# There is no really good definition of mass?

1. Feb 16, 2006

### bernhard.rothenstein

A recent paper has the title
"There is no really good definition of mass", phys.teach. 44 40 2006.
Do you think that the statement is correct?

2. Feb 16, 2006

### pervect

Staff Emeritus
Is this paper available online anywhere?

I would summarize the situation by saying that there are several reasonably good defintions of mass, rather than none.

Static space-times have Komar mass, while asymptotically flat spacetimes have ADM and Bondi masses.

For space-times which are static, asymptotically flat, and without gravitational radiation, all three masses are the same.

The ADM and Bondi masses for asymptotically flat space-times differ in that the Bondi mass of a system can decrease with time as the system emits gravitational radiation, while the ADM mass remains constant.

For a sufficiently strong defintion of "good", Noether's theorem prevents there from being any "good" (great?) defintion of mass, due to the fact that the symmetry group of GR is so large, as the group of diffeomorphisms is an infinte group rather than a finite one.

http://arxiv.org/PS_cache/physics/pdf/9807/9807044.pdf

Last edited by a moderator: Apr 22, 2017 at 8:16 AM
3. Feb 16, 2006

### dicerandom

Last edited by a moderator: Apr 22, 2017 at 8:16 AM
4. Feb 16, 2006

### Careful

Well one would expect a good definition of mass to be quasi local right ? I can speak of the mass of one particle here : neither the Bondi, ADM and Komar definition does that. The only fairly satisfying case I am aware of is mass in the context of local biffurcation horizons (a more general treatment of black holes).

5. Feb 16, 2006

### pervect

Staff Emeritus
I don't see how the mass of a single particle can possibly be made to be quasi-local.

Suppose we have a particle "at rest" far away from a black hole in an asymptotically flat time, and the same particle in a circular orbit near a black hole in an asymptotically flat space-time. The gravitational binding energy of the particle will have to subtract from its energy if energy is going to be conserved. This means that the particle is going to have a lower mass in an orbit than it will at infinity. This prevents the idea of a mass for the particle that is independent of its environment - the mass must depend on the environment of the particle (how close it is to the black hole).

When the mass of the orbiting particle is small compared to the black hole, the Komar mass of the particle makes sense as an approximation to the mass of the particle, unless I'm missing something. I guess I'm a bit concerned about how to handle the fact that the particle is not stationary. Aside from that possible issue, the system should be quasi-static, so the Komar mass concept should generally apply.

When the mass of the orbiting particle is large enough, this approximation breaks down, and one winds up with only a mass for the particle + black hole system as whole entity, and one is unable to assign a mass to the particle only.

The gravitational binding energy in this case is a large fraction of the total mass of the system. Unfortunately, because of Noether's theorem, the gravitational binding energy can't be localized as one might like, at least in standard GR. If there turns out to be some sort of preferred frame, as in Garth's SCC theory, this difficulty might be able to be worked around. So far there is no evidence that such a preferred frame exists.

6. Feb 17, 2006

### bernhard.rothenstein

Posting the question , I had in mind the lever of the paper I jsu quoted

7. Feb 17, 2006

### Careful

** I don't see how the mass of a single particle can possibly be made to be quasi-local. **

Check out the work of Ashtekhar, Lewandowski and Wald on local bifurcation horizons, you can derive the first law of thermodynamics without caring about asymptotics at all as far as I recall. Intuitively, you would expect matter to define a restframe in which you can define what particles are and consequently what mass is (this mass does not necessarily be conserved, the fact that we observe it as a constant should be seen as a constraint on the definition of particle/restframe or on the initial conditions). As a relativist you cannot deny that such concept is higly desirable; it is just how we observe our world to work.

** Suppose we have a particle "at rest" far away from a black hole in an asymptotically flat time, and the same particle in a circular orbit near a black hole in an asymptotically flat space-time. The gravitational binding energy of the particle will have to subtract from its energy if energy is going to be conserved. This means that the particle is going to have a lower mass in an orbit than it will at infinity. This prevents the idea of a mass for the particle that is independent of its environment - the mass must depend on the environment of the particle (how close it is to the black hole). **

??? In a comoving restframe, the rest mass of a point particle is perfectly well defined and constant. It is just for extended objects, that local mass densities may change and the shape of it may alter under extreme tidal forces. Under the same extreme conditions, there may even be a noticable energy loss through gravitational radiation. All this is just academic, our world is approximatly flat on scales of the solar system and beyond, meaning that the rest masses of particles defined in a suitably quasi local way remain constant on sufficiently long timescales for all practical purposes. You should not take the indistinguishability of particles as proclaimed by quantum mechanics so strict. Quantum mechanics teaches you that a particle has no realistic meaning in the sense of being a specific extended object anyway.

Your worry seems to originate from the idea that the *total energy* of the universe should remain constant. But what is total energy and what is the time function which should give me this thing ? The restframe of matter is clearly not going to do it for non static cases, so perhaps we should include a form of gravitational self energy to obtain a satisfying answer ? Or simply state that the difference = gravitational self energy ? Anyway, it should come as no real surprise that the total Hamiltonian is explicitely time dependent.

Cheers,

Careful

Last edited: Feb 17, 2006
8. Feb 17, 2006

### Garth

In GR is not the understanding that it is the 'rest' mass of a particle that is taken to be constant and it is energy that is not conserved? i.e. The problem is with the definition of energy rather than mass.

Garth

9. Feb 17, 2006

### pervect

Staff Emeritus
I did chose a rather poor example - especially if a point particle is used. The mass of a point particle is the invariant of its energy-momentum 4-vector. Probably this should be included as a separate sort of "mass" in GR, along with the other three I mentioned.

Basically I agree that mass in GR is a lot more complicated than one would like, but my POV is not that there is a lack of defintion of the concept, rather that there are too many defintions that are too easy to confuse. Unfortunately this doesn't seem to be able to be avoided. (I suppose it's theoretically possible that someone will come up with some new defintion that's superior to everything else written on the topic, but I'm a bit skeptical.)

Moving on, I will try to see if I can come up with a better example of what I was trying to say.

Let's consider instead a static planet. The Komar mass of the static planet is well defined, and time invariant. It has a couple of formulations. One of the formulations is as an intergal of the stress energy tensor. (It can also be done as a surface intergal).

I am taking the philosophical POV that it makes sense, in this context, to talk about the mass of a piece of the planet, and not just the total mass of the planet. (If the metric isn't static, there's probably no good way to talk about the mass of a piece of the planet.)

The mass of a piece of a planet is then just the appropriate intergal of the stress-energy tensor over a region that is part of the planet rather than the whole planet.

Pehraps this POV is suspect. But that's the way I was thinking about it. I'd be interested in other comments on this point. Is this a non-standard POV?

Using this POV, though, we assign a mass to each small part of the planet, and this mass will depend on the local metric. When we want to find the total mass of an extended object (the whole planet) by adding up the masses of its pieces, we have to include metric corrections to the mass of each piece, depending on where that piece was located.

I'm not familiar at all with bifrucation horizons, unfotunately.

I do agree that the "mass of the universe" isn't well defined, but that's because the universe is neither static nor asymptotically flat.

10. Feb 17, 2006

### Stingray

There are a number of definitions of mass floating around, and which one you want to use (or should use) depends on what you want to do. Mass is something we define, so when you want to use the word, it usually means that you're looking for a quantity with particular properties (e.g. conservation).

Asymptotic measures exist of course, but they are really not very satisfying IMO. They are so heavily discussed mainly because they can be. Quasilocal definitions exist in various approximation schemes, but this is also not ideal in my opinion.

For compact extended bodies, something called the Dixon mass is uniquely defined in exact GR. I think this is a wonderful thing, but it is difficult to use (or at least it seems to be), so few people talk about it. Its relation to other concepts of mass in the literature is not known except in special cases (e.g. when Killing vectors exist).

When horizons are involved, theorems derived in Ashtekar et al's dynamical horizon framework give a very natural quasilocal definition of mass for these objects as well. Again, this does not involve any approximations.

11. Feb 18, 2006

### pervect

Staff Emeritus
Yet another concept of mass I haven't run into (along with the one mentioned by Careful). Do you have any references where to read more about Dixon mass? (esp. any references in arxiv). Especially simple references....

In general terms, I prefer saying that there are "too many" definitions because it hopefully gives people the idea that there is a lot to read about the topic (mass in GR), rather than saying "there aren't any good defintion" because that makes it sound like the topic is barren.

12. Feb 18, 2006

### Stingray

Unfortunately, I don't think any papers recent enough to be on the ArXiv discuss it in any detail. Finding something simple is probably even more difficult. Your best bet is likely to be Ehlers and Rudolph, Gen. Rel. Grav. 8, 197 (1977). Go to Dixon, Proc. Roy. Soc. Lond. A314, 499 (1970) for some more detail. After that, the references get extremely dense. You'll need to be at least somewhat familiar with bitensors regardless.

The basic idea is that given (almost) any body with definite boundaries, it turns out to be possible to define a unique timelike worldline that may be called the center-of-mass. Along with this comes a unique timelike vector field defined on this worldline (which reduces to the tangent vector in simple cases). Spacelike hypersurfaces can then be defined throughout the body that are orthogonal to these vectors in an appropriate sense. This construction gives a unique foliation of the spacetime in a neighborhood of almost any object.

One can then define a "linear momentum" as a vector field on the center-of-mass line. At any "proper time," it is given by an integral of the stress-energy tensor (and some geometric quantities) over the appropriate hypersurface. The mass is then derived in the usual way from the norm of the momentum. In general, it is time-dependent. In the presence of an appropriate Killing vector, it can be related to the ADM mass (though there are additional terms that look like gravitational binding energy, rotational kinetic energy, and something else that I don't remember).

Of course there are many ways of defining momentum, but Dixon's choice is actually a part of a larger formalism of relativistic mechanics that he developed. This has many desirable properties that (with minor assumptions) can only be recovered when defining linear and angular momentum in a particular way. Actually, definitions for the entire infinite set of multipole moments follow as consequences of certain aesthetic requirements.

I don't want to go into all of the advantages of these choices, but my favorite is that the standard equation of motion $\nabla_{a} T^{ab} =0$ holds iff the linear and angular momenta satisfy particular first order ODEs along the center-of-mass line. In all other similar formalisms, stress-energy conservation gets converted to an infinite number of ODEs. So this is a huge simplification, and naturally separates the "bulk" and "internal" degrees of freedom in arbitrary bodies. It also makes finding the motion of the center-of-mass line amenable to direct analysis in full GR.

13. Feb 18, 2006

### Careful

Thanks, I was not aware of the work of Dixon either, but what you say about it more or less coincides with my intuitive idea of what it ought to be.

All the best,

Careful