of these masses would experience a decrease in rest energy due to the potential of the other one.

In the pure Newtonian model, the total potential energy, relative to when the masses are at infinite separation, is

-G m_1 m_2/(r_1 + r_2).

The total potential energy is actually not this simple expression. This expression does not include the binding energy of m1 and m2.

The problem is that if m1 and m2 are point masses, the binding energy for each mass is infinite. Thus the problem, formulated in terms of point masses, is not formulated in a term that will be acceptable to GR. The closest approach to a point mass in GR would be a black hole. A black hole is not able to be approximated by weak field approximations, however.

Thus, you need to consider both m1 and m2 as being some sphere (or some other shape) of finite density with some finite non-zero dimension to get results where you can apply linearized GR.

Then at any point, you can integrate G dM / r to get the total Newtonian potential energy at that point.

This will be a lot of work, so I'm not going to attempt it.

Note that this potential will also give the linearized approximation to the time dilation at this point. Thus at the surface of m1, there will be time dilation due to both m1 and m2. If m1 and m2 are spherical, the point on the surface of m1 closer to m2 will have more time dilation than the point on the surface of m1 further away from m2.

Also note that a first order PPN approximation won't include the effects of kinetic energy. Velocities are assumed to be small, so terms of the order of .5 m v^2 will be ignored.

I think you would be better off assuming that m1 and m2, aside from being distributed, were held apart by a lightweight rod - and ignoring the stress terms - rather than dealing with the orbiting case.

Gold Member
Thanks, Pervect, for a constructive answer.

I don't think there is any problem with the binding energy of each mass, because if each mass is spherically symmetrical, not too dense and sufficiently far away from the other one to avoid tidal effects, the binding energy should simply act like part of the rest mass, so this would merely be a constant higher order correction on the original rest mass, and would scale in the same way under time dilation or potential energy calculations. The paradox after all relates to potentials, not fields or tides.

Admittedly, the same paradox applies to the binding energy of the mass itself, in that the potential energy of the mass due to itself is half of the energy change due to its own time dilation, but for purposes of the two-body system, each body can be assumed to have an effective overall rest mass which includes any binding energy and scales linearly (at least to first order) with time dilation.

I can't ignore kinetic energy if it is present in this model, as it is of the same order of magnitude as the potential energy. In general, terms in (v/c)^2 need to be included (as this is of the same scale as Gm/rc^2) but terms in (v/c)^4 do not. I am assuming that any kinetic energy is mv^2/2 (although the m in this case could be the mass before or after applying the time dilation, as this only has a higher order effect).

However, the kinetic energy appears on both sides of the paradox comparison, so one can alternatively assume a scheme where the potential energy is extracted by means of an external framework or rod, as you suggest. I personally think that the free fall case (such as a two-body orbiting system) is simpler, because you don't have to consider where internal energy might exist within some supporting framework.

I think the key point is the concept that the effective rest energy of some part of a system at rest in the background coordinate system is given by the local rest energy scaled by the time dilation factor from the metric. This seems intuitively correct, and from gravitational red-shift calculations and the Pound-Rebka experiment it seems that this is accepted as standard. However, if this is applied to a case involving multiple sources it seems to lead to this paradox, even though such cases lie well within the region in which the linearized approximation says that potentials should add linearly. Does this assumption about the energy rely on some condition which is definitely violated in the multiple-source case, even in the weak field approximation, or is this part of the paradox valid, in which case the resolution is elsewhere?

pervect
Staff Emeritus
Thanks, Pervect, for a constructive answer.

I don't think there is any problem with the binding energy of each mass, because if each mass is spherically symmetrical, not too dense and sufficiently far away from the other one to avoid tidal effects, the binding energy should simply act like part of the rest mass, so this would merely be a constant higher order correction on the original rest mass, and would scale in the same way under time dilation or potential energy calculations. The paradox after all relates to potentials, not fields or tides.

You keep saying that there is a paradox, but there isn't any.

g_00 is equal to 1-2U in the weak field approximation. The "time dilation" factor, aka the redshift factor, the square root of g_00, is equal to 1-U. (This can be demonstrated by a taylor series approximation of the square root of 1 - (a small number).)

U is a positive number, the negative of the Newtonian potential per unit mass. (The Newtonian potential is negative, U is positive). In geometric units, U is dimensionless (energy and mass have the same units, so energy/mass is dimensionless).

For a static system, the Komar mass is multiplied by the redshift factor. This ONLY works for a static system where the Komar mass applies. That's why I suggest working out the case of the system held together by a lightweight rod. In the weak field, we can ignore the pressure terms, so the Komar mass contributed by a volume element dV is

[clarify] as per http://en.wikipedia.org/wiki/Komar_mass

(1-U) (rho+3P) dV

(1-U) is the redshift factor

rho is the density, as measured in a locally Minkowski frame
P is the pressure as measured in a locally Minkowski frame, assumed to be zero
dV is the volume as mesured in a locally Minkowski frame

but since P=0 because of the weak field approximation, this is (1-U) rho dV or (1-U) dM

I.e. the mass element dM gets multipled by the redshift factor (1-U) for a total contribution of (1-U) dM to the Komar mass integral.

So when we integrate the Komar mass of the system we get

Komar mass$$= \int dM - \int U dM$$

The first term is just the rest mass of the system.

The second term is the intergal of the Newtonian potential energy per unit mass * dM, so it is equal to the total gravitational binding energy of the composite system.

Thus in the linear approximation, the total Komar mass of the system is the rest mass of the system minus the Newtonian binding energy, as expected.

And that's all there is to it. You are imagining a problem where there isn't any.

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Gold Member
You keep saying that there is a paradox, but there isn't any.

...(calculations snipped)...

Thus in the linear approximation, the total Komar mass of the system is the rest mass of the system minus the Newtonian binding energy, as expected.

And that's all there is to it. You are imagining a problem where there isn't any.

I'm already happy that the Komar mass gives the expected result, as does the mathematically equal MTW method. I don't think your assumption in the quick calculation that P=0 is valid, as it is the 3P term which actually makes it work; the integral needs to take into account the fact that the potential varies with location, but when it's calculated correctly I'm sure it works. This is covered in Newtonian terms in that MTW exercise.

However, neither the Komar mass formula nor the MTW method (which effectively includes an extra time dilation factor) seems physically meaningful to me, and neither can be extended to the case of multiple masses.

My original point (the paradox) was that when multiple sources are present, the total energy of a configuration as calculated using the GR linear approximation, applying the time dilation factor to the SR energy, is less than the energy of the original source masses at infinity by twice the potential energy, rather than only once.

As far as I can see, this paradox is a direct consequence of standard theory, and I'd like to find an answer within the scope of that theory. However, as GR itself does not support the concept of the location of gravitational energy in curved space, I can understand that there is a tendency to answer this paradox by just quoting this fact about GR. There may well be a reason why this spills over to the linear approximation case, but if so I'd like to see the reason; even in GR it is normally possible to define pseudotensor quantities which behave like a conserved energy flow in a particular frame of reference, and look like energy in the field as far as that frame is concerned.

The only mathematical solution I've found to the paradox which does not cause any obvious physical inconsistencies at this level of approximation is, as previously mentioned, to assume that there is energy in the field analogous to the electrostatic energy in Coulomb theory but opposite in sign, which exactly cancels out the excess potential energy deficit and makes it up to the correct overall value. It also means that energy is totally conserved in this model and flows continuously through space, which seems surprisingly neat. As far as I can tell, this scheme also gives the same result as the Komar mass, MTW mass calculation and ADM mass for the special cases where those apply. If this idea is consistent with GR, this "energy" in the field obviously cannot be "energy" in the GR tensor sense, because that is zero outside masses. However, it could be a frame-dependent pseudotensor quantity without conflicting with GR.

If this idea can be shown to be consistent with GR, it might be a useful way to understand the energy flow in the linear approximation. That's the direction which I'm hoping to pursue in the near future.

If it can be shown to be inconsistent with GR, then I'd like a GR explanation of the paradox, including some other way to describe the energy distribution in the linear approximation which copes consistently with multiple sources.

Of course, if the idea can be shown to be inconsistent with GR but there isn't an alternative GR explanation for the paradox, then I'd consider that to be yet another niggling piece of evidence that suggests there might be something a bit wrong with GR, but that would be outside the scope of this forum.

pervect
Staff Emeritus
The Komar mass approach will NOT assign any energy to the field, but works only for static systems. It's by far the simplest approach to energy in GR, though.

The ADM approach will work for any asymptotically flat space-time, but is far more complicated. I believe that the ADM approach does wind up assigning a negative energy to "empty space", but I haven't worked it out in detail myself, and I don't have any textbook references that work out the details either.

So which approach is right? Why, both of them. There is fundamentally no way to assign energy to a specific location in GR, due to Noether's theorem.

MTW and Wald also work out energy using the psuedotensor approach. I believe that the psuedotensor gives results equivalent to the ADM approach (as both are defined for asymptotically flat space-times), but on a quick search I couldn't find a specific reference for this, so it's possible I may be mistaken.

MTW's pseudotensor approach is on pg 465, the energy density is called $$T^{\mu\nu}_{eff}$$

Wald's approach using psuedotensors is on pg 84-86, with more relevant notes on pg 292.

Note the following:
wald pg 85 said:
Furthermore, t_ab is not even gauge invariant.... This reflects the above mentioned fact that there is no meaningful notion of the local stress-energy of the gravitational field in general relativity

I interpret this remark to mean that there is no single meaningful notion of the location of energy in GR, i.e. we have different notions depending on our gauge choice.

In reviewing your calculations from six months ago, you'll never get anywhere using point masses in GR, and your statement

However, in the time dilation model, the potential due to each of the two masses still gives rise to the full time dilation at the location of the other

is just wrong, the time dilation in the linearized theory will be given by the Newtonian potential U from both masses.

Note that the Newtonian potential U of a point mass is infinite - that's why you just can't use point masses and get a meaningful number.

Or as I said earlier

Thus, you need to consider both m1 and m2 as being some sphere (or some other shape) of finite density with some finite non-zero dimension to get results where you can apply linearized GR.

pervect
Staff Emeritus
I'm pretty sure that the energy balances out to the first order as far as linearized theory goes.

Note that a completely general system will emit gravitational waves, for instance if you have a planet orbiting a sun the general solution will include energy emitted in the form of gravitational waves.

If you want to explore the gravitational wave emission energy, you should use the Bondi mass instead of either the ADM mass or the Komar mass.

Gold Member
The Komar mass approach will NOT assign any energy to the field, but works only for static systems. It's by far the simplest approach to energy in GR, though.

Agreed.

I'm not so familiar with the ADM approach, but as it seems that none of the approaches work for two separate sources there isn't anything to compare against.

I'll have another look at your references on the pseudotensor approach, thanks.

Perhaps I should have mentioned Noether's theorem specifically rather than just saying that "GR itself does not support the concept of the location of gravitational energy in curved space", but I am of course aware of it.

Note that the Newtonian potential U of a point mass is infinite - that's why you just can't use point masses and get a meaningful number.

I'm aware of course that point masses can't be used in calculations because of the infinite self-interaction. I'm not assuming that my masses are points; I'm considering objects of finite density, such as spheres which are small compared with the distance between the masses, and assuming that their self-energy is taken into account in the overall mass assigned to each object, which already depends on its own potential. I'm then assuming that each other object will add its additional potential, so when I talk about the potential energy extracted when bringing the masses into some configuration, I'm assuming that each mass is now affected by the others, but that the effect on itself is unchanged, at least to that level of approximation.

I must admit that I'm beginning to suspect that this idea is unfortunately going to be provably inconsistent with GR; for example, if there were positive energy in the field and a central mass was decreased by twice its potential energy of formation, then outside the weak field limit that would prevent black holes (as for exponential metrics like the spherical solution in Watt-Misner theory).

Gold Member
I'm pretty sure that the energy balances out to the first order as far as linearized theory goes.

Note that a completely general system will emit gravitational waves, for instance if you have a planet orbiting a sun the general solution will include energy emitted in the form of gravitational waves.

If you want to explore the gravitational wave emission energy, you should use the Bondi mass instead of either the ADM mass or the Komar mass.

The case in which the paradox arises is the trivial case of having two or more sources, and I think it's sufficient to describe the case where they are instantaneously at rest relative to one another. If there's a problem with "instantaneously", then feel free to introduce strings or rods to hold them in place, but that makes things more complicated. I'm not interested in gravitational waves; that's an effect which is orders of magnitude weaker than this paradox.

Gold Member
After various internet searches, I've now discovered that Walter Thirring has been along this path before (in 1959), and more recently various other people studying "field theories of gravity" have come to the same general conclusion that I have:

For a field theory of gravity to make sense in a flat space relativistic approximation, there must be energy in the field with a positive density of 1/(8 pi G) (del phi)^2 and the interaction energy within the masses themselves must be (phi rho) rather than a half of this.

Unfortunately, this definitely doesn't agree with GR, because a positive energy density in the field prevents black holes. I mentioned this suspicion earlier in this thread, but one of the papers I found confirms it.

Here are some of the papers I found on-line on the subject:

"Field Theory of Gravitation: Desire and Reality", Yurij V. Baryshev, gr-qc/9912003

"On Energy-Momentum Tensors of Gravitational Field", A.I.Nikishov, gr-qc/9912034

(There is another later version of this paper available on the internet, presumably as published in some Russian journal, with various typographical improvements and corrections).

"Problems in field theoretical approach to gravitation", A.I.Nikishov, gr-qc/0410099

So far I have not found on-line copies of the referenced papers by Walter Thirring, but here are the references:

1. W.E.Thirring, Fortschr. Physik. 7, 79 (1959).
2. W.E.Thirring, Ann. Phys. (N.Y.) 16, 96 (1961).