Does relativistic mass increase with velocity?

[PeterDonis]

if we cannot discuss what "amount of material" is and wheather is must be relativistically invariant or not

If you are interested in that topic, you should start a separate thread. This thread is about relativistic mass, which we have already seen is not a valid candidate for "amount of material" in relativity.

 

[DrStupid]

This thread is about relativistic mass, which we have already seen is not a valid candidate for "amount of material" in relativity.

How can we see that if we don't even know what "amount of material" means? I looked for a definition but I found nothing that would make sense in this context. Only @pervect can clarify what he is talking about. But if it turnes out to be something related to Newtonean mechanics, he (and not me) would need to do that in a different thread.

That applies to all unclear terms and statements above. Either the need to be explained or they should be kept out of the discussion.

 

[vanhees71]

It's of course hard to answer, if it is not allowed to discuss the question asked in this thread. So I stick to the definition of mass. As a hint: The "amount of substance" is defined via the mol as one of the basic units of the SI. There is not much difference whether you discuss the concept within Newtonian or relativistic physics.

The mass of a special-relativistic closed system is defined by $p_{\mu} p^{\mu} =m^2 c^2$, where $p^{\mu}$ is the four-vector of total momentum of the system. Defined in such a way it's a scalar quantity, and in the non-relativistic limit it coincides with the mass in Newtonian mechanics.

Concerning general relativity it's of course more complicated, because there the notion of "total energy and total momentum" is pretty tricky.

 

[DrStupid]

As a hint: The "amount of substance" is defined via the mol as one of the basic units of the SI. There is not much difference whether you discuss the concept within Newtonian or relativistic physics.

That's why it is obviously something completely different than mass - no matter whether you discuss the concept within Newtonian or relativistic physics.

The mass of a special-relativistic closed system is defined by $p_{\mu} p^{\mu} =m^2 c^2$, where $p^{\mu}$ is the four-vector of total momentum of the system. Defined in such a way it's a scalar quantity, and in the non-relativistic limit it coincides with the mass in Newtonian mechanics.

The non-relativistic limit of the relativistic mass also coincides with the mass in Newtonian mechanics and it is even additive. It doesn't looks like this adds something to the discussion. All we can say is that relativistic mass is equivalent to energy and should not be used anymore. That is already done. The only remaining question what "mass (matter)" means and how to address it will most probably stay open.

 

[vanhees71]

I don't know, why anybody is insisting on this confusing idea of relativistic masses (in fact when you introduce a relativistic mass it were even direction dependent, which has been said already above or recently in another thread in this forum).

From a quantum-theoretical point of view the concept of mass is indeed pretty different in relativistic and non-relativistic physics. In non-relativistic physics of an elementary particle it is introduced in the construction of the quantum theory from the analysis of the underlying space-time symmetry (Galilei-Newton spacetime) as a central charge, extending the "classical Galilei group" to the "quantum Galilei" or "Wigner-Bargmann group". In special-relativistic physics the underlying space-time symmetry is the Poincare group (and only the proper orthochronous part connected smoothly with the group identity!), which has no non-trivial central extensions and central charges. The result is that mass (squared) is a Casimir operator labelling the possible unitary irreducible representations, of which those with $m^2>0$ and $m^2=0$ lead to a physically interpretable quantum dynamics and particularly the successful local relativistic QFTs.

The conclusion, also valid for the macroscopic world, is that in special relativity there's no additional conservation law for mass, and conclusively mass is not an additive conserved quantity. All conservation laws in special realivity related via Noether to the space-time symmetry are the 10 conserved quantities energy, momentum, angular momentum, center-of-energy velocity.

So from both a mathematical and a physical point of view one should define mass in special relativity as a scalar quantity and energy and momentum as the corresponding four-vector related to the space-time-translation symmetry of Minkowski space. This avoids a lot of unnecessary misunderstandings!

 

[DrStupid]

I don't know, why anybody is insisting on this confusing idea of relativistic masses (in fact when you introduce a relativistic mass it were even direction dependent, which has been said already above or recently in another thread in this forum).

If a "relativistic mass" depends on direction is a matter of its definition. E/c² for example is independent from direction. There is no reason to start such discussions over and over again. It is sufficient to advice against the use of relativistic mass. That has been done in this thread.

 

[vanhees71]

Why don't you just use $E/c^2$ without naming it (in my opinion errorneously) "mass"? There is no need for renaming well-established quantities such that the language confuses the facts. An energy is an energy and mass is invariant mass! Obviously it's not sufficient to give the said advice, because you still insist on using "relativistic mass" in addition to the physically one and only mass, which is the invariant mass.

 

[PeterDonis]

How can we see that if we don't even know what "amount of material" means?

Because whatever it means, it can't be frame dependent, and relativistic mass is frame dependent.

And invariant mass can't be "amount of material" either, because "amount of material" should be additive, and invariant mass is not.

Please do not post in this thread again about "amount of material"; it is a thread hijack and will get you a warning and a thread ban. As I said before, if you want to discuss possible relativistic (or non-relativistic, for that matter) definitions for "amount of material", please start a separate thread on that topic.

 

[PeterDonis]

I don't know, why anybody is insisting on this confusing idea of relativistic masses

Because that is the topic of this thread. If you don't want to discuss relativistic masses, don't post in this thread. If you want to discuss what "amount of material" means, whether in relativity or Newtonian mechanics, then please, as I said to @DrStupid, start a separate thread. Don't hijack this one.

 

[DrStupid]

Why don't you just use $E/c^2$ without naming it (in my opinion errorneously) "mass"?

Because we are talking about relativistic mass.

 

[PeterDonis]

Why don't you just use $E/c^2$ without naming it (in my opinion errorneously) "mass"?

Because this thread is about relativistic mass. Pointing out that it is the same thing as $E / c^2$ is fine, although even that was not always true in historical usages of the term "relativistic mass" (I referred to longitudinal vs. transverse mass in an earlier post).

 

[anuttarasammyak]

If mass increases with velocity(v), can I say velocity is a quantity dependent on (v)?

I would rather say
If mass does not increase with velocity(v), we say velocity is a quantity dependent on v.
Such a velocity is called 4-velocity. 4-velocity explains that we cannot go beyond c.

If mass increase with velocity, we can keep using traditional velocity(v). Here mass=mass(v) explains that we cannot go beyond c.

Using the factor $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
One attribute it to velocity $u(v)=v\gamma$. The other attribute it to mass $m(v)=m\gamma$ in the momentum
$$p=\gamma \cdot m \cdot v=(\gamma \cdot v)\cdot m=(\gamma \cdot m )\cdot v$$
Two ways do not contradict but the modern way is the former, 4-velocity method, which you observe variable v is enclosed in one variable leaving the other a constant.