Gravitational energy : positive or negative ?

[notknowing]

I know that gravitational energy is a delicate topic in GR. It is true that the localisation of gravitational energy is not possible in GR but it should at least be possible to determine correctly the sign of the gravitational energy. I found the following article in arxiv.org, nl
http://arxiv.org/PS_cache/gr-qc/pdf/0508/0508041.pdf
in which it is mentioned that "it is known that gravitational energy is negative". Does someone know of a refereed paper or book where this is explained ? I would expect the gravitational energy to be positive. The energy carried by gravitational waves should also be positive, otherwise a rotating star system would speed up instead of slowing down when emitting gravitational waves. Also, if it were negative, one should expect some sort of antigravity effect with all kinds of strange consequences.
Further, does someone know of a refereed paper or book where the total energy in the (static) gravitational field is calculated correctly ? According to Noether's theorem it should be possible to find a definite and clear answer to this.

 

[pervect]

A couple of good sources for information on mass in GR are:

Wald, "General relativity"
Held, "General Relativity and Gravitation, One Hundred Years After the Birth of Einstein, Vol 2"

Misner, Thorne, Wheeler, "Gravitation" has some limited discussion. Many other introductory textbooks don't treat the topic at all.

As far as I know the concept of horizon mass as described by this author in this letter is unique to him - it's the first I've heard of it.

It's widely known that assigning a mass to the gravitational field will give different results for different observers. For instance, a freely falling observer will locally not see a "gravitational field", and hence will assign zero energy to it (as the field does not exist for him). Therfore there is no reason to expect that there will be any particular agreement on the sign of the energy of the gravitational field, and a lot of reason to avoid the idea altogether. I believe this is discussed in MTW, and used as an argument to avoid assigning an energy to the gravitational field (since different observers will not agree on any such assignment).

In Newtonian theory, its well-known that if you assemble a bunch of masses into a system, the total energy of the system decreases (one can imagine lowering the piecies into place with a cable, and one extracts work in this process, keeping total energy conserved).

This energy is known as the gravitational binding energy. The "Newtonian limit" is the easiest way to make sense of the author's paper. GR must agree with the Newtonian limit for weak fields.

The usual way of calculating the mass of a static system in terms of its components is via the Komar mass integral. This approach does not attempt to define a "gravitational field" or assign a mass to it.

It turns out that both energy and pressure are source terms for the Komar mass. Furthermore, objects deep in a gravity well will have their contribution to the mass multiplied by a "redshift factor". If one adopts an asymptotically Minkowskian metric, the "redshift factor" is just sqrt(g_00).

Thus to find the total mass, one integrates
See for instance http://en.wikipedia.org/wiki/Komar_mass

sqrt(g_00)*(rh0 + Px + Py + Pz)*dV

where rho = $$\hat{T_{00}}$$, Px = $$\hat{T_{11}}$$, Py = $$\hat{T_{22}}$$, and Pz = $$\hat{T_{33}}$$, T being the stress-energy tensor, and the "hats" indicate that I've used a local orthonormal frame-field. dV must be measured in the same frame-field that the stress-energy tensor is measured in.

Because I've used a local orthonormal frame-field, you can think of rho as being the "local" energy density, Px being the "local" pressure component in the x direction, and Py and Pz being the "local" pressure components in the y and z direction. dV is the "local" volume element. In an ideal fluid, the pressure is isotropic, and one usually sees just rho+3P, P being the isotropic pressure.

The "local" expressions for energy and pressure are measured in a "local" metric which is Minkowskian, the region being small enough that it is essentially flat so that the rules of SR apply.

The g_00 in the above expression is NOT a local metric coefficeint, but a global one, however.

The result of this expression is that matter deep in a gravity well contributes less to the total gravity because of the metric coefficient g_00. This is somewhat offset by the addition of the pressure terms - in GR, pressure (as well as energy) also causes gravity.

 

[notknowing]

Thanks for this detailed answer. I'll need some time to check some of the references. So, basically you say that such an answer can not be given (which implies that the article I referenced is not correct). But is this not in contradiction with Noether's theorem ?

 

[pervect]

Noether's theorem says that static systems will have a conserved energy. The paper you quote comes up with a conserved value for the total energy, so that's not a problem, but the paper ambitiously seeks to do more, to assign that energy a specific location.

The issues with this assignement of energy to a specific location are two-fold. These relate to the question of whether the assignement of energy is covariant, i.e. can be expressed in a manner that is independent of the observer. The paper does not offer a covariant formulation (and I rather doubt that proposal is covariant). The presentation of the paper is tied to a specific coordinate system.

The second issue is gauge invariance. In classical E&M, one can make an arbitrary choice of gauge, but this gauge choice has no impact on the energy. This is not the case in GR.

The lack of gauge invariance means that one might get stuck with several, alternate descriptions of the distribution of energy, none of which can claim to have any more merit than any other.

The paper, however, doesn't claim to have cracked these issues. It is offered in the framework of "quasi-local" energy and momentum. This is not something I'm all that familiar with, http://relativity.livingreviews.org/Articles/lrr-2004-4/ talks about it some. While it gets very technical quickly, the introduction gives some idea of the state of the art:

During the last 25 years one of the greatest achievements in classical general relativity is certainly the proof of the positivity of the total gravitational energy, both at spatial and null infinity. It is precisely its positivity that makes this notion not only important (because of its theoretical significance), but a useful tool as well in the everyday practice of working relativists. This success inspired the more ambitious claim to associate energy (or rather energy-momentum and, ultimately, angular momentum too) to extended but finite spacetime domains, i.e. at the quasi-local level. Obviously, the quasi-local quantities could provide a more detailed characterization of the states of the gravitational ‘field’ than the global ones, so they (together with more general quasi-local observables) would be interesting in their own right.

Moreover, finding an appropriate notion of energy-momentum and angular momentum would be important from the point of view of applications as well. For example, they may play a central role in the proof of the full Penrose inequality (as they have already played in the proof of the Riemannian version of this inequality). The correct, ultimate formulation of black hole thermodynamics should probably be based on quasi-locally defined internal energy, entropy, angular momentum etc. In numerical calculations conserved quantities (or at least those for which balance equations can be derived) are used to control the errors. However, in such calculations all the domains are finite, i.e. quasi-local. Therefore, a solid theoretical foundation of the quasi-local conserved quantities is needed.

However, contrary to the high expectations of the eighties, finding an appropriate quasi-local notion of energy-momentum has proven to be surprisingly difficult. Nowadays, the state of the art is typically postmodern: Although there are several promising and useful suggestions, we have not only no ultimate, generally accepted expression for the energy-momentum and especially for the angular momentum, but there is no consensus in the relativity community even on general questions (for example, what should we mean e.g. by energy-momentum: Only a general expression containing arbitrary functions, or rather a definite one free of any ambiguities, even of additive constants), or on the list of the criteria of reasonableness of such expressions. The various suggestions are based on different philosophies, approaches and give different results in the same situation. Apparently, the ideas and successes of one construction have only very little influence on other constructions.

I basically view the proposal in the letter you cited earlier as one proposal among many in an area that is still under active developoment, a proposal that some good points but lacks some highly desirable features.

 

[notknowing]

Thanks again for this clarification. The article you indicated is indeed far above my technical level. I was however struck by the first lines in the introduction :

During the last 25 years one of the greatest achievements in classical general relativity is certainly the proof of the positivity of the total gravitational energy, both at spatial and null infinity. It is precisely its positivity that makes this notion not only important (because of its theoretical significance), but a useful tool as well in the everyday practice of working relativists.

So, it seems that the positivity of the total gravitational energy is proven ?! Do you know who/when/where this was proven ? The author from the arxiv paper (I mentioned previously) mentions in his introduction just the opposite (namely that the gravitational energy is negative ). Am I missing something ?

 

[coalquay404]

Thanks again for this clarification. The article you indicated is indeed far above my technical level. I was however struck by the first lines in the introduction :

So, it seems that the positivity of the total gravitational energy is proven ?! Do you know who/when/where this was proven ? The author from the arxiv paper (I mentioned previously) mentions in his introduction just the opposite (namely that the gravitational energy is negative ). Am I missing something ?

The positive energy/positive mass conjectures and their proofs are some of the great achievements in mathematical relativity. Much initial work was done by Geroch, although a lot of his material remains unpublished. An important contribution towards the topic was Pong Soo Jang's paper "On the positivity of energy in general relativity" (I think this was vol 8 of GRG). Subsequently, Rick Shoen and S.-T. Yau published a series of three astonishing papers in Comm. Math. Phys. that finally settled the question of the positivity of mass in the case of asymptotically flat initial data. Interestingly, their variational methods allowed them to prove this not only in the case of maximal initial data, but more general cases by implementing a nice reduction method.

Finally, Witten produced a paper in (I think) 1981 that also gave a proof of the positive mass conjecture. His version was much simpler than Schoen & Yau's as it used some pretty elementary spinorial methods; it's probably understandable to a final year undergrad/first year grad student.

 

[Stingray]

To add a bit, the positive energy theorems refer to what is essentially the entire mass of the spacetime. They show that given reasonable assumptions, all spacetimes have positive mass except for Minkowski. That has zero mass, as you'd expect.

The original proofs due to Shoen, Yau, and Witten dealt with the ADM mass (as I recall). Others later proved similar results for the Bondi mass. Various reductions in the number of assumptions required have been found over the years as well. But these quantities intuitively include "bare masses," binding energies, etc. Unfortunately, they can't usually be split up into these components in any rigorous way.

I haven't taken a look at the paper you mentioned, but there are many notions of energy in use in general relativity. So you really have to define more precisely what is meant before asking whether it is positive or negative.

 

[notknowing]

To add a bit, the positive energy theorems refer to what is essentially the entire mass of the spacetime. They show that given reasonable assumptions, all spacetimes have positive mass except for Minkowski. That has zero mass, as you'd expect.

The original proofs due to Shoen, Yau, and Witten dealt with the ADM mass (as I recall). Others later proved similar results for the Bondi mass. Various reductions in the number of assumptions required have been found over the years as well. But these quantities intuitively include "bare masses," binding energies, etc. Unfortunately, they can't usually be split up into these components in any rigorous way.

I haven't taken a look at the paper you mentioned, but there are many notions of energy in use in general relativity. So you really have to define more precisely what is meant before asking whether it is positive or negative.

Thanks colquay404 and Stingray for this interesting information. I'll try to find some of the references.

 

[Chris Hillman]

Negative in Newton too

Hi, notknowing, the author of the paper you read was speaking carelessly, which led you to conclude (probably incorrectly, but I haven't looked up the eprint you cited) that there was a conflict with the positive energy theorem (which really is a -theorem- in gtr, as others have already explained, so the PET would win in any genuine conflict).

In case Pervect didn't already say this, I just thought I'd quickly point out that the gravitational binding energy is also negative in Newtonian gravitation. Suppose we start with some isolated massive object (say an idealized star) and lift successive spherical shells off the shrinking surface of our object, such that the radius of each shell tends to infinity, so that ultimately the matter is "infinitely dispersed". That would be a pretty good state of zero potential energy, so reversing this thought experiment, since we performed work in dispersing the matter in the original object, the gravitational binding energy must be negative.

Similarly for gtr or any other reasonable theory of gravitation, modulo the difficulties of defining "energy in the large" which others have already alluded to.

One can also discuss the gravitational binding energy associated with a test particle falling radially toward an isolated massive object, where we can draw similar conclusions using similar reasoning (work would be required to move the test particle back out to "spatial infinity").

Chris Hillman

 

[notknowing]

Hi, notknowing, the author of the paper you read was speaking carelessly, which led you to conclude (probably incorrectly, but I haven't looked up the eprint you cited) that there was a conflict with the positive energy theorem (which really is a -theorem- in gtr, as others have already explained, so the PET would win in any genuine conflict).

In case Pervect didn't already say this, I just thought I'd quickly point out that the gravitational binding energy is also negative in Newtonian gravitation. Suppose we start with some isolated massive object (say an idealized star) and lift successive spherical shells off the shrinking surface of our object, such that the radius of each shell tends to infinity, so that ultimately the matter is "infinitely dispersed". That would be a pretty good state of zero potential energy, so reversing this thought experiment, since we performed work in dispersing the matter in the original object, the gravitational binding energy must be negative.

Similarly for gtr or any other reasonable theory of gravitation, modulo the difficulties of defining "energy in the large" which others have already alluded to.

One can also discuss the gravitational binding energy associated with a test particle falling radially toward an isolated massive object, where we can draw similar conclusions using similar reasoning (work would be required to move the test particle back out to "spatial infinity").

Chris Hillman

Hi, I know about the Newtonian argument that the gravitational binding energy is negative. Personally, I think this has more to do how these concepts are used or defined. The splitting of total energy into potential and kinetic energy for instance is something artificial and "for nature" this does not really mean something. I can not believe that the energy or energy density in the gravitational field is really negative. You never find negative energies in nature (antiparticles can be considered to be particles with negative energy moving backwards in time but they also can be seen as particles with positive energy and opposite charge moving forward in time - if I remember correctly). Also, if the gravitational field energy was negative, one would expect some kind of anti-gravity effect on it, resulting in a very bizar situation. It is completely anti-intuitive and I really can not believe in this.

 

[Chris Hillman]

Potential is a relative thing

Hi, notknowing,

Potential energy is a relative concept: the statement that the gravitational potential energy of a massive isolated object is negative just means that the potential energy is smaller than the energy of the system in which all the matter has been "dispersed to infinity", because work is required to effect the dispersal. If you agree that it is natural to assign the latter the energy value of zero, the conclusion follows.

We are not claiming that isolated massive objects have negative Komar mass, if that is what is worrying you. See the discussion in the textbook by Carroll.

Chris Hillman