Undergrad Can we deal with relativistic mass once and for all?

[vanhees71]

Now we even try to discuss GR, where several posters in this thread still struggle with SR, and this most due to the fact that some still insist on using outdated concepts. Unfortunately even Feynman did so in the famous Feynman lectures, which are among my most highly favored general theory books, but in this point they are bad. There are many excellent books of physicists as eminent as Feynman that contain unfortunate approaches that do not help students but confuse them. That happens to any textbook writer from time to time. That's why one should read not only one book but many.

The best way to avoid trouble with what's called "equivalence principle" you can, if you have the minimum of necessary math in order to do GR, formulate it in a very simple form:

Spacetime is a 4D-Lorentzian manifold (i.e., a pseudo-Riemannian space with the fundamental form of signature (1,3)). This implies that around any point $x$ there exists a map defining coordinates $x^{\mu}$ such that
$$g_{\mu \nu}(x) \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} = \eta_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu}.$$
Such frames are called Galileian and are the best approximation of inertial frames possible in the presence of gravity, which cannot be described adequately by special-relativistic models but only in GR. Physically they can be realized by free-falling bodies that can be considered pointlike. An example is the International Space Station, freely falling in the gravitational field of the Earth (solar system). To a very good approximation the astronauts do not observe gravity and thus are to a good approximation in such a local inertial (or Galilean) reference frame.

 

[PAllen]

Scalar inertial mass is possible for 3-force, as discussed, for example in the Feynman lectures: http://www.feynmanlectures.caltech.edu/I_15.html.

Sorry, I see nothing there that gets around the fact that the ratio of 3 force to 3 acceleration is dependent on angle between force and velocity, thus not a scalar. He uses the word inertia only in one paragraph, and does not analyze different directions of force application. Seems not at all to support your claim.

 

[PAllen]

Despite what I have written on this thread earlier, I now propose that you can define a plausible scalar inertial mass for 3-force using 4-force as a model. Just as in 4-force, for the case of no mass/energy flows, invariant mass gives the ratio between 4-force and 4-accleration, you can reduce this to 3 dimensions and time derivatives (rather than proper time derivatives) by using the notion of celerity (the spatial part of 4-velocity, i.e. γv). Then, define time rate of change of celerity (celeration anyone??). Then 3-force over invariant mass gives celeration. Since celerity goes to infinity as as v approaches c, no matter how large the rate of change of celerity, you can't get bigger than infinite celerity, so speed remains below c. This attaches the limiting factor to the behavior of proper time (or spacetime geometry), not mass.

Then, to get coordinate acceleration from celeration, you divide the celeration component parallel to velocity by γ³, and the component orthogonal to velocity by γ, with none of the derivation for this relying on mass or momentum. These give you the velocity parallel and orthogonal components of coordinate acceleration.

 

[atyy]

Sorry, I see nothing there that gets around the fact that the ratio of 3 force to 3 acceleration is dependent on angle between force and velocity, thus not a scalar. He uses the word inertia only in one paragraph, and does not analyze different directions of force application. Seems not at all to support your claim.

That fact is derived as a consequence of Eq 15.10 (and the preceding equation).

 

[atyy]

Here is another reference illustrating that the relativistic mass can be considered an inertial mass:
http://www.scholarpedia.org/article/Nordtvedt_effect by Kenneth Nordvedt
"The subsequent theory of Special Relativity generalized this connection by asserting that every form of energy within a body contributes to the body's inertial mass; M=E/c²."

 

[PAllen]

That fact is derived as a consequence of Eq 15.10 (and the preceding equation).

I disagree. I see him using inertia here qualitatively, not specifying any expression for it. It says a lot of force is needed for a small change in speed, and does not derive anything about the quantitative relationship between force and acceleration. Also, his example happens to be one where force is orthogonal to velocity, which is the one case when relativistic mass behaves like inertia.

 

[PAllen]

Here is another reference illustrating that the relativistic mass can be considered an inertial mass:
http://www.scholarpedia.org/article/Nordtvedt_effect by Kenneth Nordvedt
"The subsequent theory of Special Relativity generalized this connection by asserting that every form of energy within a body contributes to the body's inertial mass; M=E/c2."

I completely agree with this statement, but invariant mass includes all forms of energy within a body, including kinetic and potential. We keep going around in circles, with you denying the seeming obvious fact that invariant mass includes kinetic energy. I claim this statement agrees with me not you. it gets back to the same issue we had with the Carlip paper, where you interpret as saying something very different from what me and others here think it says.

 

[SiennaTheGr8]

Despite what I have written on this thread earlier, I now propose that you can define a plausible scalar inertial for 3-force using 4-force as a model. Just as in 4-force, for the case of no mass/energy flows, invariant mass gives the ratio between 4-force and 4-accleration, you can reduce this to 3 dimensions and time derivatives (rather than proper time derivatives) by using the notion of celerity (the spatial part of 4-velocity, i.e. γv). Then, define time rate of change of celerity (celeration anyone??). Then 3-force over invariant mass gives celeration. Since celerity goes to infinity as as v approaches c, no matter how large the rate of change of celerity, you can't get bigger than infinite celerity, so speed remains below c. This attaches the limiting factor to the behavior of proper time (or spacetime geometry), not mass.

Then, to get coordinate acceleration from celeration, you divide the celeration component parallel to velocity by γ³, and the component orthogonal to velocity by γ, with none of the derivation for this relying on mass or momentum. These give you the velocity parallel and orthogonal components of coordinate acceleration.

Quite right! In this context, celerity is arguably a more "natural" quantity than velocity. In symbols, with $c=1$ and constant $m$:

$\mathbf F = m \mathbf A$,

where $\mathbf F = \gamma (\mathbf f \cdot \mathbf v, \mathbf f)$ is the four-force (lowercase $\mathbf f$ being the three-force) and $\mathbf A = \gamma(\dot \gamma, \dot{\mathbf{w}})$ is the four-acceleration ($\mathbf w = \gamma \mathbf v$ is the celerity). From this we have:

$\mathbf f = m \dot{\mathbf w} = m \dfrac{d}{dt} (\gamma \mathbf v)$.

So yes, plain old invariant mass is the constant of proportionality relating three-force and "celeration" (!). Also, there's this nice relation (again, for constant $m$):

$\mathbf f \cdot \mathbf v = m \dot \gamma$.

 

[SiennaTheGr8]

$\mathbf f = m \dot{\mathbf w} = m \dfrac{d}{dt} (\gamma \mathbf v)$.

...

$\mathbf f \cdot \mathbf v = m \dot \gamma$.

And then also $\dot \gamma = \dot{\mathbf w} \cdot \mathbf v$

 

[Herman Trivilino]

So if I measure the mass of a moving electron, it would seem heavier, because I multiply it's rest mas with factor gama.

This depends upon what you mean by "seem heavier". To me, an object of mass $m$ has a mass equal to $m$, not $\gamma m$.

How do you feel about $\gamma^3 m$?

But I think it might be useful to describe it's inertial mass.

One can argue that that inertial mass ought to be $m$, or $\gamma m$, or $\gamma^3 m$. It's not a well-formed concept in relativistic physics.

 

[atyy]

Here is yet another reference (p110) of https://www.amazon.com/dp/0198567324/?tag=pfamazon01-20 by Rindler:
"m = γ (u)m₀, (6.4)
p = mu. (6.5)
The formalism thus leads us naturally to this quantity m, which we shall call the relativistic inertial mass (or usually just ‘mass’), and to p, which we shall call the relativistic momentum (or usually just ‘momentum’)."

Compare this with the Nordvedt reference in post #35 and the Feynman reference in post #29.

 

[PAllen]

Here is yet another reference (p110) of https://www.amazon.com/dp/0198567324/?tag=pfamazon01-20 by Rindler:
"p = mu. (6.5)
The formalism thus leads us naturally to this quantity m, which we shall call the relativistic inertial mass (or usually just ‘mass’), and to p, which we shall call the relativistic momentum (or usually just ‘momentum’)."

Compare this with the Nordvedt reference in post #35 and the Feynman reference in post #29.

What??! This is explicitly defining relativistic inertial mass to be invariant mass!

[edit: It took a while to get permission to see the whole more context from that link. Rindler is using u differently from most authors, to refer to regular velocity not 4-velocity. So, yes, this book purports to call relativistic mass inertial mass. It does not change the fact that it does not meet any normal definition of scalar inertia. It is as if the author is saying, we shall call a duck a chicken. ]

 

[atyy]

What??! This is explicitly defining relativistic inertial mass to be invariant mass!

I added Eq (6.4) to post #41 to clarify the notation.

 

[Jan Nebec]

To me, an object of mass $m$ has a mass equal to $m$, not $\gamma m$.

Well...what I'm saying is, that it is a fact that an object with a velocity approaching $c$ will seem harder and harder to accelerate because of the increase of it's inertial mass? And this is a mass calculated with multiplying the rest mass with $\gamma$ or $\gamma^3$, depens of the situation...that's my oppinion.

 

[PAllen]

Well...what I'm saying is, that it is a fact that an object with a velocity approaching $c$ will seem harder and harder to accelerate because of the increase of it's inertial mass? And this is a mass calculated with multiplying the rest mass with $\gamma$ or $\gamma^3$, depens of the situation...that's my oppinion.

That’s defensible. But think about the following:

Go to the rest frame of the body. There its inertia is unambiguously invariant mass. Accelerate it to .1 c, relativistic affects still small at this speed. Repeat this process a million times. The result is not a speed of 100,000 c in the original frame, it is still less than c. The explanation of this process is all about velocity addition, which has nothing to do with mass or inertia.

 

[Herman Trivilino]

Well...what I'm saying is, that it is a fact that an object with a velocity approaching $c$ will seem harder and harder to accelerate because of the increase of it's inertial mass?

If you define inertial mass to be $\gamma^3 m$, then yes.

And this is a mass calculated with multiplying the rest mass with $\gamma$ or $\gamma^3$, depens of the situation...that's my oppinion.

But those are only two of the many many choices, one for every possible angle between the direction of the observer's motion and the direction of the force. So if you happen to choose $\gamma^3 m$ then you will have what you've call inertial mass in your first sentence above. But if you choose some other value then you won't!

 

[vanhees71]

Here is another reference illustrating that the relativistic mass can be considered an inertial mass:
http://www.scholarpedia.org/article/Nordtvedt_effect by Kenneth Nordvedt
"The subsequent theory of Special Relativity generalized this connection by asserting that every form of energy within a body contributes to the body's inertial mass; M=E/c²."

You don't really proposse to take this seriously? Either you do Newtonian gravity with Newtonian mechanics or you do relativistic gravity, which is as far as we know, and which becomes more and more established by all kinds of recent high-precision observations in astrophysics, GR. Then there's no inertial vs. gravitational mass anymore but just (invariant!) mass. The sources of the gravitational field in GR is the energy-momentum-stress tensor of the matter fields and nothing else. Everything is generally co-variant as it should be within GR. To deal with non-covariant quantities in GR is completely meaningless. In SR you might get along with some non-covariant notions like "relativistic mass" for a while, but there's good reason for not doing so, because it's leading to errors as many historical examples show. It becomes, e.g., a complete mess if you use old-fashioned notions of temperature and other thermo-statistical quantities rather then defining them in the modern way as scalars (for temperature). In GR everything else becomes completely uninterpretable (i.e., gauge dependent).

 

[vanhees71]

Hello!

I've been reading about relativistic mass for last few days and it leads me to even more confusion.
Supposing, we are assuming SR.

1. Why some people say that relativistic mass leads to confusion? As far as I learned, relativistic mass tells me the mass of an object, that is moving relative to me. So if I measure the mass of a moving electron, it would seem heavier, because I multiply it's rest mas with factor gama. I don't see any problems here?

Again, mass in the modern sense of the word (modern meaning the way one should introduce SR from the very beginning since Minkowski's ground-breaking talk/paper of 1908) is the same quantity as in non-relatistic physics, and it's a Lorentz scalar. I've given clear arguments how it is defined in classical relativistic point mechanics by using the covariant equation of motion
$$m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2}=K^{\mu},$$
where $K^{\mu}$ is the Minkowski four-force vector (a Lorentz vector), $\tau$ the proper time of the particle with invariant mass $m$.

If you say "measure" you have to say how. The historical way (admittedly done using the old-fashioned idea of all kinds of "relativistic masses", which are not only speed but even direction dependent, i.e., very confusing!) was to first measure the trajectories of electrons in external electric and magnetic (static) fields. The Lorentz four-force reads
$$K^{\mu}=\frac{q}{c} F^{\mu \nu} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},$$
where $F^{\mu \nu}$ is the antisymmetric Faraday tensor, containing the electric and magnetic field components as measured in the intertial frame of reference where the experiment is done. $q$ is the charge of the test particle (electron), which is a Lorentz-scalar as mass (you can remember that any intrinsic numbers concerning point particles, like mass, charge, magnetic moment are defined in the instantaneous inertial rest frame of the particle and then written down in covariant form to make it usable in any inertial reference frame right away).

As you can see, what can be measured by measuring the trajectories is only the ratio $q/m$ (which by definition is of course a Lorentz scalar either), and indeed that was what J. J. Thomson did in his famous experiments which lead to the discovery of the electron. His electrons were however so slow, that the non-relativistic approximation was sufficient. Then around 1900 plenty of models of the electron popped up. As far as I know there were Einstein's relativistic point-particle mechanics as detailed (in a quite messy way using the relativistic mass!) and Abraham's mechanical model of the electron, predicting different equations of motion, but the sensitivity needed were corrections of order $v^2/c^2$ to decide between the two models. This was taken up by Kaufmann et al who measured the trajectories of electrons in homogeneous electric fields very accurately. In the beginning the precision was not good enough, but finally the results (maybe also by measurements of other physicists; maybe there's something more detailed in the Wikipedia which has a nice article on the experimental tests of SR) lead to the conclusion that Einstein's SR is right.

The challenge to determine the electron's charge was solved by Millikan with his famous oil-drop experiment. Together with the measurements of the trajectory in electromagnetic fields, leading to precise values of $q/m$ this also implied the value for the electron mass.

2. Only problem that I see here is, that if someone uses relativistic mass instead of rest mass, he should carefully define velocity and frame of reference. Because for example, rest masses of 1kg and 10kg can both get relativistic mass of 100kg, so we have to define their velocity relative to observer to get full description.

That's another reason to use the invariant mass. It's much easier to talk about it since you don't need to bother about the frame of reference used to define it, because it's a scalar!

3. However, relativistic mass can not be used to calculate gravitational attraction of an object! So it is useless in this respect. But I think it might be useful to describe it's inertial mass. But since we are told that inertial mass and gravitational mass are the same, I am confused, because increase of relativistic mass refers only to it's inertial mass. Am I right?

Indeed, to calculate the trajectory of test particles in gravitational fields, you need GR and the corresponding equation of motion, which is the equation of geodesics in the corresponding pseudo-Riemannian space-time manifold. You should only learn about GR when you have a clear modern (i.e., covariant) understanding of SR.

 

[Dadface]

I join the ranks of the confused in this thread, but In my case I'm mega confused. I know we can define a unit of mass in terms of a lump of metal stored somewhere in Paris. But what is mass? There seems to be different labels used. So far, in this thread, I have seen the following:

• Relativistic mass
• Scalar inertial mass
• Mass
• Inertial mass
• Gravitational mass
• Invariant mass

The common word used is mass? But what is mass? Can it be defined in words?

 

[A.T.]

...an object with a velocity approaching $c$ will seem harder and harder to accelerate ...

Only if the energy for the acceleration is not stored on board. For example, a relativistic wind up car would gain "inertial mass" when you wind it up, not when it accelerates.

 

[Jan Nebec]

But those are only two of the many many choices...

Well yes, the exact formula can be found here: https://encrypted-tbn0.gstatic.com/...gTfYw8diKXFmta07ozKASLLhZFQPugMPPkr01AntmQ_tU

 

[Jan Nebec]

The explanation of this process is all about velocity addition, which has nothing to do with mass or inertia.

Velocity addition only tells me that relative velocity between objects can never exceed the speed of light. But it doesn't quite explain why objects resist acceleration more and more when they gain speed?

 

[Mentz114]

Velocity addition only tells me that relative velocity between objects can never exceed the speed of light. But it doesn't quite explain why objects resist acceleration more and more when they gain speed?

Possibly because there is loss of simultaneity between action and reaction which makes the reaction 'act' more than the action. The same thing is observed between the front and back of the accelerating spaceship.

 

[PeroK]

Velocity addition only tells me that relative velocity between objects can never exceed the speed of light. But it doesn't quite explain why objects resist acceleration more and more when they gain speed?

They don't resist acceleration as they gain speed. Instead, energy is frame dependent, as it is in classical physics, by the way.

The problem with relativistic mass is that it suggests that the energy is somehow related to an intrinsic change in the object itself. This can lead students up the wrong path.

Intrinsic changes shouldn't be frame dependent. Failure to grasp this is why you and @Dadface, among others, are tying yourself in theoretical and experimental knots.

 

[vanhees71]

I join the ranks of the confused in this thread, but In my case I'm mega confused. I know we can define a unit of mass in terms of a lump of metal stored somewhere in Paris. But what is mass? There seems to be different labels used. So far, in this thread, I have seen the following:

• Relativistic mass
• Scalar inertial mass
• Mass
• Inertial mass
• Gravitational mass
• Invariant mass

The common word used is mass? But what is mass? Can it be defined in words?

Here's my take on these notions:

Relativistic mass is sometimes used in SR, particularly in older books. It's not used in contemporary physics research anymore and it is usually leading to confusion. In GR it's not even definable.

Scalar inertial mass, inertial mass, mass, invariant mass: Scalar inertial mass is an expression I've never heard before. Usually one talks about mass or invariant mass. In my community if you talk simply about mass everybody understands "invariant mass".

Inertial/Gravitational mass: rarely used. In SR there is only inertial mass anyway. Gravitational mass doesn't make sense in relativistic physics. The only relativistic theory of gravitation, which stands all observational tests today, is GR, and there you have invariant mass as the only well-defined concept of mass, and the source of gravity is unambigously the energy-momentum-stress tensor of matter and radiation.

 

[PeroK]

@Jan Nebec SR fundamentally is about the nature of time and space, not about the properties of physical objects.

You can explain a particle accelerator and the limit of $c$ entirely through an appropriate model of spacetime.

If you look for an answer in the nature of the particles themselves - through a change in mass or inertia - you are looking in the wrong place.

 

[Dadface]

Think of a relativistic object smashing into a planet and causing a huge crater. Now imagine an identical object, but where before impact the speed increased by an infinitely small amount, this resulting in a gargantuan increase of kinetic energy. I think it may be generally agreed that in this event there will be much more damage done and a much bigger crater formed. But without the concept of relativistic mass it's difficult to get an intuitive feeling as to why this is the case.

Without a mass increase we would have two equal mass objects traveling at speeds which differ by an infinitesimally small amount but where one object can cause, for example, a million times more damage than the other. Apart from a tiny larger speed what else is different, if anything, about the faster object?

I used to think that at non relativistic speeds an accelerating object displayed an increase of kinetic energy primarily as an increase of speed, I further thought that as the speed approached c the increase of mass effect became more dominant. I think I still believe that but I'm here trying to follow other viewpoints.

 

[PeroK]

Think of a relativistic object smashing into a planet and causing a huge crater. Now imagine an identical object, but where before impact the speed increased by an infinitely small amount, this resulting in a gargantuan increase of kinetic energy. I think it may be generally agreed that in this event there will be much more damage done and a much bigger crater formed. But without the concept of relativistic mass it's difficult to get an intuitive feeling as to why this is the case.

Without a mass increase we would have two equal mass objects traveling at speeds which differ by an infinitesimally small amount but where one object can cause, for example, a million times more damage than the other. Apart from a tiny larger speed what else is different, if anything, about the faster object?

I used to think that at non relativistic speeds an accelerating object displayed an increase of kinetic energy primarily as an increase of speed, I further thought that as the speed approached c the increase of mass effect became more dominant. I think I still believe that but I'm here trying to follow other viewpoints.

The same is true for classical KE: $\frac12 mv^2$.

If the speed is large enough, a tiny change in speed represents a huge change in KE.

 

[jbriggs444]

Without a mass increase we would have two equal mass objects traveling at speeds which differ by an infinitesimally small amount but where one object can cause, for example, a million times more damage than the other. Apart from a tiny larger speed what else is different, if anything, about the faster object?

Energy. And momentum.

 

[Dadface]

The same is true for classical KE: $\frac12 mv^2$.

If the speed is large enough, a tiny change in speed represents a huge change in KE.

Agreed but consider the classical case. If the speed increases from v to v+dv the fractional increase of KE is given by

f = 2vdv/v + dv squared/v

In other words as v increases the fractional increase of classical KE per unit increase of speed, decreases. In the relativistic case the fractional change of KE approaches infinity as v approaches c