Why is the relativistic mass a rejected concept?

[atyy]

A fast moving object will length contract. Its density will increase If its density is given by its invariant mass divided by volume. Will it turn into a black hole?

 

[PAllen]

A fast moving object will length contract. Its density will increase If its density is given by its invariant mass divided by volume. Will it turn into a black hole?

Nah, more like a black pancake. I have to be done with this thread. Several major reasons why relativistic mass leads to confusion have been presented. As I see, not a single reason for its value has been presented. At best, strained defenses that it can be used without error - which is, of course true.

 

[harrylin]

[..] on the first page the user ZealScience asked a sensible question I've often wondered about and seems very important for this discussion, [..]
let me rephrase it succintly:
[..] Let's say a particle A has invariant mass m, but later we find out that the particle itself is rather a shell with three smaller particles [..] in it, bouncing around at high speeds. [..] the "invariant mass" of A is actually [..] the sum of the relativistic masses [..].

Okay maybe we can still make the concept well-defined, but at least it's not fundamental [..]

Good observations, you hit the nail on its head
Indeed, "invariant" mass is simply a standardised relativistic mass.

 

[pervect]

The point of http://arxiv.org/abs/gr-qc/9909014v1 is that in the weak field, slow motion approximation, for a box of stuff, it is the "energy" or "relativistic mass" that couples to gravity. Eq 15 sets up the scenario, and the following discussion shows that the virial theorem for bound systems results in the coupling being given by "rest mass+kinetic+potential energy" (so we get the result kinda by accident). Interestingly, in Eq 20-22 he gives an argument that the cancellations are not accidental but result from general covariance.

I've been giving this paper a careful re-read, and I want to see if you agree with my understanding of eq 15.

My interpretation of what Carlip is saying here is that if we look at the contribution of the presence of matter (I use the term loosely, to include light as well as solid matter) to the total Lagrangian that said contribution due to matter, which he calls the coupling, is going to be proportional to the "gravitational mass" of the matter and can be used to define said gravitational mass.

He talks about inertial mass, but I think all he really does is assume that that's equal to the total energy E, without justifying it explicityly.

Is this close to your interpretation?

 

[atyy]

I've been giving this paper a careful re-read, and I want to see if you agree with my understanding of eq 15.

My interpretation of what Carlip is saying here is that if we look at the contribution of the presence of matter (I use the term loosely, to include light as well as solid matter) to the total Lagrangian that said contribution due to matter, which he calls the coupling, is going to be proportional to the "gravitational mass" of the matter and can be used to define said gravitational mass.

He talks about inertial mass, but I think all he really does is assume that that's equal to the total energy E, without justifying it explicityly.

Is this close to your interpretation?

Yes, that's my understanding.

I don't have anything definite worked out for the inertial mass, but I would suggest trying http://arxiv.org/abs/astro-ph/0006423, Eq 21, with the first term producing the inertial mass and the second term being related to Carlip's terms.

 

[atyy]

Carlip, http://arxiv.org/abs/gr-qc/9909014
Straumann, http://arxiv.org/abs/astro-ph/0006423

It seems that Carlip takes phi constant in going from his Eq 15 to 16.

But in trying to get to something like Carlip's Eq 16 from Straumann's Eq 21, it seems I need to take grad(phi) constant, which makes sense, but doesn't quite seem to match up with Carlip's steps.

 

[pervect]

Yes, that's my understanding.

I don't have anything definite worked out for the inertial mass, but I would suggest trying http://arxiv.org/abs/astro-ph/0006423, Eq 21, with the first term producing the inertial mass and the second term being related to Carlip's terms.

I was thinking that the inertial mass would probably come from the pure-matter terms in the Lagrangian, the gravitational mass from the coupling terms. Though I haven't worked it out in any detail.

This would leave the pure field terms, which would persumably give the "mass" or energy of the field. But of course only for the linear model, not in the general theory!

 

[PAllen]

On the topic of the utility of relativistic mass, I wanted, and still want to say no more. However, this thread has gone in a different direction. I wonder if you (Pervect or Atyy) have any comments on my outline of derivation in post #87? Under the simplifying assumptions of those two MTW sections, it seems to follow almost immediately that invariant mass of a swarm of partices is the same as its active gravitational mass (the thing called M in section 19.1).

 

[atyy]

On the topic of the utility of relativistic mass, I wanted, and still want to say no more. However, this thread has gone in a different direction. I wonder if you (Pervect or Atyy) have any comments on my outline of derivation in post #87? Under the simplifying assumptions of those two MTW sections, it seems to follow almost immediately that invariant mass of a swarm of partices is the same as its active gravitational mass (the thing called M in section 19.1).

I don't have a direct comment off the top of my head. But the funny thing is that we know that in some exact solutions, the "mass" measured at infinity is certainly not the relativistic mass (eg. http://books.google.com/books?id=qhDFuWbLlgQC&source=gbs_navlinks_s, p259).

 

[PAllen]

I don't have a direct comment off the top of my head. But the funny thing is that we know that in some exact solutions, the "mass" measured at infinity is certainly not the relativistic mass.

No, it isn't the relativistic mass; it is (given the simplifying assumptions) the invariant mass, which includes only kinetic energy in the center of momentum frame, plus rest mass of the particles (with the feature that it can be computed directly in any frame). Really, the system rest mass of the swarm of particles.

 

[PAllen]

I don't have a direct comment off the top of my head. But the funny thing is that we know that in some exact solutions, the "mass" measured at infinity is certainly not the relativistic mass (eg. http://books.google.com/books?id=qhDFuWbLlgQC&source=gbs_navlinks_s, p259).

Nice that that book page was available on google. The explanation given is that the difference is the self gravitational binding energy. Well, duh, that is excluded by the assumptions of MTW section 19.1. This particular derivation is very general as to e.g. relative speed of the elements contributing to T, but explicitly excludes significant self gravitation. Thus, it says, you can't use it for a star.

 

[atyy]

Ok, here's a thought - does the invariant mass include potential energy?

 

[PAllen]

Ok, here's a thought - does the invariant mass include potential energy?

No.

My transform argument to show the irrelevance of relativistic mass for a rapidly moving body covers any 'isolated body' however structured or massive (concluding that only center of momentum mass / energy *contributes* to coordinate independent features of curvature). However, invariant mass is useful only assuming no E/M, and no significant self gravitation. It is just norm of the sum of the component 4-momenta.

Now, I did find that a true total 4-momentum vector can be defined for any isolated body, in AF spacetime, in GR (no exceptions apparently; proof due to Moller). However, I have no idea under what conditions the norm of this vector could be taken to be an active gravitational mass. I have an intuition that, since such a total 4-momentum must take account of E/M and gravitational self-energy, it's norm might meaningful as a gravitational mass under some conditions - but then, if it were that simple, it would be well known result. So maybe it just doesn't work at that level of generality.

 

[atyy]

I think one reason for not requiring coordinate invariance is that we are talking about phi, which has no meaning. In Newtonian physics, only grad(phi) has meaning. If you change reference frames, you change the kinetic energy of everything by a constant, which doesn't affect grad(phi).

 

[Drakkith]

A fast moving object will length contract. Its density will increase If its density is given by its invariant mass divided by volume. Will it turn into a black hole?

No. It is not length contracting in its own frame. The density of the object is not increasing.

 

[PAllen]

No. It is not length contracting in its own frame. The density of the object is not increasing.

I hope you realize that was a joke, poking at my insistence that if only we stop using relativistic mass, newbie questions about fast moving objects will turn into black holes will go away. Atyy certainly doesn't believe that a black hole would form. Getting the joke, and the point, I responded, "no that's a black pancake, not a black hole".

 

[Drakkith]

I hope you realize that was a joke, poking at my insistence that if only we stop using relativistic mass, newbie questions about fast moving objects will turn into black holes will go away. Atyy certainly doesn't believe that a black hole would form. Getting the joke, and the point, I responded, "no that's a black pancake, not a black hole".

It makes so much sense now!

I guess that's what I get for not reading the whole thread lol.