An observation on potential energy in GR

[Jonathan Scott]

There are various different ways to try to add up the effective total mass of a configuration of masses in GR in such a way that it comes out to be equal to the total of the source mass as measured locally minus the gravitational binding energy. One of these is for example the Komar mass, which effectively treats the pressure within a system in equilibrium as effectively contributing to the gravitational mass. However, this is clearly not valid in a dynamic situation. Also, as recently discussed in another thread there is a recent paper by Ehlers and others which says that when you calculate the details for a spherical source, the effect of the pressure terms on the field does not act like additional rest mass but rather cancels out, although an adjustment of exactly the same magnitude arises for other reasons. (So far I have not made a lot of progress on understanding the implications of that paper).

I've only recently spotted that there's another very simple way of calculating the effective total mass which gives the expected Newtonian answer directly, at least in the weak field approximation. The answer is to integrate the local mass or energy density times the square root of the local time dilation factor (relative to the distant time rate), which is equivalent to applying half of the fractional time dilation. This causes the effective energy of each piece of local mass or energy to be adjusted downwards by exactly half of its potential energy relative to every other piece of local mass or energy in the configuration, accounting correctly for the binding energy between every pair of pieces. This method does not need to resort to treating pressure as contributing to the effective rest mass nor does it need to include some sort of "gravitational field energy" to adjust the total.

This may admittedly seem an odd thing to do in a GR context. For background, I should mention that I spotted it when investigating the implications of a Machian alternative in which the Einstein field equations are still assumed to hold (at least to post-Newtonian accuracy) around a single dominant central mass but the constant G in those equations is effectively an abbreviated notation for the gravitational effect of all distant masses, excluding the local dominant central mass and other smaller bodies. (I presume that it would not be acceptable under PF rules to give any further details of the Machian alternative theory here, so I'm only mentioning it for context. I will however say it's not the same as Brans-Dicke).

For this Machian scheme to match GR theory and experiment in this limited case (especially to the precision needed for the PPN beta parameter to give the correct perihelion precession for Mercury) the effective local value of G (which would apply for example to local lab experiments) has to vary as the square root of the time dilation factor. That means that within this scheme, the total gravitational source strength of a system, taking full account of internal gravitational binding energy, is simply the integral of Gm for all masses, provided that the locally varying value of G is used rather than the constant value which excludes the effect of the local central mass.

In GR, G itself is defined as constant, but it appears that the same method of summing the effective total energy using the square root of the time dilation applied to the local mass might be a useful tool. However, I don't at present have a strong mathematical justification for it apart from the fact that it clearly trivially matches the Newtonian result in the weak field case (and so far I don't have an equivalent expression for strong fields).

Does anyone know if there is already a known method of evaluating the effective total gravitational mass within standard GR by using the square root of the time dilation factor in this way (or equivalently, half of the fractional time dilation factor), and if so can they point me to further information?

 

[mfb]

Does anyone know if there is already a known method of evaluating the effective total gravitational mass within standard GR by using the square root of the time dilation factor in this way (or equivalently, half of the fractional time dilation factor), and if so can they point me to further information?

I don't know, but I would be surprised. It looks like fast-moving masses would give completely wrong answers in that calculation.

 

[Jonathan Scott]

I don't know, but I would be surprised. It looks like fast-moving masses would give completely wrong answers in that calculation.

What makes you think that? The relevant factor for a fast-moving mass would of course be applied to the total energy, not the rest mass, and that still matches the Newtonian result as far as I can see. I'm not expecting the rule to cover extremely dynamic situations such as those where the effects of retarded time make a significant difference.

 

[Jonathan Scott]

On revisiting this model and trying to pin down the definitions, I've found that the two parts of this model relating to the square root of the time-dilation factor (correct total energy taking into account potential energy, and effective variation of local G relative to Newtonian value) are not the same thing, but rather combine, in a rather neat way, which also eliminates some other problems such as the need for a fixed reference potential.

That is, if the following two factors are both applied when calculating the gravitational effect of a source mass, the result appears to be compatible with Newtonian energy conservation and other considerations of symmetry (roughly that force on A due to B is same as force on B due to A), at least in a weak field scenario:

1. The effective total mass $M$ of a collection of masses is the sum of each rest mass multiplied by the square root of the gravitational time-dilation factor (excluding the effect of the mass itself) at its location, that is $M = \sum \Phi_i^{1/2} m_i$. It should be easy to see that the effective total mass defined in this way comes out the same regardless of whether the masses are divided into smaller component masses and so on, as it correctly accounts for the gravitational binding energy at all levels. In Newtonian terms it is equivalent to assigning half of the potential energy for the interaction between each pair of particles to each of the particles.

2. The effective gravitational source strength of a collection of one or more masses is given by its total mass as above multiplied again by the square root of the time-dilation factor but at the observation location (where the time-dilation factor now includes the effect of the relevant source masses themselves). That is, $GM = G_s \Phi_o^{1/2} M$ where $G$ is the effective gravitational constant relative to the coordinate system, $G_s$ is the standard gravitational constant and $\Phi_o$ is the time-dilation factor at the observation location.

The time-dilation factor is expected to be relative to the time coordinate of the chosen coordinate system, but it can include an arbitrary multiplicative constant without affecting the physics of the combined result, as when the two square root terms are multiplied the overall result is scaled by the same constant.

If one assumes an isotropic coordinate system around a source mass, the second of the above factors varies with the potential of the observation point but the first does not. If we take the standard Newtonian potential and calculate the corresponding time-dilation multiplicative potential, applying the second factor, we get the following:

$$1 - \frac{GM}{rc^2} \approx 1 - \frac{G_s M}{rc^2} \left (1 - \frac{G_s M}{2 rc^2} \right ) = 1 - \frac{G_s M}{rc^2} + \frac{1}{2} \left ( \frac{G_s M}{rc^2} \right ) ^2$$

Those who are familiar with the Schwarzschild solution in isotropic coordinates (as used for example in the PPN formalism) will note that this expression for the multiplicative potential has the same PPN $\beta$ parameter as GR, so if this expression is taken as the time and space scale factor for the metric the result appears to be completely compatible with solar system experiments (although this result is only suggestive without a more complete formal correspondence between this model and an appropriate coordinate system).

I suspect this model may not be fully compatible with the theory of GR, in that I think the effective variation of $G$ with location in this idea is something which could be physically measured, but for GR $G$ is a universal constant. However, as the variation would only show up as a tiny effect in situations involving multiple source masses, which are non-linear and difficult to handle at all in GR, I can't be sure anyway.

I think that the above clearly "works" as a model, and gives some insights into possibilities for a relativistic explanation of the location of potential energy (at least in the weak field case). So far I've not spotted any obvious inconsistencies with standard physics, but that's no guarantee that I won't spot one as soon as I've posted this. However, without some sort of theoretical tie-in it isn't particularly meaningful. If anyone knows of any theoretical work which seems to be along these lines, I'd be interested to hear about it. I have various ideas myself, but as non-mainline speculation they are outside the scope of these forums.

 

[PeterDonis]

a Machian alternative in which the Einstein field equations are still assumed to hold (at least to post-Newtonian accuracy) around a single dominant central mass but the constant G in those equations is effectively an abbreviated notation for the gravitational effect of all distant masses, excluding the local dominant central mass and other smaller bodies

Have you read Sciama's paper on the origin of inertia, which constructs a theory along these lines? Here is a previous PF thread discussing it:

https://www.physicsforums.com/threads/sciamas-machian-origin-of-inertia.573332/

 

[PeterDonis]

One of these is for example the Komar mass, which effectively treats the pressure within a system in equilibrium as effectively contributing to the gravitational mass. However, this is clearly not valid in a dynamic situation.

Neither is your suggestion, because the concepts of "time dilation factor" and "gravitational potential energy" are not well-defined in a dynamic situation. More precisely, in order for the Komar mass to be well-defined, the spacetime must be stationary (it must have a timelike Killing vector field); but that is also exactly the required condition for the concepts of "time dilation factor" and "gravitational potential energy" to be well-defined. So your scheme has the same range of applicability as the Komar mass.

 

[Jonathan Scott]

Have you read Sciama's paper on the origin of inertia, which constructs a theory along these lines? Here is a previous PF thread discussing it:

https://www.physicsforums.com/threads/sciamas-machian-origin-of-inertia.573332/

Yes, thanks, many years ago. It's a beautiful paper, and very appealing, but so far I've not come across any viable theory which fully incorporates the idea (and Brans-Dicke only works when the Machian part is effectively turned off).

 

[PeterDonis]

so far I've not come across any viable theory which fully incorporates the idea

Yes, unfortunately, AFAIK Sciama never published the follow-up paper that is referred to in the abstract of the one we have; for whatever reason, he apparently didn't pursue this line of inquiry.

 

[Jonathan Scott]

Neither is your suggestion, because the concepts of "time dilation factor" and "gravitational potential energy" are not well-defined in a dynamic situation. More precisely, in order for the Komar mass to be well-defined, the spacetime must be stationary (it must have a timelike Killing vector field); but that is also exactly the required condition for the concepts of "time dilation factor" and "gravitational potential energy" to be well-defined. So your scheme has the same range of applicability as the Komar mass.

No, this covers the Newtonian dynamic case as well, in that it gives in general the same results as Newtonian theory including relativistic kinetic and potential energy (assuming a sufficiently weak field that linear approximations are valid). For example, the distant gravitational field of a pair of objects in orbit around one another is the same as that for a single mass whose total energy is equal to the mass of the system taking kinetic and potential energy into account.

 

[Jonathan Scott]

Yes, unfortunately, AFAIK Sciama never published the follow-up paper that is referred to in the abstract of the one we have; for whatever reason, he apparently didn't pursue this line of inquiry.

I heard that Sciama did follow up Machian ideas for a while, but was put off by observations suggesting that the Whitrow-Randall relation was out by a couple of orders of magnitude (this was before the invention of dark matter), and apparently never went back to it.

Edit: added missing word

 

[PeterDonis]

this covers the Newtonian dynamic case as well

Only if you wave your hands and claim that you can approximately define a "time dilation factor" and "gravitational potential energy" for a non-stationary spacetime that is nevertheless almost stationary, which is what I take you to mean by "the Newtonian dynamic case".

Newtonian theory including relativistic kinetic and potential energy (assuming a sufficiently weak field that linear approximations are valid)

And assuming that all relative velocities are much smaller than the velocity of light. Otherwise the spacetime is not "almost stationary". So we can't really include "relativistic kinetic and potential energy"; we have to remain in the non-relativistic regime.

the distant gravitational field of a pair of objects in orbit around one another is the same as that for a single mass whose total energy is equal to the mass of the system taking kinetic and potential energy into account.

Do we actually know this experimentally?

 

[Jonathan Scott]

Only if you wave your hands and claim that you can approximately define a "time dilation factor" and "gravitational potential energy" for a non-stationary spacetime that is nevertheless almost stationary, which is what I take you to mean by "the Newtonian dynamic case".

I don't think there's even a problem with relativistic speeds, provided that potentials are not changing so rapidly that light-speed delays are important. The kinetic energy aspect certainly works for any speed.

No, this covers the Newtonian dynamic case as well, in that it gives in general the same results as Newtonian theory including relativistic kinetic and potential energy (assuming a sufficiently weak field that linear approximations are valid). For example, the distant gravitational field of a pair of objects in orbit around one another is the same as that for a single mass whose total energy is equal to the mass of the system taking kinetic and potential energy into account.

Do we actually know this experimentally?

Not as far as I know.

It is clearly a desirable feature for consistency with Newtonian theory. Potential energy can be converted to and from "real" energy inside a closed system and we would not expect such a conversion to change the total energy or distant field of a system. Of course, I'm really more concerned with the multiplicative equivalents of potential and kinetic energy, which relate to the coordinate time dilation factor due to gravity and the gamma factor due to velocity, which according to the conventional definitions remain in balance for free fall motion in a static potential.

Various textbooks make the point that the Komar mass pressure term ensures that the gravitational effect of a composite body "correctly" takes account of binding energy.

Richard Tolman pointed out in one of his famous paradoxes that relying on pressure to give the right result meant that in a dynamic situation where the pressure was temporarily varying inside a star this could apparently theoretically affect the distant field, which did not seem right (especially as in the symmetrical case it would violate Birkhoff's theorem), and others have tried to show that in certain special cases this effect doesn't arise in practice.

 

[PeterDonis]

I don't think there's even a problem with relativistic speeds, provided that potentials are not changing so rapidly that light-speed delays are important.

If there are relativistic speeds involved, the system is not almost stationary, so there isn't even an approximate sense in which the time dilation factor and gravitational potential energy are well-defined.

The kinetic energy aspect certainly works for any speed.

Does it? Remember we are not talking about kinetic energy of test objects; we are talking about having two or more significant gravitating masses in the system, whose relative motion is relativistic. Can you show that your hand-waving approximations still work for this case? Bear in mind that linearized gravity certainly does not work for this case.

 

[Jonathan Scott]

For non-relativistic speeds, this model is basically Newtonian gravity plus the minor modification to the potential and the assumption that half of the potential energy comes off each interacting mass, which makes very little difference to the results but provides a form of global conservation and localization of total energy, allowing Newtonian theory to be effectively extended a bit to a post-Newtonian level of approximation. And being multiplicative, it follows relativistic kinetic energy rather than just the low speed approximation. Unlike models with energy in the field, it does not provide a continuity theorem for energy and momentum, but the total is preserved at some level of approximation.

I don't have details for the limitations of this model, although it's clear for example that combinations of relativistic particles whose center of mass is moving slowly are equivalent to the same total energy.

 

[PeterDonis]

allowing Newtonian theory to be effectively extended a bit to a post-Newtonian level of approximation

This still looks like hand-waving to me. I don't see anything that guarantees it will work the way you are claiming.

being multiplicative, it follows relativistic kinetic energy rather than just the low speed approximation.

I don't understand what this means.

combinations of relativistic particles whose center of mass is moving slowly are equivalent to the same total energy.

This looks like hand-waving to me as well. Can you actually show any math? Remember that we know Newtonian gravity doesn't work for relativistic particles.

 

[Jonathan Scott]

I'm assuming Special Relativity locally (so for example a reflective box full of photons effectively has rest mass) and scaling time and (inversely) space using the time-dilation factor based on the Newtonian potential, and I definitely assume that any major sources are static or slow-moving. The potential post-Newtonian accuracy of results only specifically applies to the comparison with the Schwarzschild solution (for example for Mercury's perihelion precession), where plugging the modified Newtonian time-dilation expression into the isotropic metric gives the same result as GR (which however as I said before may not be particularly significant, just suggestive).

For the "multiplicative" point I mean that instead of considering kinetic energy = (total energy - rest energy) and potential energy = (total energy * (time-dilation factor - 1)) and saying that the additive sum is constant, I'm actually thinking in terms of total energy = rest energy * gamma factor * gravitational time-dilation factor, where the two factors vary in matching opposite ways to give constant total energy in the static field free fall case. This is of course equivalent, but the gamma factor version is not limited to non-relativistic speeds.

The main point of the model is that as far as I can see it provides a self-consistent answer to the question "where is the energy?" in a Newtonian sense which extends naturally to a relativistic setting, without requiring "energy" in the field (although it may well be that some other model will give similar results with a different concept of "energy" in the field, including a continuity equation, and in the past I've assumed that was a more promising direction).

I don't have enough theory behind it to answer general cases; that's really what I'm looking for. I was a bit thrown by the second factor, which I didn't expect and which doesn't match my previous intuitions about energy, but appears to be required for other consistency reasons. From working through some simple cases it appears it could be valid. I guess that if this isn't a recognised approach I will have to continue working through this on my own. It's quite possible that when I have time to work through more complex cases it will turn out not to be self-consistent after all. Note also that it's not immediately clear how to evaluate the potentials which affect the source masses when those potentials are themselves defined in terms of the source masses, so again this is just a first-order correction at present.

I updated this old thread somewhat hastily today because I found that my earlier model was apparently incomplete, although I find the new model with the second factor more difficult to understand, and it may well need further fixing. For example, it says that the effective gravitational mass of a test mass seen from near to its own location (but relative to a global isotropic coordinate system) is modified firstly by the square root of the time-dilation factor at its location, and then again by the same square root factor at the observation location, so overall it seems to vary with the usual time-dilation factor as in the conventional model of the total energy. (This is one of the things "fixed" by the second factor). This suggests that rather than the second factor being part of the conversion from "mass" to "gravitational mass" (which makes it an adjustment to the effective value of G) it should be treated as part of the "effective mass" adjustment, relative to a given location.

I'd guess that according to the forum rules some discussion of the usefulness and limitations of a specific model based on a particular set of stated rules is acceptable, but I was not actually expecting to discuss or defend the details of the model itself, and at the moment I haven't even yet caught up with my own private notes on the subject from when I created the thread in 2014. I think that the first factor seemed an unexpectedly neat way of describing the location of potential energy and worth sharing. The second factor crept up on me uninvited, but it fixes some paradoxes and I think it's trying to tell me something useful, so I've updated the thread to mention it. I was hoping someone had been this way before, as I don't have a lot of time for physics between my day job and running my orchestras, and I've sometimes had some very useful suggestions on these forums (although occasionally it has taken me years to discover that someone had indeed been there before).

 

[PeterDonis]

scaling time and (inversely) space using the time-dilation factor based on the Newtonian potential

Which, once again, means that the time dilation factor and the "Newtonian potential" must be well-defined. Strictly speaking, they are only well-defined in a stationary spacetime. A spacetime with more than one gravitating mass is not stationary. I'm giving you some leeway by allowing a spacetime that is "almost stationary" to have a time dilation factor and Newtonian potential that are "almost well-defined", but that's already somewhat hand-waving. You certainly can't just assume that those quantities are well-defined or even "almost well-defined" in any spacetime you choose.

This is of course equivalent, but the gamma factor version is not limited to non-relativistic speeds.

Yes, it is, because of the "almost stationary" requirement above. A spacetime in which two or more gravitating masses are moving at relativistic speeds relative to each other is not even "almost stationary". The fact that your expression involving the gamma factor is formally valid at relativistic speeds is irrelevant. Your entire method depends on the Newtonian potential being well-defined or at least "almost well-defined", and is simply not valid in regimes where that is not the case.

I am closing the thread at this point because it seems to me to be verging on personal theory.