Understanding Mass and Density in Relativity Theory: A Critical Analysis

[sweet springs]

Hi.
1)In relativity theory mass is not additive so we cannot divide it to contributions from different parts in a system. In this sense "mass density" volume integration of which is mass is a false idea, isn't it ?

2)In a system that is a tiny tiny fraction of space would contain only one or zero particles. Mass of the system divided by given particle mass is one or zero. Is number density is given integrating the results from all these tiny tiny systems ? Is there another good definition or way of measurement of number density ?

 

[.Scott]

1)In relativity theory mass is not additive so we cannot divide it to contributions from different parts in a system. In this sense "mass density" volume integration of which is mass is a false idea, isn't it ?

If each mass is traveling at the same velocity, they are additive. Even if they are not, the masses will be additive - but not the same as their masses when at rest relative to an observer in an inertial reference frame.

2)In a system that is a tiny tiny fraction of space would contain only one or zero particles. Mass of the system divided by given particle mass is one or zero. Is number density is given integrating the results from all these tiny tiny systems ? Is there another good definition or way of measurement of number density ?

When you get down to tiny spaces - such as one quadrant of a hydrogen atom, there will be a probability of finding an electron there. That probability will be the limit to the amount of information in that system. So your notion of there being on 0 or 1 particles is not true. Of course, if you tested for where the electron is, you would find it in a specific place - but even then, you would simply have some information about what the density was at the moment of the measurement.

 

[PeterDonis]

Even if they are not, the masses will be additive

Only if "mass" here means relativistic mass. But relativistic mass as a concept causes more problems than it solves. It's better to just say the energies are additive (provided we stick to a single inertial frame).

 

[PeterDonis]

In relativity theory mass is not additive so we cannot divide it to contributions from different parts in a system.

You can't if you restrict "contributions" to the rest mass of each part of the system. But if you allow the energy of each part, in the system's center of mass frame, to contribute to the system's rest mass, then you can indeed add up all the contributions to get the system's total rest mass.

In this sense "mass density" volume integration of which is mass is a false idea, isn't it ?

It is if you only integrate rest mass density. But if you integrate energy density in the center of mass frame, as above, it works. (But see below for a clarification.)

Is number density is given integrating the results from all these tiny tiny systems ?

If we are talking about classical physics, and we are working in the system's center of mass frame, then yes, you can obtain number density by integrating this way--although "averaging" would be a better term since the integral isn't just summing up the densities of each part, it's averaging them because it's a density. (The same applies to integrating energy density as I described above.)

 

[PeterDonis]

if you integrate energy density

you can obtain number density by integrating this way

Another caution here is that this only really works in the simple way I described it in flat spacetime--i.e., if we ignore the effects of gravity. Trying to use these techniques to obtain the externally observed masses of self-gravitating bodies like planets or stars can be done, but it's more complicated and has some significant limitations.

 

[Buzz Bloom]

If each mass is traveling at the same velocity, they are additive. Even if they are not, the masses will be additive - but not the same as the rest masses.

Hi Scott:

I have seen many times in threads at these forums authoritative mentors saying that the term "rest mass" is no longer a term that is used in respectable physics discussions. The term "mass" is used to mean what "rest mass" was previously used to mean. I have sometimes also seen the statement that the term "mass" does not have a single well defined meaning. The meaning is different in different contexts.

The following is what I have been able to understand from these threads as the most common meaning for "mass".

Mass is independent of any chosen coordinate system or reference frame. With respect to a reference frame, the total mass equivalent of a particle includes it's mass plus the mass equivalent of the energy carried by the particle. However, what is called the "potential energy" of a particle or body due to its position relative to a gravitational field is not included.​

The following is an example of the variability of the concept of "mass" which occurs in discussing the mass of a proton.

A proton consists of quarks and gluons. Quarks have mass, but gluons don't. The mass of a proton is greater than the total mass of it's gluons. The additional mass of the proton is the mass equivalent of the energy carries bu the quarks and gluons.​

If I have this concept wrong, I hope someone will correct me.

Regards,
Buzz

 

[PeterDonis]

I have seen many times in threads at these forums authoritative mentors saying that the term "rest mass" is no longer a term that is used in respectable physics discussions.

Be very careful in trying to interpret what you see being said in PF threads. Context is very important. The term "rest mass" is still "respectable", as long as it's clear what it means. The term "invariant mass" is often used instead of "rest mass" to make the meaning clearer; and for brevity, the term "mass" without qualification generally means "invariant mass" (or "rest mass", which is the same thing). But you really need to look at the context of the discussion to be sure.

I have sometimes also seen the statement that the term "mass" does not have a single well defined meaning.

Again, context is important. Yes, "mass" without qualification can mean one of at least two things: invariant mass, or "relativistic mass". But often the context will tell you which one is meant.

Mass is independent of any chosen coordinate system or reference frame.

This is true of invariant mass. It is not true of relativistic mass.

With respect to a reference frame, the total mass equivalent of a particle includes it's mass plus the mass equivalent of the energy carried by the particle.

Now you're using the term "mass equivalent". What is that supposed to mean? What you are describing has a much simpler name: "energy".

what is called the "potential energy" of a particle or body due to its position relative to a gravitational field is not included.

As I said in my previous post, bringing gravity into it adds a whole new set of complications. A better way of expressing what you are saying here would be that, in a local inertial frame, the total energy of a particle does not include gravitational potential energy.

The following is an example of the variability of the concept of "mass" which occurs in discussing the mass of a proton.

No, it's an example of how the invariant mass of a system includes the energy, in the system's center of mass frame, of its constituents. There's no "variability" involved.

 

[.Scott]

I have seen many times in threads at these forums authoritative mentors saying that the term "rest mass" is no longer a term that is used in respectable physics discussions.

In this case, I was not referring to a particle's rest mass. But you have a point. The term "rest mass" already refers to something. So I have changed the wording in that post.

Clearly, if you bring each component of a collection to rest relative to an inertial observer, then the masses will be additive. And while in motion, the mass observed from any inertial reference frame will be the addition of the mass of each component as observed from that same inertial reference frame.

Overall, I think it is easier to describe how they are additive that in what way they are not additive.

 

[Buzz Bloom]

Now you're using the term "mass equivalent". What is that supposed to mean? What you are describing has a much simpler name: "energy".

Hi Peter:

Thank you for your clarifying post.

I suppose I am confused by the fact that the units of mass and the units of energy are not the same: E = Mc², or M = E/c². The makes it confusing for me to use energy as a synonym when I am am talking about mass. Is there something wrong in using the term "mass equivalent"?

Regards,
Buzz

 

[Dale]

In relativity theory mass is not additive so we cannot divide it to contributions from different parts in a system. In this sense "mass density" volume integration of which is mass is a false idea, isn't it ?

Yes, definitely. Even in flat spacetime this would be wrong and would systematically under estimate the mass of the system.

Intuitively, it is wrong for the same reason that for two vectors A and B
$|A+B|\ne|A|+|B|$

 

[SiennaTheGr8]

I suppose I am confused by the fact that the units of mass and the units of energy are not the same: E = Mc², or M = E/c². The makes it confusing for me to use energy as a synonym when I am am talking about mass. Is there something wrong in using the term "mass equivalent"?

Regards,
Buzz

There's nothing wrong with it if you understand what you're doing. But the "relativistic mass" concept has fallen out of favor because it's redundant and tends to confuse learners.

If you'd like to use the same units for mass and energy, I might suggest choosing ENERGY units, and using the term "rest energy" for $E_0 = mc^2$ (aka "proper energy" or "invariant energy"). Thus:

$E = E_0 + E_k$

$E = \gamma E_0$

$E_0^2 = E^2 - (pc)^2$

 

[PeterDonis]

I suppose I am confused by the fact that the units of mass and the units of energy are not the same

This is no more mysterious than the fact that the units of time and the units of distance are not the same. In relativity, that's just a matter of historical accident--we chose our common system of units before we learned about relativity. But in both cases, there is a simple conversion factor from one set of units to the other: $c$ for time to distance, $c^2$ for mass to energy. Since this conversion factor is invariant--it's the same for all observers, all reference frames--it is not telling you anything about physics; it's just telling you about humans' historical choices of units.

Is there something wrong in using the term "mass equivalent"?

Not "wrong", just unnecessarily vague and confusing.

 

[sweet springs]

Thanks for all your kindness.
3)Distribution of mass or where is mass? is a false idea. Metric $g_{ij}$ is a function of coordinate $(x_0,x_1,x_2,x_3)$. Thus it seems inappropriate to relate mass and metric or gravity, doesn't it?

 

[PeterDonis]

Distribution of mass or where is mass? is a false idea.

No, it isn't. There is a well-defined stress-energy tensor at every event in spacetime. That tensor includes information about the "distribution of mass".

Metric $g_{ij}$ is a function of coordinate $(x_0,x_1,x_2,x_3)$. Thus it seems inappropriate to relate mass and metric or gravity, doesn't it?

No, because the metric and the stress-energy tensor are related by the Einstein Field Equation.

 

[sweet springs]

I say about mass re:OP and you say about stress-energy tensor. I agree with you say about stress-energy tensor.

 

[pervect]

I say about mass re:OP and you say about stress-energy tensor. I agree with you say about stress-energy tensor.

The problem I have is I'm not sure which concept of mass you're using. If the context is SR, the main choices are invariant mass and relativistic mass. If the context is GR, the main chocies are ADM, Bondi, and Komar mass.

The multiplicity of defnitions is a hint that the problem is rather deep.

 

[Mister T]

Clearly, if you bring each component of a collection to rest relative to an inertial observer, then the masses will be additive.

Only if there is no interaction between those components. If they exert forces on each other the energy you would need to transfer to (or from) the collection to assemble it contributes to (or reduces) the mass of the collection.

And while in motion, the mass observed from any inertial reference frame will be the addition of the mass of each component as observed from that same inertial reference frame.

With the additional restriction that this motion doesn't involve motion of the components relative to each other.

 

[.Scott]

Only if there is no interaction between those components.

That's true. If I have two 10-gram magnets and I bring them together N to S, I will get less than 20 grams. But if I bring them together N to N, I will get more than 20 grams.

But don't try to measure these differences with your bathroom scale.

 

[sweet springs]

If the context is GR, the main chocies are ADM, Bondi, and Komar mass.

Thanks. I do not know them yet. Ref. OP does some of them have density, distribution or locality?

 

[pervect]

Thanks. I do not know them yet. Ref. OP does some of them have density, distribution or locality?

Not that I can recall. What's your objection to using the stress-energy tensor? I don't understand what you're trying to do, so I can't give you much feedback.

 

[sweet springs]

Thanks.　　I would like to understand more about concept of mass in TOR.

Mass in TOR belongs to a system. Though a　system occupies a finite or infinite space region, distribution or spatial location of mass of the system within the region are false idea. In this sense mass in TOR has no locality.

In GR, as Einstein equation $G_{ij}=kT_{ij}$ shows, something with spatial (plus time) distribution, actually T, is required.

So I suspect mass in TOR and metric or gravity are not in good compativility in contrast to Newton's close relation $$F=G\frac{mM}{r^2}$$.

 

[pervect]

In Newtonian terms, the basic issue that makes the concept of mass in General Relativity difficult is that it's not possible to localize "gravitational field energy", but it's also not possible to ignore it either.

This makes what you want to do impossible by any scheme known so far.

By "localize gravitaional field energy", it's commonly understood to mean to do so in a coordinate independent manner. Which implies that it's possible to formulate it as some sort of tensor. But there aren't any suitable tensor candidates.

MTW has a short section on why it's not possible to localize the energy of the gravitational field, for instance.

People have used things called "pseudo tensors" to define mass in GR, but those are gauge dependent (which in the case of weak field gravity is equivalent to coordinate dependence, though I think it means something different for strong field gravity).

The even shorter version I can give you is that you've picked a very difficult problem to become interested in. Basically I can warn you it's difficult, I can give you a few pointers as to some approaches that have been taken, but I probably can't even cover all the attempts to define mass in GR.

Noether's theorem came about as one of the more fruitful attempts to solve the issue of mass in GR, the difficulties with mass in GR were first noted by Hilbert. A short quote from the Wiki may be helpful here:

In 1918, David Hilbert wrote about the difficulty in assigning an energy to a "field" and "the failure of the energy theorem" in a correspondence with Klein. In this letter, Hilbert conjectured that this failure is a characteristic feature of the general theory, and that instead of "proper energy theorems" one had 'improper energy theorems'.

This conjecture was soon proved to be correct by one of Hilbert's close associates, Emmy Noether. Noether's theorem applies to any system which can be described by an action principle. Noether's theorem associates conserved energies with time-translation symmetries. When the time-translation symmetry is a finite parameter continuous group, such as the Poincaré group, Noether's theorem defines a scalar conserved energy for the system in question. However, when the symmetry is an infinite parameter continuous group, the existence of a conserved energy is not guaranteed. In a similar manner, Noether's theorem associates conserved momenta with space-translations, when the symmetry group of the translations is finite-dimensional. Because General Relativity is a diffeomorphism invariant theory, it has an infinite continuous group of symmetries rather than a finite-parameter group of symmetries, and hence has the wrong group structure to guarantee a conserved energy. Noether's theorem has been extremely influential in inspiring and unifying various ideas of mass, system energy, and system momentum in General Relativity.

See http://cwp.library.ucla.edu/articles/noether.asg/noether.html for some of the notes about Hilbert finding energy theorems in GR "improper" and how Emily Noether became involved.

 

[sweet springs]

Thanks. I say about non-locality of mass in TOR and you say about that of gravitational energy-momentum. I assume these are essentially two different issues though they might have some similar features. Non-locality of mass in TOR is noticed even in SR.

 

[sweet springs]

Though we cannot say where within the system mass in TOR is located or distributed, anyway can we say it is located within the space where the system occupies and not in the space outside the system ? Spacial existence of mass wow! might be of philosophy or of wrong category but I am interested in it in relation to 2) of my OP.

 

[Mister T]

@sweet springs I think you're chasing it down a rabbit hole. I can imagine trying to locate the mass of a point particle by defining my system as the particle itself, such as an electron. But a definite location of a point particle is something that exists only in classical physics. You would have to ignore quantum physics in a realm where it's essential to not ignore it.

 

[sweet springs]

Thanks.

I can imagine trying to locate the mass of a point particle by defining my system as the particle itself, such as an electron.

Yea this is the point. The position of such particle itself system should give the position of particle mass in TOR.

As for a body of finite size also, body itself system could give the position of body mass in TOR in condition that:
- size of the body or position error is very small, practically zero, comparing the length of concern
- properties inside the system are not of our interest.
A case of example is M in Schwartzshild radius formula $$r_g=\frac{2GM}{c^2}$$.
M is mass in TOR of the celestial body itself system whose radius is practically negligible with spatial distance of our concern and whose below surface condition is not of our interest.

 

[pervect]

Thanks. I say about non-locality of mass in TOR and you say about that of gravitational energy-momentum. I assume these are essentially two different issues though they might have some similar features. Non-locality of mass in TOR is noticed even in SR.

Well, we are back where we started, I think. Which is me pointing out that there isn't a single definition of mass in General Relativity. But you are talking about it as if there is. We can ask "which one", but you don't answer :(.

I don't see how to stop going around in circles at this point. Perhaps something will come to me, but most likely we'll have to leave it where it's at.

 

[PeterDonis]

we'll have to leave it where it's at.

Good idea. Thread closed.