Does mass become infinite near the speed of light?

[Orodruin]

p=m·v is not a relativistic formula made look Newtonian but the original definition of momentum

Yes, but this definition presumes you to have a notion of mass if you want to take it as a definition of momentum. There are two notions of mass in Newtonian mechanics, inertial mass and gravitational. Neither of those have a straight forward generalisation to special relativity. Gravitation is the domain of GR and inertial mass is direction dependent. If you want to keep the Newtonian version of the momentum formula you use it to define (relativistic) mass and not momentum and then you need to separately determine what momentum is.

Nobody claimed that it is a good argument for using relativistic mass. It's just the origin of relativistic mass.

I know its origin. It does not make it better as an archaic concept which has largely been abandoned.

 

[DrStupid]

Yes, but this definition presumes you to have a notion of mass if you want to take it as a definition of momentum.

No, the notion of mass is not required for but given by such basic definitions and assumptions.

There are two notions of mass in Newtonian mechanics, inertial mass and gravitational. Neither of those have a straight forward generalisation to special relativity.

The classical inertial mass (quantity of matter) is given by Def. 2 (p=m·v), Lex 2 (F=dp/dt), Lex 3 (F1+F2=0), isotropy, principle of relativity and the transformation. This implicit definition works with both Galilean and Lorentz transformation but the results are different.

The gravitational mass is given by Newton's law of gravitation together with the other conditions mentioned above. That doesn't work with Lorentz transformation.

inertial mass is direction dependent

That would violate isotropy.

If you want to keep the Newtonian version of the momentum formula you use it to define (relativistic) mass and not momentum and then you need to separately determine what momentum is.

I can use it to define mass and momentum.

 

[Orodruin]

No, the notion of mass is not required for but given by such basic definitions and assumptions.

Sorry, but this is just silly. If you do not have a notion of mass, defining momentum in terms of it is a vacuous statement. Make up your mind. You either define momentum through the relation or you define mass through the relation - you cannot have it both ways.

That would violate isotropy.

But you do not have isotropy! The mass is moving! This is a well-known result in special relativity and was known very early on. This is the very result that led to the concepts of transversal and longitudinal (relativistic) mass - both now defunct.

I can use it to define mass and momentum.

No, you can't. You cannot use one relation to define two new concepts. What you can do is to use the second law to define momentum (it is what force is the change in and the momentum of something with zero velocity is zero). Then you can define mass through $p = mv$. You do lose out on $\vec F = m\vec a$ (this is the mass which would be directionally dependent) and $E_k = mv^2/2$ if you do so. There is no a priori reason of preferring one over the other.

 

[DrStupid]

If you do not have a notion of mass, defining momentum in terms of it is a vacuous statement.

No, it isn't because the definition of momentum doesn't exist in vacuum. I already mentioned the additional conditions which also needs to be fulfilled.

This is the very result that led to the concepts of transversal and longitudinal (relativistic) mass - both now defunct.

Don't confuse relativistic mass with longitudinal mass.

You cannot use one relation to define two new concepts.

Please read my posts more carefully. I do not use only one relation.

 

[Orodruin]

Please read my posts more carefully. I do not use only one relation.

This is what you implied when you said this:

p=m·v is the definition in Newtonian mechanics

This statement by itself does not define mass and momentum. With regards to the rest it needs to be taken as a whole and you need the definition of momentum in terms of the second law before doing p = mv if you want p = mv to define mass.

Don't confuse relativistic mass with longitudinal mass.

By longitudinal, I mean transversal and vice versa. The relativistic mass is equal to the transversal mass, but this has to do with how force relates to acceleration - which is the alternative definition of inertial mass.

No, it isn't because the definition of momentum doesn't exist in vacuum. I already mentioned the additional conditions which also needs to be fulfilled.

Then you are not defining momentum through p = mv, you are defining mass through that relation. Either way works but you simply cannot define two quantities with one relation.

 

[DrStupid]

I said this:

The classical inertial mass (quantity of matter) is given by Def. 2 (p=m·v), Lex 2 (F=dp/dt), Lex 3 (F1+F2=0), isotropy, principle of relativity and the transformation.

Discussions make no sense if such an important information is ignored.

By longitudinal, I mean transversal and vice versa. The relativistic mass is equal to the transversal mass, but this has to do with how force relates to acceleration - which is the alternative definition of inertial mass.

Longitudinal and relativistic mass are completely different things. We should discuss them separately.

Then you are not defining momentum through p = mv, you are defining mass through that relation. Either way works but you simply cannot define two quantities with one relation.

See above.

 

[Orodruin]

Discussions make no sense if such an important information is ignored.

But this is something you added later! If you considered it so important, why did you not state it from the beginning?

 

[vanhees71]

Well there is progress in physics from Galilei/Newton to Einstein/Minkowski, and for a good reason we nowadays define momentum using the fundamental discovery of the relation between conservation laws and symmetries of physical laws by Noether to unanimously define the observables. This concept works for both classical and quantum physics.

From this perspecitve the most simple way to (heuristically) derive energy and momentum of a free particle in special relativity is to define an action, built with the velocity $\vec{v}=\mathrm{d} \vec{x}/\mathrm{d} t$ only. This action should be Lorentz invariant, and the unique way to write it thus is
$$A[x]=-m c^2 \int \mathrm{d} t \sqrt{1-\dot{\vec{x}}^2/c^2}$$
with $m=\text{const}$ the (invariant) mass of the particle and $c$ the limiting speed of Minkowski space (aka speed of light in vacuo). From this the momentum follows from Noether's theorem applied to spatial translation invariance:
$$\vec{p}=\frac{\partial L}{\partial \dot{\vec{x}}}=\frac{m \dot{\vec{x}}}{\sqrt{1-\dot{\vec{x}}^2/c^2}}$$
and energy from Noether's theorem applied to temporal translation invariance:
$$E=H=\dot{\vec{x}} \cdot \vec{p}-L=\frac{m \dot{\vec{x}}^2}{\sqrt{1-\dot{\vec{x}}^2/c^2}}+m c^2 \sqrt{1-\dot{\vec{x}}^2/c^2}=\frac{m c^2}{\sqrt{1-\dot{\vec{x}}^2/c^2}}.$$
It's no surprise that due to the Lorentz covariance of the formalism
$$(p^{\mu})=(E/c,\vec{p})$$
builds a Lorentz four-vector, and mass is a scalar, independent of the velocity of the particle:
$$p_{\mu} p^{\mu}=m^2 c^2.$$

 

[DrStupid]

But this is something you added later! If you considered it so important, why did you not state it from the beginning?

No, not this way! These are some of my previous statement:

Alternatively mass could be defined implicit by other laws and definitions.

You can't simply choose a definition because it must be consistent with related laws and definitions which are universally accepted at the same time.

it results from the use of these equations

p=m·v is the definition of momentum in classical mechanics and the properties of mass and the equation for kinetic energy result from this definition (among other things).

I highlighted the plural for you. I made clear from the beginning that there is not only one relation that leads to relativistic mass. So don't try to blame me for your faults!

 

[Orodruin]

I highlighted the plural for you. I made clear from the beginning that there is not only one relation that leads to relativistic mass. So don't try to blame me for your faults!

Seriously, chill. Just because you claimed p = mv was the definition of momentum when you are using it as the definition of mass is no reason to get upset. The fact remains that you cannot define two concepts with one relation. This is explicitly claimed by you:

I can use it to define mass and momentum.

but later in the post where you specify which definition of Newtonian mechanics you use it is clear that you have already defined momentum! Hence, p = mv is not your definition of momentum.

Apart from that, I suggest you read vanhees recent post.

 

[DrStupid]

This is explicitly claimed by you:

end of discussion