Does mass become infinite near the speed of light?

[avito009]

I read somewhere that at 90% the speed of light the mass doubles. So does mass only nearly double at the speed of light and does mass not become infinite at the speed of light? I thought nothing with mass can travel at the speed of light because mass would become infinite at light speed.

Also as per my understanding the relativistic mass increases at higher speeds but this increse in mass is temporary and when at rest the mass again returns to the normal value. Is this correct?

Please answer these two questions.

 

[Matterwave]

The concept of relativistic mass has largely fallen into disuse over the past years. It tends to lead to confusions more than it leads to any genuine physical insight. What can be said is the kinetic energy of a massive object approaches infinity when that object approaches (but it can't reach) the speed of light. The "relativistic mass" as defined as $m_{rel}\equiv\gamma m$ where $m$ is the invariant mass of the object, also approaches infinity when the object approaches the speed of light.

 

[aditya ver.2.0]

Correct.
In simple words, at nearly the speed of light (speed of light can not be attained), the energy required to accelerate a body further becomes quite awfully large. To explain this phenomena, it is said that the mass of the body increases near the speed of light, but in actual the kinetic energy approaches infinte.

 

[vanhees71]

As Matterwave said, it's the energy which becomes infinite, not the mass, which stays constant, i.e., independent of the speed of the particle (as long as there are no intrinsic excitations of the particle involved).

 

[A.T.]

it's the energy which becomes infinite, not the mass,

To understand this, it helps to consider a relativistic wind up car:

Does its relativistic mass (energy) increase, as it accelerates?

 

[vanhees71]

This is a very interesting example. The mass decreases here. Remember again that mass is always "invariant mass", i.e., the mass of the object as measured in its momentary restframe. In the beginning some spring inside the car is tensioned and thus has potential energy which you provided winding it up. Thus there is more energy saved in the car at rest than when the spring is not tense, and thus there's an additional mass given by \Delta m=\Delta E/c^2. Now you relaease the car and it gets accelerated. It gains kinetic energy and looses the potential energy. At any moment thus its mass decreases while its kinetic energy increases by the corresponding amount according to the famous mass-energy relation. At the end it has lost the mass \Delta m.

 

[ShayanJ]

I think textbook authors should abandon the concept of relativistic mass. Its of almost no use and produces too much trouble. I'm sure there can be other explanations for why nothing with mass can reach the speed of light. One thing that seems fine to me is to say that in SR, Newton's second law changes from \vec a=\frac{\vec F}{m} to \vec a=\frac{1}{\gamma m}(\vec F-\frac{\vec v \cdot \vec F}{c^2}\vec v).

 

[harrylin]

I read somewhere that at 90% the speed of light the mass doubles. So does mass only nearly double at the speed of light and does mass not become infinite at the speed of light? I thought nothing with mass can travel at the speed of light because mass would become infinite at light speed.

Also as per my understanding the relativistic mass increases at higher speeds but this increse in mass is temporary and when at rest the mass again returns to the normal value. Is this correct?

Please answer these two questions.

1. As already clarified by others, although the "rest mass" stays the same ("invariant"), indeed the "relativistic mass" increases with speed; and this increase in inertia can be measured in for example cyclotrons*. But what you perhaps did not realize is that this increase is very non-linear; "doubling at 90%" and "infinite at 100%" are not in disagreement with each other! The related non-linearity of speed increase as well as the increased kinetic energy were nicely illustrated in a demonstration video:
https://www.physicsforums.com/threa...and-video-bertozzi-the-ultimate-speed.770488/

2. Your second question: yes, at rest we simply measure the rest mass. Note that if the mass is for example a ball that travels with you inside a high speed rocket, and you try to throw it away from you, you would also not feel any effect as relative to you the object is in rest (that's "relativity").

*See https://en.wikipedia.org/wiki/Cyclotron#Relativistic_considerations

 

[A.T.]

Remember again that mass is always "invariant mass"

I explicitly asked about "relativistic mass" or "energy". But now I think it's better to ask:

Does the inertia of the wind up car increase as it accelerates? Does it require more force to accelerate, the faster it goes?

The point of this wind up car example is to constrain this very common way of putting it:

...indeed the "relativistic mass" increases with speed; and this increase in inertia can be measured ...

And to provide a better understanding: The increase in inertia comes from external energy input, not from speed itself. If all the energy for acceleration is stored on board, there is no increase in inertia during acceleration.

 

[harrylin]

I explicitly asked about "relativistic mass" or "energy". But now I think it's better to ask:

Does the inertia of the wind up car increase as it accelerates? Does it require more force to accelerate, the faster it goes? [..]

That's an an interesting one! :) I assume that the relativistic synchrotron equations do not apply to the car in your example (if it could go fast enough of course) - the constraint for the usual equations and discussions is that the object's rest mass (and thus also its rest energy) remains constant (ceteris paribus as one used to say in old days).

[edit]:

And to provide a better understanding: The increase in inertia comes from external energy input, not from speed itself. If all the energy for acceleration is stored on board, there is no increase in inertia during acceleration.

Right - while I fear that that example could make matters overly complex for the OP, your precision about external energy input may be helpful indeed!

 

[Ducatidoug]

Well a question concerning life at the speed of light. Since no object with any mass can reach the speed of light then only objects or waves of 0 mass exist there which I would assume to mean that only pure energy exists at that realm. Also time and distance do not exist correct? So if there is no mass is there any gravity? Of course we have gravity and it does affect light by bending it. So matter does interact with mass less photons and neutrinos. Is it possible that the big bang is the result of massive amounts of pure energy slowing down (for whatever reason) to the point of becoming an explosion of matter and also creating the dimension of time and space?

 

[Ibix]

"Pure energy" and "realm" don't really have a meaning in physics. Also you appear you appear to beusing "dimension" in its science fiction sense of "a separate universe" rather than its trchnical sense of "direction". The existence of matter or energy pre-supposes the existence of spacetime as the background on which they exist, at least in our current physical models. Finally, there is no way to slow anything that travels at the speed of light.

Your questions don't make much sense, I'm afraid.

 

[Idunno]

According to observers watching a spaceship get close to the speed of light, the mass of the ship gets greater, making it harder to accelerate, thus it cannot go faster than light, as the energy needed to accelerate approaches infinity. Fine.

However, what about the people in the ship? According to them, their mass is still the same, the rest of the universe is traveling close to the speed of light, and the mass of the rest of the universe has increased. According to the people in the ship, what is the problem with using a bit more fuel and accelerating a little bit more?

 

[PAllen]

According to observers watching a spaceship get close to the speed of light, the mass of the ship gets greater, making it harder to accelerate, thus it cannot go faster than light, as the energy needed to accelerate approaches infinity. Fine.

However, what about the people in the ship? According to them, their mass is still the same, the rest of the universe is traveling close to the speed of light, and the mass of the rest of the universe has increased. According to the people in the ship, what is the problem with using a bit more fuel and accelerating a little bit more?

None. They can apply 1 g acceleration (for example) forever (until running out of propulsion). They will see Doppler increase without bound. An object they throw out of the ship will always move away 1 g acceleration initially, no matter how long they've been firing their rockets. However, If they stop accelerating periodically to have inertial basis to measure speed of objects going by, they simply find that they are ever close to c, but never reaching it. This is part of the limitation is using 'increasing mass' as the explanation of the c as a limiting velocity.

The way to see algebraically what happens is suppose when already .9c with respect to stars, the rocket drops a space buoy and continues accelerating till that space buoy is moving at .9 away from the rocket. This is perfectly possible. You might think that means the stars are moving at 1.8c. That is wrong. velocities don't add that way. The correct rule (for collinear motion) is (u+v)/(1 + uv/c²). Thus, when the buoy is moving .9c away from the rocket, the stars are moving away at .9945c. Understood as a feature of spacetime, with non-Galilean rules of algebra and geometry, you see that increasing 'relativistic mass' is completely irrelevant.

 

[ZapperZ]

Well a question concerning life at the speed of light. Since no object with any mass can reach the speed of light then only objects or waves of 0 mass exist there which I would assume to mean that only pure energy exists at that realm. Also time and distance do not exist correct? So if there is no mass is there any gravity? Of course we have gravity and it does affect light by bending it. So matter does interact with mass less photons and neutrinos. Is it possible that the big bang is the result of massive amounts of pure energy slowing down (for whatever reason) to the point of becoming an explosion of matter and also creating the dimension of time and space?

See, this is one of the many reasons why we try to refrain from using the term "relativistic mass", because it creates this kind of confusion and this kind of problems where people seems to think that this is REAL mass that can cause real gravitational effects. In fact, the questions in this thread are the POSTER CHILD on why this term should not be used.

Please read our FAQ on why this is really not an accurate description (it is more accurate to consider relativistic momentum), and why, even Einstein, stopped using this term later on in his life.

Zz.

 

[Idunno]

None. They can apply 1 g acceleration (for example) forever (until running out of propulsion). They will see Doppler increase without bound. An object they throw out of the ship will always move away 1 g acceleration initially, no matter how long they've been firing their rockets. However, If they stop accelerating periodically to have inertial basis to measure speed of objects going by, they simply find that they are ever close to c, but never reaching it. This is part of the limitation is using 'increasing mass' as the explanation of the c as a limiting velocity.

The way to see algebraically what happens is suppose when already .9c with respect to stars, the rocket drops a space buoy and continues accelerating till that space buoy is moving at .9 away from the rocket. This is perfectly possible. You might think that means the stars are moving at 1.8c. That is wrong. velocities don't add that way. The correct rule (for collinear motion) is (u+v)/(1 + uv/c²). Thus, when the buoy is moving .9c away from the rocket, the stars are moving away at .9945c. Understood as a feature of spacetime, with non-Galilean rules of algebra and geometry, you see that increasing 'relativistic mass' is completely irrelevant.

Thank you very much for your answer. I've been wondering about this for awhile. To paraphrase you, the explanation that you cannot go faster than light because the mass is increasing is a limited explanation, as it is not the explanation that observers on the accelerating spaceship would use... That fair?

 

[PAllen]

Thank you very much for your answer. I've been wondering about this for awhile. To paraphrase you, the explanation that you cannot go faster than light because the mass is increasing is a limited explanation, as it is not the explanation that observers on the accelerating spaceship would use... That fair?

Fair enough. Personally, I wouldn't use the increasing mass explanation at all, but some great physicist have used it, within its limitations (e.g. Richard Feynman). Einstein, however abandoned relativistic mass soon after 1905.

 

[vanhees71]

I don't know, how often one has to repeat this argument. Today, we define mass as a scalar. Sometimes one emphasizes this by calling it "invariant mass". Relativistic mass was an idea from the very beginning of the development of relativity before its full mathematical structure has been understood. This idea of a velocity-dependent mass is an unnecessary confusion. Everything related with it is fully described by the total energy of a system, which is the time component of the four-momentum vector.

 

[Herman Trivilino]

This is a very interesting example. The mass decreases here. Remember again that mass is always "invariant mass", i.e., the mass of the object as measured in its momentary restframe. In the beginning some spring inside the car is tensioned and thus has potential energy which you provided winding it up. Thus there is more energy saved in the car at rest than when the spring is not tense, and thus there's an additional mass given by \Delta m=\Delta E/c^2. Now you relaease the car and it gets accelerated. It gains kinetic energy and looses the potential energy. At any moment thus its mass decreases while its kinetic energy increases by the corresponding amount according to the famous mass-energy relation. At the end it has lost the mass \Delta m.

An excellent way to explain the mass-energy equivalence!

 

[Herman Trivilino]

Thank you very much for your answer. I've been wondering about this for awhile. To paraphrase you, the explanation that you cannot go faster than light because the mass is increasing is a limited explanation, as it is not the explanation that observers on the accelerating spaceship would use... That fair?

That's certainly a valid reason, but the real reason is more general. Consider the equation $E=\gamma mc^2$ where $\gamma$ is defined as the ubiquitous $(1-\frac{v^2}{c^2})^{-\frac{1}{2}}$. The quantity $\gamma m$ increases beyond all bounds as $v$ appraoches $c$. Choosing to call $\gamma m$ the mass is a just that, a choice. It is not a consequence of the postulates, but the introduction of an arbitrary re-definition of the word mass. Einstein used it briefly after 1905 but then quickly abandoned it. Unfortunately it was favored in some but not all arenas for almost the following 90 years or so, at which time it started disappearing from the lexicon.

 

[DrStupid]

It is not a consequence of the postulates, but the introduction of an arbitrary re-definition of the word mass.

What was the original definition?

 

[Herman Trivilino]

What was the original definition?

The original definition of mass? I don't know. In Newton's time it was "quantity of matter". The modern definition involves comparison to a standard body.

 

[vanhees71]

Mass is a Casimir operator of the Poincare group, i.e., a scalar. Any other historical definition (in the early days before Minkowski they even had "longitudinal" and "transverse" masses and all kinds of very confusing quantities, which are of no other use than to confuse the students) is obsolete. I don't know, why so many people insist on making physics even more complicated than it is in its most natural form ;-).

 

[DrStupid]

In Newton's time it was "quantity of matter".

That wasn't a definition but Newton's name for mass. The definition was mass=density*volume but that isn't very helpful. It is used the other way around as a definition of density. Alternatively mass could be defined implicit by other laws and definitions. But in special relativity such a definition would result in relativistic mass.

The modern definition involves comparison to a standard body.

The reference is required for the measurement (in addition to the definition).

 

[PAllen]

That wasn't a definition but Newton's name for mass. The definition was mass=density*volume but that isn't very helpful. It is used the other way around as a definition of density. Alternatively mass could be defined implicit by other laws and definitions. But in special relativity such a definition would result in relativistic mass.

No it wouldn't unless you choose a definition that leads to this. For instance, if I want mass to be invariant, I define it operationally as resistance to acceleration measurd in the MCIF of the body; or in relation to a unit mass, I define it s the ratio of momentum of the given mass to a comoving unit mass. The latter idea is simply that mass is the contribution to momentum due to the 'content of the body' and not its speed.

[edit: or mass is what give 4-momentum from 4-velocity. You only get 'relativistic mass' with what I would consider definitions that are ill conceived in the context of SR.]

 

[DrStupid]

No it wouldn't unless you choose a definition that leads to this.

Physics doesn't work that way. 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. If you change a definition you need to check and if necessary change everything else as well.

 

[PAllen]

Physics doesn't work that way. 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. If you change a definition you need to check and if necessary change everything else as well.

Physics absolutely works that way. With a new paradigm (SR, GR, QM) you get new definitions. All you require is that new definiitons produce the same value as old for the cases when the old theory is a good approximation. That is certainly true of any SR mass definitions that I would consider well conceived.

My proposed definitions ARE consistent with the modern treatment of SR. They are the only natural definitions in the language of laws expressed using 4-vectors.

 

[DrStupid]

Physics absolutely works that way. With a new paradigm (SR, GR, QM) you get new definitions.

You are confirming my last sentence but you seem to ignore the second one or you don't consider that with his paper from 1905 Einstein changed the transformation only. Everything else remained unchanged for several years. That's where the relativistic mass came from. It is the good old quantity of matter in combination with Lorentz transformation and not a new definition. It has not been introduced in order to use classical equations under relativistic conditions but it results from the use of these equations because they have not yet been redefined (using a new concept of mass) that time.

All you require is that new definiitons produce the same value as old for the cases when the old theory is a good approximation.

That's not a matter of approximation. We are talking about different formalisms for the same physics. The resulting values are identical.

 

[PAllen]

You are confirming my last sentence but you seem to ignore the second one or you don't consider that with his paper from 1905 Einstein changed the transformation only. Everything else remained unchanged for several years. That's where the relativistic mass came from. It is the good old quantity of matter in combination with Lorentz transformation and not a new definition. It has not been introduced in order to use classical equations under relativistic conditions but it results from the use of these equations because they have not yet been redefined (using a new concept of mass) that time.

My definitions all amount to 'good old quantity of matter'. If two objects with the same speed have different momentum, the mass ratio is the momentum ratio. Whether historically, different definitions were used, it does not follow that those definitions were inevitable or required.

 

[DrStupid]

My definitions all amount to 'good old quantity of matter'. If two objects with the same speed have different momentum, the mass ratio is the momentum ratio.

That applies to both rest mass and relativistic mass but only the latter is consistent with the classical definition of momentum and Lorentz transformation. Using rest mass instead requires a corresponding redefinition of momentum.

 

[PAllen]

That applies to both rest mass and relativistic mass but only the latter is consistent with the classical definition of momentum and Lorentz transformation. Using rest mass instead requires a corresponding redefinition of momentum.

And I claim any attempt to define momentum as mv is ill conceived in SR.

 

[DrStupid]

And I claim any attempt to define momentum as mv is ill conceived in SR.

The classical definition works as well as the relativistic definition. With the corresponding concepts of mass they are in fact identical.

 

[TriPle]

I think textbook authors should abandon the concept of relativistic mass. Its of almost no use and produces too much trouble. I'm sure there can be other explanations for why nothing with mass can reach the speed of light. One thing that seems fine to me is to say that in SR, Newton's second law changes from \vec a=\frac{\vec F}{m} to \vec a=\frac{1}{\gamma m}(\vec F-\frac{\vec v \cdot \vec F}{c^2}\vec v).

I believe that Newton stated his second law as F = dp/dt

 

[Ibix]

Mass is just a word. In a sense, it's not wrong to use it to mean whatever mathematical term you want.

However.

Rest mass is invariant (like Newtonian mass) while relativistic mass is not. Rest mass is part of the source term for gravity (like Newtonian mass) while relativistic mass is not. Rest mass is the resistance of a body to a change of motion independant of that motion (like Newtonian mass) while relativistic mass is not.

There is no single relativistic mass that you can put into all relativistic equations to make them "look like" their Newtonian equivalents (see, for example, longitudinal and transverse mass).

Also, relativistic mass (as usually constituted) is $P^0$, or the total energy of the object (give or take a factor of $c$). Why not let energy be one thing and mass another, rather than having two terms for one thing?

@vanhees71 gave more modern reasons to prefer rest mass. I'd be interested to know if Newtonian mass emerges in a similar way from the Cartan's geometric formulation of Newtonian physics.

 

[pervect]

From what I recall from one of Max Jammer's book on the history of the concept of Mass, "Concepts of Mass in Classical and Modern Physics", Newton's original concept of mass, based on even earlier roots, was "quantity of material".

Energy doesn't fit this definition, of course - it's not a material substance.

As has been discussed at length elsewhere (and seemingly ignored as much as it's discussed), relativistic mass is just another name for energy - which is different from the classical idea of "quantity of material".

 

[DrStupid]

Rest mass is invariant (like Newtonian mass) while relativistic mass is not.

That's the only reason to favor rest mas over relativistic mass. Everything else is far-fetched.

 

[PAllen]

That's the only reason to favor rest mas over relativistic mass. Everything else is far-fetched.

What's far fetched about defining:

Mass is the thing that scales momentum and kinetic energy. For a given speed, what determines momentum and KE is mass.

From that, if you measure at high speeds, you get that momentum is not mv, and KE is not mv²/2, as they appear to be at lower speeds. A calorimeter would suffice to measure the energy carried by projectile.

To me, this is utterly natural, and all else if far fetched. (What a useful, objective term: far fetched).

These definitions are natural in Newtonian physics. With them, the (ultimately approximate) relation mv is derived, not defined.

 

[Ibix]

I'd say "the mass term in the stress energy tensor is not relativistic mass", "$P^0$ already has a name" and "relativistic mass must mean more than one thing to make relativistic formulae look Newtonian" were simple statements of fact. All are reasons to favour "mass means invariant mass". See "if I move fast enough, do I turn into a black hole?", and "photons must have mass because they have energy and energy is mass" to name but two sources of confusion.

 

[DrStupid]

What's far fetched about defining:

Mass is the thing that scales momentum and kinetic energy. For a given speed, what determines momentum and KE is mass.

1. That's not sufficient to define a frame-independent mass. Relativistic mass also fits to this definition.
2. I was talking about the reason for the definition. Not about the definition itself.

These definitions are natural in Newtonian physics. With them, the (ultimately approximate) relation mv is derived, not defined.

It's the other way around. 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).

"the mass term in the stress energy tensor is not relativistic mass"

In the stress-energy-tensor there is no mass term but an energy-term (more precisely energy density) which is proportional to relativistic mass and not to rest mass.

"$P^0$ already has a name"

And so does relativistic mass.

"relativistic mass must mean more than one thing to make relativistic formulae look Newtonian"

Relativistic mass doesn't need to make relativistic formulae look Newtonian.

"if I move fast enough, do I turn into a black hole?"

That's not a problem of relativistic mass but of Newton's law of gravitation. Newtonian gravity doesn't work in relativity - neither with relativistic mass nor with rest mass.

"photons must have mass because they have energy and energy is mass"

With relativistic mass instead of rest mass this statement is correct. Such problems do not result from relativistic mass but from the existence of different concepts of mass - no matter which of them is preferred.

That's what I mean with far-fetched reasons. Rest mass is preferred because it is frame-independent in relativity - nothing more and nothing less.

 

[PAllen]

1. That's not sufficient to define a frame-independent mass. Relativistic mass also fits to this definition.
2. I was talking about the reason for the definition. Not about the definition itself.

No, it doesn't. By scaling energy, for example, I mean I fire a standard projectile at some speed, and test mass at the same speed. I find the energy ratio is the same independent of speed. I can measure the energy with a calorimeter. I define this speed independent ratio (for momentum or energy) as mass without recourse to weight. I can then find that weight is proportional to the same quantity.

It's the other way around. 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 am proposing NOT to define momentum this way; it is NOT necessary to do it this way. You can certainly measure energy as a function of speed with just the mass definition given and find that the Newtonian formula is only approximate. I am not arguing that mine is the only sensible definition, but I strenuously arguing against any claim that it is unnatural, let along far fetched.

 

[PAllen]

Analogous to my energy example, one can measure momentum given some constant force that can be applied. Momentum is then defined by the time needed for the constant force to stop a body. Again one finds that the ratio of a test body's momentum to standard body's momentum (for objects at the same speed), is independent of speed. Mass is the scaling factor, for any given speed. Then, you can find, at high speeds, the mv is no longer an accurate formula for momentum.

 

[DrStupid]

I am proposing NOT to define momentum this way; it is NOT necessary to do it this way.

You can't change what already happened. Momentum has been defined that way in classical mechanics and the relativistic mass is a result of this definition.

 

[PAllen]

You can't change what already happened. Momentum has been defined that way in classical mechanics and the relativistic mass is a result of this definition.

Classical mechanics has been reformulated several different ways over its history. I am proposing a natural formulation in which energy and momentum expressions are derived rather than definitions. This redefinition is natural, even classically. Classically it would be expected to be equivalent to other formulations of classical mechanics. When two equivalent formulations become inequivalent with new measurements, you have a choice. All I claim is carrying over mv as a definition is NOT the only natural choice.

 

[pervect]

You can't change what already happened. Momentum has been defined that way in classical mechanics and the relativistic mass is a result of this definition.

Momentum is not universally defined as p=mv. In the Lagrangian formulation of classical mechanics, momentum is defined as the partial of the Lagrangian L(x, v) with respect to the velocity v [where x is a generalized coordinate and v=dx/dt is a generalized velocity]. This is actually a more general definition of momentum, one that works for angular momentum as well as linear momentum. This is a consequence of the Lagrangian forumaltion working with any choice of coordinates, it doesn't require any specialized coordinates. So if one uses angular coordinates with this formulation, one gets angular momentum, and when one uses linear coordinates, one gets linear momentum.

 

[DrGreg]

The modern* approach to relativity is to work with coordinate-independent 4-dimensional vectors and tensors. Rest mass arises very naturally in this approach. The relevant equations are:

4-velocity:
$$ \textbf{V} = \frac{\mbox{d}\textbf{x}}{\mbox{d}\tau} $$
4-momentum:
$$ \textbf{P} = m \textbf{V} $$
4-force:
$$ \textbf{F} = \frac{\mbox{D}\textbf{P}}{\mbox{d}\tau} $$
where $m$ is rest mass and $\tau$ is proper time (both coordinate-independent invariants). What could be a more natural 4-d generalisation of 3-d Newtonian physics than that?*Actually it's been around for a long time now!

 

[DrStupid]

Momentum is not universally defined as p=mv.

You are right. I should have been more precise: p=m·v is the definition in Newtonian mechanics.

 

[DrStupid]

All I claim is carrying over mv as a definition is NOT the only natural choice.

And all I claiming is that this definition has been carried over into special relativity before new formalisms were established.

 

[Orodruin]

Relativistic mass doesn't need to make relativistic formulae look Newtonian.

Why this unhealthy obsession with making things look Newtonian. There is no good reason they should apart from in the Newtonian limit. Using relativistic mass is just a way of taking a part of a covariant object and splitting it into two quantities which do not transform covariantly. Now why would you want to do that - historical context aside.

That's not sufficient to define a frame-independent mass. Relativistic mass also fits to this definition.

No it does not. Velocity does not transform covariantly.

Relativistic mass doesn't need to make relativistic formulae look Newtonian.

But this is exactly what you have been arguing by wanting to hold on to p = mv!

And all I claiming is that this definition has been carried over into special relativity before new formalisms were established.

This I would just call a historical curiosity, not a good argument for using relativistic mass.

 

[vanhees71]

Mass is a Casimir operator of the Poincare group. With the quantity that generates space-time translations, forming a Lorentz vector, called the four-momentum vector, it's defined as $p_{\mu} p^{\mu}=m^2$ (natural units, $c=1$). That's the clearest theoretical definition I can think of within special relativity, and it clearly defines "invariant mass".

 

[DrStupid]

Why this unhealthy obsession with making things look Newtonian.

I'm not aware of such an obsession.

No it does not. Velocity does not transform covariantly.

Nobody claimed something like that.

But this is exactly what you have been arguing by wanting to hold on to p = mv!

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

This I would just call a historical curiosity, not a good argument for using relativistic mass.

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