Is relativistic mass not an appropriate concept?

[SamRoss]

Is relativistic mass not an appropriate concept?

 

[Dale]

Appropriate for what? It is usually deprecated in favor ofthe invariant mass, but "appropriate" implies some specific purpose.

 

[bcrowell]

It's old-fashioned. The fact that it's gone out of style most likely indicates something about its lack of utility or convenience.

 

[Meir Achuz]

It is unfortunate that some physicists still use it.
It goes against the grain of SR, by almost denying that
m^2 is an invariant.

 

[SamRoss]

It is unfortunate that some physicists still use it.
It goes against the grain of SR, by almost denying that
m^2 is an invariant.

As far as I can tell, the idea of relativistic mass, which I am now convinced is counterproductive, is brought up through application of special relativity to the conservation of momentum. When this is done, it is found that the formula for momentum should be P=\gammamv. The problem arises when people lump the \gamma with the m, even though there is no physical basis for doing this, thus inventing a "rest mass" and "relativistic mass". These two expressions are then erroneously compared with rest energy and relativistic energy, which do have a physical basis which is derived from the relativistic Doppler Effect.

Is this a correct way of looking at things?

 

[lugita15]

Personally, I like the concept of relativistic mass, which actually arose before Einstein came up with Special Relativity.

A bit of history: In the late nineteenth century, physicists realized that charged particles would more difficult to accelerate than uncharged particles, since accelerating charges emit energy in the form of electromagnetic radiation. Thus, in addition to ordinary "mechanical" inertia there would have to be an electromagnetic contribution to the mass. Using Maxwell's equations alone, Abraham found that the electromagnetic mass varies as \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}. Lorentz believed that all forces were electromagnetic in origin, so he assumed that all mass would vary in this way. He interpreted this result as saying that the mass of an object depends on its motion with respect to the aether. Then Einstein came along and showed that the aether was unnecessary, but the formula for how mass changes with velocity remained.

My point is that you shouldn't be so quick to dismiss a notion that occurred naturally in the development of relativity. I don't see why rest mass is so overwhelmingly preferred in textbooks. Just as we talk about how length and time change in different inertial frames, why can't we also mention that mass changes? Rest mass shouldn't be taught exclusively, just as proper time and proper length shouldn't be taught exclusively.

 

[SamRoss]

Personally, I like the concept of relativistic mass, which actually arose before Einstein came up with Special Relativity.

A bit of history: In the late nineteenth century, physicists realized that charged particles would more difficult to accelerate than uncharged particles, since accelerating charges emit energy in the form of electromagnetic radiation. Thus, in addition to ordinary "mechanical" inertia there would have to be an electromagnetic contribution to the mass. Using Maxwell's equations alone, Abraham found that the electromagnetic mass varies as \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}. Lorentz believed that all forces were electromagnetic in origin, so he assumed that all mass would vary in this way. He interpreted this result as saying that the mass of an object depends on its motion with respect to the aether. Then Einstein came along and showed that the aether was unnecessary, but the formula for how mass changes with velocity remained.

My point is that you shouldn't be so quick to dismiss a notion that occurred naturally in the development of relativity. I don't see why rest mass is so overwhelmingly preferred in textbooks. Just as we talk about how length and time change in different inertial frames, why can't we also mention that mass changes? Rest mass shouldn't be taught exclusively, just as proper time and proper length shouldn't be taught exclusively.

(1) I would not look to nineteenth century physicists for confirmation regarding twentieth century physics.

(2) Einstein was not a proponent of relativistic mass. In a 1948 letter to Lincoln Barnett, Einstein wrote:

"It is not good to introduce the concept of the mass M = m/(1-v2/c2)1/2 of a body for which no clear definition can be given. It is better to introduce no other mass than 'the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion." I found this quote at http://www.weburbia.com/physics/mass.html It may also be found in The Concept of Mass (Mass, Energy, Relativity) by Lev Okun.

 

[Dickfore]

Is relativistic mass not an appropriate concept?

I will answer your question as soon as you define what you mean by the term that is bolded But, seriously, until you tell me how you measure that physical quantity, we cannot discuss it.

 

[SamRoss]

The paper by Lev Okun that I mentioned above may be found at http://books.google.com/books?id=Oj...&resnum=5&ved=0CC0Q6AEwBA#v=onepage&q&f=false

It is number 5 on the Contents page there.

 

[lugita15]

(1) I would not look to nineteenth century physicists for confirmation regarding twentieth century physics.

(2) Einstein was not a proponent of relativistic mass. In a 1948 letter to Lincoln Barnett, Einstein wrote:

"It is not good to introduce the concept of the mass M = m/(1-v2/c2)1/2 of a body for which no clear definition can be given. It is better to introduce no other mass than 'the rest mass' m. Instead of introducing M, it is better to mention the expression for the momentum and energy of a body in motion." I found this quote at http://www.weburbia.com/physics/mass.html It may also be found in The Concept of Mass (Mass, Energy, Relativity) by Lev Okun.

I don't know about you, but I see mass as the measure of resistance to acceleration. Thus mass should be the ratio of (longitudinal) force to acceleration, or equivalently the ratio of momentum to velocity. And this naturally leads to relativistic mass.

But my basic point remains. I don't know anyone who would only teach proper time and not time dilation. So why only teach rest mass and not mass increase? They're exactly analogous.

 

[SamRoss]

I don't know about you, but I see mass as the measure of resistance to acceleration. Thus mass should be the ratio of (longitudinal) force to acceleration, or equivalently the ratio of momentum to velocity. And this naturally leads to relativistic mass.

But my basic point remains. I don't know anyone who would only teach proper time and not time dilation. So why only teach rest mass and not mass increase? They're exactly analogous.

The view you just presented would contradict what some would call relativistic mass. In Einstein's paper, On the Electrodynamics of Moving Bodies, in the section entitled Dynamics of the Slowly Accelerated Electron, Einstein defines mass similarly to how you did and obtains Longitudinal Mass = m(1-v^2/c^2)^-3/2, NOT m(1-v^2/c^2)^-1/2. He cautions, "With a different definition of force and acceleration we should naturally obtain other values for the masses".

In the very same section, Einstein proceeds to derive the relativistic kinetic energy. Look closely and you will notice that although he presents an integral with dv he treats m as constant, taking it out of the integral sign. m does not depend on v!

 

[Dickfore]

In the same paper, he also shows how the radius of curvature R of the trajectory of a particle is determined by the component of the force perpendicular to the instantaneous velocity F_{\bot}:

<br /> \frac{v^{2}}{R} = \frac{F_{\bot}}{m} \, \sqrt{1 - \frac{v^{2}}{c^{2}}}<br />

 

[SamRoss]

Right, and in that derivation he is using the fact that the force on the electron alters relativistically, not the mass

 

[jason12345]

Relativistic mass can give the appearance to beginners that it's an add-on to SR, rather than the LTs of the equation of motion of a particle in its proper frame giving rise to a gamma factor in another frame.

 

[lugita15]

Relativistic mass can give the appearance to beginners that it's an add-on to SR, rather than the LTs of the equation of motion of a particle in its proper frame giving rise to a gamma factor in another frame.

Actually, you can do the derivation the other way around; you can start with just the relativistic mass formula, and then derive all of special relativity including the Lorentz transformations. See http://www.mathpages.com/home/kmath635/kmath635.htm" .

 

[Dickfore]

Right, and in that derivation he is using the fact that the force on the electron alters relativistically, not the mass

of course. Come to think of it, if there is "relativistic mass", then shouldn't there be "relativistic charge" as well?

 

[pervect]

of course. Come to think of it, if there is "relativistic mass", then shouldn't there be "relativistic charge" as well?

Argh! Longitudinal mass, transverse mass, and relativistic mass are bad enough, now you want longitudinal charge, transverse charge, and relativistic charge? :-)

Oh - wait - you mean you'd rather have none of them, and stick with just invariant mass. Ummm - nevermind. ::goes back to sleep::

 

[atyy]

Relativistic mass is "inertial mass" in the special relativistic generalization of Newton's second law.

By the principle of equivalence, it is "gravitational mass".

By the equivalence of relativistic mass and energy, this implies that energy should be "gravitational mass".

Searching for generally covariant equations for a relativistic theory of gravity, this suggests that a tensor whose components include the relativistic mass should be "gravitational mass".

The stress-energy tensor is such a tensor.

Once you get there, you can discard mass.

Even the invariant mass of particles, since there are no particles in general relativity.

 

[SamRoss]

Actually, you can do the derivation the other way around; you can start with just the relativistic mass formula, and then derive all of special relativity including the Lorentz transformations. See http://www.mathpages.com/home/kmath635/kmath635.htm" .

lugita 15,

It would be beneficial to read past the first paragraph of stuff you find on a webpage. Here are the first few paragraphs of the link that you just posted...

"Does Relativistic Mass Imply Special Relativity?

In a collection of essays on the subject of special relativity (“Six Not-So-Easy Pieces”), Richard Feynman presents the formula for relativistic mass

M=m(1-v^2/c^2)^-1/2

and then remarks that

For those who want to learn just enough about it so they can solve problems, that is all there is to the [special] theory of relativity – it just changes Newton’s laws by introducing a correction factor to the mass.

Unfortunately he gives no explicit explanation of this assertion. Later he discusses how the relativistic mass formula can be derived from the Lorentz transformation, but that’s the reverse of what’s needed to support the above claim, i.e., he needs to show that the Lorentz transformation follows from the relativistic mass formula. Incidentally, Feynman was well aware of the questionable nature of such claims. For example, in Vol II of his Lectures on Physics (section 26) he wrote

Whenever you see a sweeping statement that a tremendous amount can come from a small number of assumptions, you always find that it is false. There are usually a large number of implied assumptions that are far from obvious if you think about them sufficiently carefully. "

In any event, even if you did start from relativistic mass and derive the Lorentz equations mathematically, your result would be dubious because your original postulate of relativistic mass would not be based on empirical evidence. Contrary to this, the postulate of the constancy of the velocity of light is well founded in the Michelson-Morley experiment as well as being contained within Maxwell's equations. In addition, the assertion of the relativity postulate is demonstrated with the equivalent phenomena found by passing a magnet through a coil, first considering the magnet to be moving, then the coil. These two postulates lead directly to the Lorentz equations.

 

[SamRoss]

Relativistic mass can give the appearance to beginners that it's an add-on to SR, rather than the LTs of the equation of motion of a particle in its proper frame giving rise to a gamma factor in another frame.

I have to disagree with you there. The Lorentz Transformations involve space and time. When we apply them to the equations of motion we certainly do get gamma factors, but that is because a relativistic acceleration must be accounted for because acceleration is, as we all know, the second derivative of space with respect to time. Check out http://en.wikiversity.org/wiki/Special_Relativity#Acceleration_transformation Scroll down to "General Method" and you will find a derivation of relativistic acceleration. Compare this with Einstein's treatment of the equations of motion in On the Electrodynamics of Moving Bodies Section 10
http://www.fourmilab.ch/etexts/einstein/specrel/www/ (keeping in mind that u=0 in the instant he is considering) and you will find that he is, as I said, taking a relativistic acceleration into account and leaving the mass invariant. He would have no reason to do otherwise.

 

[SamRoss]

For those keeping track, of the four Science Advisors who have taken part in this discussion, not one has been a cheerleader for relativistic mass. I am satisfied that my original question has been answered in the negative. Thanks to all.

 

[atyy]

For those keeping track, of the four Science Advisors who have taken part in this discussion, not one has been a cheerleader for relativistic mass. I am satisfied that my original question has been answered in the negative. Thanks to all.

See Rindler's latest textbook or Penrose's Road to Reality.

Or Schutz http://books.google.com/books?id=qhDFuWbLlgQC&dq=schutz+relativity&source=gbs_navlinks_s "the density of total mass energy T⁰⁰"

 

[jtbell]

Note that Rindler was born in 1924, so his original training in physics was basically in the 1940s, before many physicists had stopped using the concept of "relativistic mass" in their work.

 

[John232]

My own mind experiment that I try to use when dealing with the realativistic mass problem is that I assume objects are traveling at relativistic speeds to Earth already. I think it is safe to assume that these objects are out there and the whole time I can sit here feeling the same weight as I always do. Or, any experimentor can conduct his own experiments on Earth without an object traveling at a relativistic speed that would alter the results of their own experiment. I am sure this would be the case if in fact there are objects traveling at relativistic speeds to Earth, because that is what is happening.

But, the only problem is that if there are multiple objects traveling at relativistic speeds to each other and all those objects measure a different mass due to relativistic mass increase then they would all measure a different amount of mass for every other object traveling at a relative speed at the same time. I don't even want to try and describe that all mathmatically...

How could all the objects travel the same trajectories for each observer if they measure a different amount of mass for every other object? Every observer would end up in a completely different situtation. I would like to think you could just assign them indepentdent time variables so that they could be measured to travel the same speed to two relativistic observers but Einstein did that already in deriving SR.

 

[Daverz]

I don't think that defining

m = \gamma m_0

is necessarily going to cause brain damage, though.

 

[pervect]

Relativistic mass is "inertial mass" in the special relativistic generalization of Newton's second law.

Only sometimes. Sometimes you need "longitudinal mass". Always using relativistic mass as a substitute for Newtonian mass will NOT consistently give you the right answer in dynamics problems in relativity.

By the principle of equivalence, it is "gravitational mass".

Komar mass, which is defined differently than relativistic mass because it includes pressure, is a much better substitue for gravitational mass than relativistic mass IF you have a static system. Though Komar mass doesn't work quantitatively for the common question about the gravitational effects of a relativistic fly-by any more than relativistic mass does because that's not a static system.

By the equivalence of relativistic mass and energy, this implies that energy should be "gravitational mass".

Searching for generally covariant equations for a relativistic theory of gravity, this suggests that a tensor whose components include the relativistic mass should be "gravitational mass".

The stress-energy tensor is such a tensor.

Once you get there, you can discard mass.

Even the invariant mass of particles, since there are no particles in general relativity.

That's pretty much the modern view, though GR doesn't really throw out mass totally. Sometimes one is interested in the mass of a large system in GR. This gives rise to the ADM, Bondi, and Komar masses, each of which is subtly different, and none of which is the relativistic mass.

And - we don't really need two different words for energy (energy and relativistic mass). Especially not with three other sorts of "mass" waiting in the wings.

 

[lugita15]

lugita 15,

It would be beneficial to read past the first paragraph of stuff you find on a webpage. Here are the first few paragraphs of the link that you just posted...

"Does Relativistic Mass Imply Special Relativity?

In a collection of essays on the subject of special relativity (“Six Not-So-Easy Pieces”), Richard Feynman presents the formula for relativistic mass

M=m(1-v^2/c^2)^-1/2

and then remarks that

For those who want to learn just enough about it so they can solve problems, that is all there is to the [special] theory of relativity – it just changes Newton’s laws by introducing a correction factor to the mass.

Unfortunately he gives no explicit explanation of this assertion. Later he discusses how the relativistic mass formula can be derived from the Lorentz transformation, but that’s the reverse of what’s needed to support the above claim, i.e., he needs to show that the Lorentz transformation follows from the relativistic mass formula. Incidentally, Feynman was well aware of the questionable nature of such claims. For example, in Vol II of his Lectures on Physics (section 26) he wrote

Whenever you see a sweeping statement that a tremendous amount can come from a small number of assumptions, you always find that it is false. There are usually a large number of implied assumptions that are far from obvious if you think about them sufficiently carefully. "

In any event, even if you did start from relativistic mass and derive the Lorentz equations mathematically, your result would be dubious because your original postulate of relativistic mass would not be based on empirical evidence. Contrary to this, the postulate of the constancy of the velocity of light is well founded in the Michelson-Morley experiment as well as being contained within Maxwell's equations. In addition, the assertion of the relativity postulate is demonstrated with the equivalent phenomena found by passing a magnet through a coil, first considering the magnet to be moving, then the coil. These two postulates lead directly to the Lorentz equations.

As I said before, relativistic mass has a perfectly sound basis in experiment, and the concept was invented before relativity itself.

 

[Dale]

Relativistic mass is "inertial mass" in the special relativistic generalization of Newton's second law.

A better generalization is in terms of the four-momentum. The four-vector approach extends to GR much better, and it makes the geometry much clearer.

By the principle of equivalence, it is "gravitational mass".

By the equivalence of relativistic mass and energy, this implies that energy should be "gravitational mass".

No, (as you mention below) in GR the source of gravity is a tensor, not a scalar. The equivalence principle does not imply otherwise.

Searching for generally covariant equations for a relativistic theory of gravity, this suggests that a tensor whose components include the relativistic mass should be "gravitational mass".

The stress-energy tensor is such a tensor.

Yes.

Once you get there, you can discard mass.

Even the invariant mass of particles, since there are no particles in general relativity.

Sure there are particles, just classical particles and not quantum particles.

 

[atyy]

Only sometimes. Sometimes you need "longitudinal mass". Always using relativistic mass as a substitute for Newtonian mass will NOT consistently give you the right answer in dynamics problems in relativity.



Komar mass, which is defined differently than relativistic mass because it includes pressure, is a much better substitue for gravitational mass than relativistic mass IF you have a static system. Though Komar mass doesn't work quantitatively for the common question about the gravitational effects of a relativistic fly-by any more than relativistic mass does because that's not a static system.



That's pretty much the modern view, though GR doesn't really throw out mass totally. Sometimes one is interested in the mass of a large system in GR. This gives rise to the ADM, Bondi, and Komar masses, each of which is subtly different, and none of which is the relativistic mass.

And - we don't really need two different words for energy (energy and relativistic mass). Especially not with three other sorts of "mass" waiting in the wings.

A better generalization is in terms of the four-momentum. The four-vector approach extends to GR much better, and it makes the geometry much clearer.

No, (as you mention below) in GR the source of gravity is a tensor, not a scalar. The equivalence principle does not imply otherwise.

Yes.

Sure there are particles, just classical particles and not quantum particles.

Agree, with most of these technical points, with minor quibbles. I never use the relativistic mass to calculate in special relativity, and there is really no fundamental notion of localized mass in GR. But what I'd like to hear is how do you motivate that the stress-energy tensor is the generalization of "gravitational mass" when searching for a generalization of Newton's gravity consistent with special relativity? Of course since GR is the more complete theory, the logical route if GR -> special relativity, but I'm asking for a heuristic for special relativity -> GR.

 

[Dickfore]

Einstein simply defined mass as the coefficient of proportionality between the force imparted on the particle (and he only considered electromagnetic forces) and the acceleration in an inertial reference frame that is moving with the same velocity as the particle according to a "stationary reference frame". (and in this co-moving frame, the only electromagnetic force is the electric force). This is in accord with Newton's Second Law which is a good first approximation at speeds much lower than the speed limit c. By the way, this would be what is called a "rest mass" because it is defined in the rest frame of the particle.

Using the transformations for the components of acceleration, and the components of the electric field, and the assumption of invariance of electric charge (which he argued on the basis of its operational definition), he inferred:

1. the expression for the kinetic energy;
2. the expression for the Lorentz force;
3. the curvature of a particle's trajectory in a magnetic field.
His expressions always include the rest mass.

From the expression for kinetic energy and the transformation rules of energy and momentum contained within a "packet" of monochromatic electromagnetic waves, he derived his most famous equation: "That if the energy content of an object changes by an amount \Delta E, then this is accompanied by a change in mass \Delta m = \Delta E/c^{2}."

 

[Dale]

how do you motivate that the stress-energy tensor is the generalization of "gravitational mass" when searching for a generalization of Newton's gravity consistent with special relativity?

AFAIK there is no generalization of Newtonian gravity which is consistent with SR. I always make the point that the source of gravity in GR is the stress-energy tensor precisely to encourage the understanding that GR is not merely an extension of Newtonian gravity.

 

[pervect]

I'd agree that there doesn't appear to be any fundamental definition of mass in GR, which gives rise to the large number of non-fundamental definitions we do have (and these definitions require additional assumptions to calculate at all, such as a static metric, or asymptotic flatness, another reason they are not fundamental).

As far as motivation goes, the approach I tend to use (borrowed from the textbooks I learned from) is to initially talk about gravity as the result of geodesic deviation, rather than a force.

I'll also refer readers to Baez's "The Meaning of Einstein's equation" which ties the the second derivative of the volume of a sphere of coffee grounds following geodesics to the enclosed density and pressure.

I'll tend to use the Newtonian idea of mass more to motivate Komar mass based on Wald's treatment - though this is more of an advanced topic than something I'd introduce right away, unlesss the idea of mass was brought up specifically. The Komar mass also ties in nicely with Baez's treatment, however.