# Double gravitational potential energy paradox

Gold Member

## Main Question or Discussion Point

I've recently noticed a very simple paradox in semi-Newtonian
gravity which seems to say that there must be some rest energy
in the field. However, this obviously conflicts with the GR
position. I'm wondering if anyone can spare a moment to clarify

In basic Newtonian gravity, the change in potential energy of a
system when a pair of objects of mass m_1 and m_2 are moved from
a distance r_1 apart to distance r_2 (and brought to rest) is of
course:

U = - G m_1 m_2 (1/r_2 - 1/r_1)

However, both of these objects can be considered separately to
have decreased in potential energy by this amount, relative to
one another, which seems to imply a total change in the potential
energy of twice the external change. In basic Newtonian theory,
this is not a problem, as the potential is relative, and is of
course subject to an arbitrary additive constant anyway.

If we go a little further and consider clock rates and scalar
potentials, we see that the clock rate of each of the objects,
and hence its rest energy, is decreased by the factor of the form
(1 - Gm/rc^2) due to the potential of the other. This has the
effect of decreasing the total energy of each of the two objects
by the above amount of potential energy and means that the total
energy of the two objects has most definitely decreased by twice
the potential energy change of the system as a whole.

This means that in order for energy to be conserved, half of the
internal potential energy change must have been balanced by a
change elsewhere in the system. As far as I can see, the only
possible candidate for "elsewhere" is the gravitational field.

The relative amount of the energy that must be in the field (to
first order) is exactly the opposite of the (negative) amount
that would be present in a corresponding electrostatic
configuration, and therefore I assume that to complete this
semi-Newtonian model, a field energy density g^2/(8 pi G) in the
same mathematical form as the Maxwell energy density of the
electrostatic field would exactly match the requirements.

Does this seem valid? If so, how does this fit with the
standard interpretation that GR denies the existence of
rest energy within the field? I feel that although this
idea involves approximations and simplifications, I cannot
see any obvious reason why it should not carry over to GR.

Note that despite the similarities, this semi-Newtonian model has
some quite distinct differences from the Coulomb electrostatic
model. In the Coulomb theory the potential energy is taken to be
half of the integral of the charge density times potential, and
this is then mathematically shown to be equal to the integral of
the field energy density, on the assumption that the potential
tends to zero at a distance. In the semi-Newtonian gravitational
case the approximate potential is of the form (1 - Sum(Gm/rc^2))
summed for all local masses, which tends to 1 rather than 0, so
there is an extra term in the integral and it doesn't have a
factor of a half. It comes out as follows:

energy of masses within potential + 2 * energy of field

= original total amount of mass

or to match the description at the start of this note:

energy of masses within potential + energy of field

= original total amount of mass - energy extracted

Jonathan Scott

Related Special and General Relativity News on Phys.org
D H
Staff Emeritus
In basic Newtonian gravity, the change in potential energy of a pair of objects of mass m_1 and m_2 are moved from
a distance r_1 apart to distance r_2 (and brought to rest) is of
course:

U = - G m_1 m_2 (1/r_2 - 1/r_1)
So far, so good. That is the potential energy of the system of two masses.

This next step is plain wrong:
However, both of these objects can be considered separately to
have decreased in potential energy by this amount, relative to
one another, which seems to imply a total change in the potential
energy of twice the external change.
You have introduced a factor of two error. The rest of your post propagates this factor of two error.

Gold Member
No, I'm not that stupid; that's exactly what I consider to be the paradox. I agree that the potential energy of the system as a whole did NOT change by twice the external change. However, in a static relativistic model each of the two bodies is red-shifted due to the potential of the other, so it has lost an amount of energy equal to the potential energy, and that definitely makes a factor of two.

pervect
Staff Emeritus
The bad news first - it is not possible to assign energy to the "gravitational field" in a consistent manner. The problem is that attempts to do so give results for the energy which are dependent on the gauge choice.

See for instance http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as 'the failure of the energy theorem '. In a correspondence with Klein , he asserted that this 'failure' is a characteristic feature of the general theory, and that instead of 'proper energy theorems' one had 'improper energy theorems' in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.
Field theories with a finite continuous symmetry group have what Hilbert called proper energy theorems'. Physically in such theories one has a localized, conserved energy density; and one can prove that in any arbitrary volume the net outflow of energy across the boundary is equal to the time rate of decrease of energy within the volume. As will be shown below, this follows from the fact that the energy-momentum tensor of the theory is divergence free. In general relativity, on the other hand, it has no meaning to speak of a definite localization of energy. One may define a quantity which is divergence free analogous to the energy-momentum density tensor of special relativity, but it is gauge dependent: i.e., it is not covariant under general coordinate transformations. Consequently the fact that it is divergence free does not yield a meaningful law of local energy conservation. Thus one has, as Hilbert saw it, in such theories improper energy theorems.'
Because GR is diffeomorphism invariant, it's symmetry groups are not finite in the appropriate sense.

You might also try http://en.wikipedia.org/wiki/Mass_in_general_relativity (an article I wrote for the wikipedia on the topic of mass in general relativity. Like all wikipedia articles, it is only wikipedian-reviewed, not peer-reviewed). I think it may be slightly easier to understand, though this is still tricky stuff.

The good news is that GR does have self-consistent concepts of global mass, namely the Komar mass and the ADM mass. These concepts have requirements which must be met before they can be applied - the Komar mass is applicable in stationary space-times, the ADM mass is applicable in asymptotically flat space-times.

To apply these to your examples, though, we either need to put the test mass in orbit around the primary mass (which makes the problem non-static, forcing us to use the ADM mass), or we have to add in some sort of support structure to hold the test mass in place.

I believe I've seen something else that may be related to your factor of 2 problem, but I don't quite understand the details well enough to be sure:

http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu30.html [Broken]

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Gold Member
Thanks very much for a helpful reply and in particular for reminding me that GR has been proved to have a problem with the location of energy; I also thought that Newtonian gravity had a problem with it until I came up with this idea that the double potential energy change for a relative displacement between two objects is exactly matched by a gain in the field energy.

I agree that to model the two-object case in GR we would need a frame to hold the masses apart. Unfortunately, I don't know how to do an exact two-body solution of that type in GR. (Does anyone?)

However, that model was simply the case that triggered the question. In the more general case of a static mass assembled from smaller masses, my simple field energy model says that the total field energy is exactly equal to the effective potential energy released in creating the system (relative to infinity), that the local energy density is g^2/(8 pi G), and that energy is conventionally conserved. I'm fairly sure that the Schwarzschild solution of GR does NOT say that.

It seems therefore that either GR is wrong (which I'm quite prepared to entertain, but that would seem to need extraordinary evidence) or that my very straightforward idea as to how semi-Newtonian gravity must work as a first-order approximation (giving rise to a field energy) is wrong.

Let's try the easy one first. Can people follow my logic about the double potential energy requiring a field energy, and if so, does it seem to be robust, or is there some sort of flaw in it?

Jonathan Scott

Your logic is right, you've identified a paradox in GR that apparently has been noticed before. Nobody here would ever even consider revising Einstein. i say, if you can find a good solution, go ahead and revise GR. If no one was prepared to revise the previous works of the greats then we wouldn't even be in a heliocentric solar system right now.

Just be prepared to be verbally flogged, scoffed at, and have all your credibility vanish until, finally, someone important, somewhere important, agrees with you, which could happen post mortem, so expect no glory.

Your logic is right, you've identified a paradox in GR that apparently has been noticed before. Nobody here would ever even consider revising Einstein. i say, if you can find a good solution, go ahead and revise GR. If no one was prepared to revise the previous works of the greats then we wouldn't even be in a heliocentric solar system right now.

Just be prepared to be verbally flogged, scoffed at, and have all your credibility vanish until, finally, someone important, somewhere important, agrees with you, which could happen post mortem, so expect no glory.
Actually just be prepared to be scoffed at full stop until you've done the groundwork, when I first came here I was assaulted on all sides by ideas and even got in a few broadsides when I finally gathered the right.

But at the end of the day, unless you really know what your talking about, you're gonna be scoffed at, and even then it's going to be healthy scoffing, the onus is always on you to prove a controversial idea(and it aint because it aint right, it's because if it is you need to make it airtight) Develop a hard shell move on and learn the deal, just as Einstein had to et al. When you have your airtight theorem subjected to peer review, come back and drown the misbegotten SOBs in a world of theory Anyway joking aside, one thing you learn from this place is, in all humility you have a lot to learn. And a lot more than you think. They aint scoffing for the sake of it, they've seen it all before.:tongue2: First learn humility.

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pervect
Staff Emeritus
Thanks very much for a helpful reply and in particular for reminding me that GR has been proved to have a problem with the location of energy;
GR does have ways to describe the total energy of a system, at least in special cases - i.e. in stationary or asymptotically flat space-times.

There isn't, however, any known universal notion of energy in GR that applies to an arbitrary space-time.

Unfortunately, while these techniques will give you the total energy of a system, they may not give that energy a specific location. So you can give a total number for the energy of a system, but you might not be able to describe in a covariant matter "where" that energy is located.

Aside from the other references I've already mentioned, you might want to check out the sci.physics.faq which also talks about some of this stuff:

http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".
As usual, I'm only posting enough of the FAQ to attempt to stimulate some interest - readers who are really interested should read the whole link.

pervect
Staff Emeritus
Your logic is right, you've identified a paradox in GR that apparently has been noticed before. Nobody here would ever even consider revising Einstein. i say, if you can find a good solution, go ahead and revise GR. If no one was prepared to revise the previous works of the greats then we wouldn't even be in a heliocentric solar system right now.

Just be prepared to be verbally flogged, scoffed at, and have all your credibility vanish until, finally, someone important, somewhere important, agrees with you, which could happen post mortem, so expect no glory.
I would suggest taking at least one semester in GR before attempting to revise it....

Maybe even TWO semesters :-)

Note that after even if you think you've successfully revised GR (not an easy task!), PF guidelines say that you'll have to get it published in a peer-reviewed journal before we'll talk about it here - i.e. this is not a forum for unreviewed personal theories, it's a forum to describe mainstream relativity (and PF in general is a forum to describe mainstream science).

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Gold Member
I studied GR at a basic level as part of my university physics courses many years ago, but although I understood in principle how to "push the buttons" to do the math, I didn't feel I was anywhere near understanding how it worked. I felt then (as I do now) that the gap between basic Newtonian gravity and GR is unnecessarily large, and that GR makes it difficult to even ask questions about what happens in trivial cases, let alone get answers.

To get a better intuitive understanding of the concepts, I took the Schwarzschild solution in isotropic coordinates and converted it back to a semi-Newtonian model which I could understand in a more intuitive way. I found for example that if you take the trivial action S = integral of mc^2 d tau but treat c as a variable which varies as the square of the GR scalar potential, you can use the Euler-Lagrange equations to derive the GR perihelion precession for Mercury. However, that simplification only works for isotropic coordinates, which need spherical symmetry.

After trying to work out various "consistency" requirements for a fully general gravity theory (for example, it must treat every mass in the universe in the same way) I came up with my own theory in 1986, based on the Machian principle that G = 1/sum(m/rc^2) for everything in the universe (or more generally a four-vector expression similar to the Lienard-Wiechert electromagnetic potential). It was later pointed out to me that Dennis Sciama had published an illustrative idea in the 1950s on the same basis, and Brans-Dicke theory had started from the same idea too. However, this theory fails the PPN beta test.

Within a couple of hours of recently discovering this field energy idea (on 14th Feb) I realized that it appears to be exactly what is needed to take my existing preferred but non-viable basis of a gravity theory and convert it to a potentially viable theory.

If you take the GR spherical solution in the PPN formalism and make two second-order changes, you can calculate the effects predicted by my modified Machian theory:

1. Replace G by 1/(Sum of m/rc^2) for everything in the universe as seen from the observation point. This has the effect of making it very slightly variable with position, changing the PPN beta from 1 to 1.5, in conflict with experiments.

2. Assume a field energy density of g^2/(8 pi G), modelled as if it were simply an additional spherically symmetrical distribution of ordinary matter. This reduces the PPN beta from 1.5 back to 1.

I wrote this up in a rather over-enthusiastic article which can currently be found on my web site at this URL:

I tried to submit a copy to the arXiv too (under gr-qc), but it was rejected by the moderators, so I'm obviously not going to have an easy time of it.

(Note added afterwards: OK, forget the Machian theory for now; I changed my way of describing what happens and as a result introduced a sign error in the beta change, which isn't a good sign).

I suspect that one can do quite a lot with this idea using a semi-Newtonian model in flat background coordinates, using the standard mathematics for potentials, source densities and so on. Unfortunately, I only have a few days of leave left before I have to return to the day job.

In the mean time, I would still be very interested to hear comments from others about whether the logic about the double potential energy requiring field energy makes sense, and any other implications that this idea might have.

Jonathan Scott

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Gold Member
After a bit of searching, it appears that the "factor-of-two anomaly" mentioned in the article on the Komar mass is described in more detail in section 3.2.2 of that document, and appears to be related to something to do with angular momentum. See the last paragraph of section 3.2.2 at the following URL:

http://relativity.livingreviews.org/open?pubNo=lrr-2004-4&page=articlesu4.html [Broken]

I don't understand it enough to know whether it's relevant, but my guess would be that it isn't.

Jonathan Scott

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Gold Member
The equation at the end of my article which started this thread can be rephrased as follows:
$$\Sigma m_i c^2 \phi_i + \mbox{field energy} = \Sigma m_i c^2 - \mbox{field energy}$$
That is, the total energy in the volume (for a large enough volume) can EITHER be calculated as being due to the sum of the relevant masses as affected by the internal potential PLUS the field energy, OR as the sum of the rest energy for the total amount of matter MINUS the field energy. I find it interesting that in the latter interpretation the only self-interaction is in the (negative) field energy term, making the theory essentially linear (!) and very similar to the electrostatic Coulomb interaction. However, the fact that this equality is only an approximation which needs integration over a "large enough volume" may somewhat undermine the direct usefulness of that result.

pervect
Staff Emeritus
I studied GR at a basic level as part of my university physics courses many years ago, but although I understood in principle how to "push the buttons" to do the math, I didn't feel I was anywhere near understanding how it worked. I felt then (as I do now) that the gap between basic Newtonian gravity and GR is unnecessarily large, and that GR makes it difficult to even ask questions about what happens in trivial cases, let alone get answers.
Wow - here I was joking about girlwonder giving you some bad advice, and it turns out she had sussed out the situation before I did.

I'll definitely agree that the is a large gap between GR and Newtonian gravity, and that it can be very hard to properly formulate or get answers to simple questions. I would disagree with characterizing this as "unnecessary".

I wrote this up in a rather over-enthusiastic article which can currently be found on my web site at this URL:

I tried to submit a copy to the arXiv too (under gr-qc), but it was rejected by the moderators, so I'm obviously not going to have an easy time of it.
Oh-oh, I was serious about it being a PF rule not to discuss unpublished theories in the main forums. In your case, though, it sounds like you may have a candidate for submission to the independent research (IR) forum.

I'm not sure if it will be accepted there or not, but that's the place we've reserved at PF for discussion of well-formulated personal theories.

I'll send a few more comments by E-mail, though note that currently Real Life is cutting into my PF time, so I don't have a lot of time for extended discussion.

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Gold Member
Sorry, I shouldn't have risen to the bait and wandered off topic. I did find girlwonder's reply quite entertaining, but I'm not that optimistic! My gravity theory is my own problem and doesn't seem to work at the moment anyway.

What I'd really like to know from the experts is specifically whether standard relativistic gravity theory does predict, as I've been guessing, that two masses will "red-shift" each other by the same potential energy and hence end up with their overall energy decreased by twice the external energy change (requiring some other change to balance the energy). If GR agrees with that, I'd be happy to know I haven't broken it but interested to know if it were possible to see in some sense where the mass went. If GR doesn't agree with that, I'd love to know what it would actually predict. I hope that's a reasonable question for this forum.

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pervect
Staff Emeritus
That's a reasonable question for this forum.

If you have two large objects, however, one needs to consider not the metric for one large object (the well-known Schwarzschild metric), but the metric for two large objects.

This is an example of how easy questions in GR can be difficult to answer, though if you make the two objects charged so that they repel each other, I think there might be an electrovacuum solution that covers it. I don't know of anyone who works out the energy of such a system in detail, though it should be possible to do using the Komar mass formula. I currently don't have the time to do much with this problem myself. Note that you'd have to include the stress-energy in the electrostatic fields in accounting for the energy of the system.

Your question can be and has been addressed in a different way, however, in the question of accounting for the energy in assembling a large object out of a bunch of smaller ones, as I mentioned by email.

This is worked out in MTW's textbook "Gravitation" on pg 604-606, and one recovers the usual formula for Newtonian gravitational binding energy in the non-relativistic limit when assembling a sphere of a constant density material - there is no factor of 2 discrepancy.

This is getting long, so let me recap.

1) There is no general formula for a conserved energy in GR for an arbitrary metric.

2) There are formulas for conserved energies in special cases.

3) The simplest special case in GR that has a conserved energy is the static system, which has a concept of mass given by the Komar mass formula. (I have a short write-up based on Wald's treatment of this in the wikipedia. http://en.wikipedia.org/wiki/Komar_mass. While of course I didn't make any intentional errors, it should be considered as wikipedian-reviewed and not peer-reviewed.

If you want a textbook treatment of Komar mass, see Wald, "General Relativity", which was my main source for the article.

4) The Komar mass formula is very similar to your verbal description, except that you didn't mention the pressure terms. (I'm not sure whether you were just simplifying the analysis, or didn't know about them). One basically integrates rho+3P, where rho is the energy density and P is the pressure, and additionally derates this term by the "red shift factor" to infinity. This is just a short verbal description and not intended to be very precise, see the article I mentioned for a more precise formulation.

Be warned - the Komar formula ONLY applies to stationary systems! (and the derivation in the wiki article is simplfied even more, to only cover static systems). Your example of two masses was not a stationary system, because there was nothing to hold it together.

4a) If one has two separate large masses, and one drops them together, while one cannot use the Komar formula, one can use the ADM formula as long as one has asymptotic flatness. This will give a conserved energy for the entire system during the whole "drop" process. (This bookeeping process includes accounting for energy in gravitational waves). The ADM formula, however, has no resemblance at all to the verbal formula you sketched out previously. Asymptotic flatness is expected from solving Einstien's field equations, BTW, as long as the two masses are isolated, i.e. "alone in the universe".

And I'll add the following observation:

5) Someone else on this board (Garth) has a published, peer-reviewed, but non-standard theory which is not GR, which addressed energy conservation using a "Machian" POV which you might find interesting called "Self Creation Cosmology". In a very short time, GPB should provide experimental results which can distinguish between this theory and GR. (Both theories can't be right, and GPB should tell us which one is correct.)

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Gold Member
Thanks very much. From your suggestions about the Newtonian binding energy considerations, I think I now have a much simpler illustration of how the paradox arises, involving only one body, so it can be matched with the Schwarzschild solution.

Consider assembling a thin shell of mass m and radius r by lowering the mass from infinity. The work done is the integral of -Gm/r dm which is Gm^2/2r. When this is complete, the whole shell is at a red-shift of the potential (1-Gm/rc^2) so the effective energy differs from the original energy by Gm^2/r, which is twice as much. This seems to mean that the shell has lost twice as much energy as was extracted, hence the paradox. Is this OK so far, or have I made some mistake already?

My suggestion is that the missing energy must have gone "into the field", which seemed a meaningful thing to suggest in a semi-Newtonian model, and the amount exactly matches the amount for a similar Coulomb model except for the sign, so it suggests that the mathematics of this "field" should be similar to that for electrostatics.

However, I don't know where it goes in the Schwarzschild solution. The vacuum solution seems to say there is nowhere it could be outside the shell. So where did it go? Did I miss something?

pervect
Staff Emeritus
Thanks very much. From your suggestions about the Newtonian binding energy considerations, I think I now have a much simpler illustration of how the paradox arises, involving only one body, so it can be matched with the Schwarzschild solution.

Consider assembling a thin shell of mass m and radius r by lowering the mass from infinity. The work done is the integral of -Gm/r dm which is Gm^2/2r.
Surely the integral of (GM/r) dm is GM^2/r, because M is not a function of r.

The last time I worked this (Newtonian) problem in

https://www.physicsforums.com/showpost.php?p=1010882&postcount=32

(which references)

http://scienceworld.wolfram.com/physics/SphereGravitationalPotentialEnergy.html

I also got GM^2/r for the Newtonian self-energy of a hollow spherical shell. (Some of the discussion goes on to use this result to find the gravitational self-energy of a sphere).

There's more to be said about the formula for the Komar mass of a hollow spherical shell, but basically if we ignore the pressure terms and approximate $\sqrt{1-2U}$ as 1-U for small U (both consistent with the idea of doing a Newtonian analysis) your answer of M = (1-GM/r) M_infinity is a reasonable approximation. Thus we have:

Komar mass $\approx$ mass at infinity - Newtonian potential binding energy

as we would expect in the Newtonian limit.

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Gold Member
Surely the integral of (GM/r) dm is GM^2/r, because M is not a function of r.
I must admit to spuriously switching the sign in my version of the above, because I changed from the external work being done (negative) to the decrease in the internal energy (positive).

However, I think integral m dm still has the factor of 1/2 on the result, i.e. m^2/2. That is, assuming that I haven't got far enough to fall foul of Douglas Adams' observation:

There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something more bizarrely inexplicable.
There is another theory which states that this has already happened.

pervect
Staff Emeritus
I don't see how. As the links describe, we find the total Newtonian potential energy by doing a mass-weighted average of the potential energy of each part of the hollow sphere.

The potential energy of any point on the hollow sphere will be -GM/r.

Thus the weighted average will also be $$\int \frac{-GM}{r} dm$$, the same expression you had, but the correct evaluation of this is -GM^2/r, there is no factor of 1/2.

Gold Member
Ah - I think we're doing different integrations. I should have noticed that the mixture of M and m in your reply was significant.

If you consider the final hollow sphere of mass M and integrate the effect of its own potential on itself, you get -GM^2/r. I assume that's what you've been doing. That's also what the Schwarzschild solution gives as the first order approximation result.

If however you start with a sphere of the same radius but mass zero, and lower mass on to it from infinity a bit at a time, you get the integral from m=0 to M of -Gm/r dm where the m in this case is the mass so far, so the total energy is given by -GM^2/2r with the factor of 1/2. There's the paradox. OK?

pervect
Staff Emeritus
To work the problem simply, we need to lower everything at the same time, so that our diagram is always spherically symmetrical.

As an alternative, work out the Komar mass for a sphere. We know from several sources

http://en.wikipedia.org/wiki/Gravitational_binding_energy
http://scienceworld.wolfram.com/physics/SphereGravitationalPotentialEnergy.html

what the formula should be for the binding energy of a sphere.

As I mentioned, MTW ("Gravitation") works a problem like this (using a much more sophisticated expression for the Komar mass which includes the pressure terms) and doesn't find any factor of 2 discrepancy.

Gold Member
Thanks you very much; from rereading MTW box 23.1 I now understand that the source of the paradox is entirely in the pressure term. In the case where the central object is stable, this gives rise to exactly the same energy as the external potential energy.

What really confused me in this case is the fact that I had been taught long ago that the nominal first order "rest mass" of the central object is m(1-Gm/rc^2), without mentioning that this excludes the internal energy. The effective rest mass including the internal energy is however m(1-Gm/2rc^2), matching the potential energy.

ZapperZ
Staff Emeritus