Those who use relativistic mass and why

[pmb_phy]

Pete,

That was a good tip: put 'DW' in my ignore list. The topic looks much better now.

Quart

Hi quart

Don't get me wrong. I never put people on my ignore list except for people who spam threads with the same comments they've posted a thousand times before without end. It is always a good idea to hear somone elses views (in fact you're almost always a better person for it) dw has badly abused that idea by repeating himself to the same person, the same comments a times while ignoring proof under all occasions. Its irritating after the first 100 times.

E.g. to show you what I mean I took a gander at his last one for purposes of illustration. This is the 1,000 th time that he's claimed that the position 4-vector is not 4-vector. I explained to dw why his claim is wrong 1,000 times. He ignores it 1000 times and then stgarts repeating himself

In this case R = (ct, x, y, z) is a Lorentz 4-vector. Its defined as the displacement displacement from a chosen event which is defined as the "origin of coordinates". This is standard stuff found everywhere and in nearly all relativity/em texts (e.g. Ohanian, J.D. Jackson, Thorne and Blanchard etc). Yet dw can't seem to learn it. (sigh)

To be precise, define

\bold X^P \equiv (ct_P, x_P, y_P, z_P) = Event P

\bold X^Q \equiv = (ct_Q, x_Q, y_Q, z_Q) = Event Q

\Delta \bold X \equiv \bold X^P - \bold X^Q = (ct_P, x_P, y_P, z_P) - (ct_Q, x_Q, y_Q, z_Q) = (c\Delta t, \Delta x, \Delta y, \Delta z)

x \equiv x_P - x_Q = \Delta x

y \equiv y_P - y_Q = \Delta y

z \equiv z_P - z_Q = \Delta z

Now define event Q as the "Origin" of the coordinate system. This means, for example, that x is the x-component of a displacement R from something called the "origin" and is written as

\bold R \equiv \Delta \bold X = (ct, x, y, z)

That is the template of all Lorentz 4-vectors.

A previous example was when he claimed that what I was using couldn't be readily used to translate to GR. Thus he took my explanation of what is equivalent of defining and describing the components of 4-vectors and has ignored the numerous times where I've used it in equations in GR wiuth 4-vectors. Here is a perfect example

http://www.geocities.com/physics_world/gr/grav_force.htm

In that derivation you can see how the relativistic mass falls out of a derivation which starts with all 4-vectors. See Eq. (8a) in above link. I assume you'll understand why I'll ignore dw's claims on its correctness when he tries to respond to this right?


Smart move that you took quart.

Pete

 

[DW]

Pete,

That was a good tip: put 'DW' in my ignore list. The topic looks much better now.

Quart

I guess fake physics looks better to you.

 

[DW]

While It doesn't hurt to hear somone elses views, in fact you're a better person for it, dw abuses that notion by repeating his same bogus claims a thousand times while ignoring proof.

You are the one ignoring the proof I and others have given that you are wrong.

This is the 1,000 th time that he's claimed that the position 4-vector is not 4-vector.

See, as I said you never stopped reading my posts. And no, position is not a four-vector.

I explained to dw 1,000 times that R = (ct, x, y, z) is a Lorentz 4-vector.

No it is not.

Its defined as the displacement ...

No it is not. The four-element coordinate position is not a displacement and is not a vector. Displacement is a vector. Position is not.

 

[Garth]

All that has gone before seems to illustrate the point in my last post.
Is it the case that one way to harmonise the two points of view would be to accept that energy and mass are equivalent concepts and 'relativistic mass' can simply be renamed "Total energy"? Of course particles have other properties too, inertia and other charges, in which case the defining characteristic of 'proper mass' would seem to be its inertia.
Finally, are there any thoughts about my point that the 'relativistic mass' concept, although not in favour with DW and the standard convention in particle physics, may actually be the more fundamental point of view?

 

[pmb_phy]

The relativistic Lagrangian does indeed contain the mass of a particle if I adopt the convention that the norm of the 4-momentum is the mass.

That is only true for the covariant Lagrangian. It is not true for the relativistic, non-covariant Lagranmgian. The (relativistic, non-covariant) Lagrangian for a charged particle in an EM field is given by (Reference: Classical Electrodynamics - 2nd Ed., J.D. Jackson, page 574, Eq. (12.9)}

L = \mu c^2 \sqrt{1 - v^2/c^2} + q\Phi - \frac{q}{c}\bold v \bullet \bold A

where mu is the particle's proper mass. However

m = \gamma \mu

where m is the particle's inertial mass. Solving for mu and substituting into the above equation gives

L = \frac{mc^2}{1 - v^2/c^2} + q\Phi - \frac{q}{c}\bold v \bullet \bold A

The Lagrangian can therefore be expressed in terms of either the proper mass or the relativistic mass and there is no significant difference inherent in the Lagrangian which prefers one over the other. The only diference is that the later Lagrangian is not defined for v = c. That's a direct result of m not being defined for v = c.

Pete

 

[pmb_phy]

You ..

Me? Yeeesh! You sure have a lot to learn about the relativity community. Fine. Okay, me. But also Einstein, Wheeler, Thorne, Rindler, D'Inverno, Sartori, D'Inverno, Mould, Peacock, Guth, etc. etc. etc. etc. etc. etc. etc. etc.

...are wanting to replace the m in that form of Newton's second law with Planck's variable mass concept,...

In 1905 Einstein attempted to write the equations of a charged particle in an EM field in the form F = ma. That led to his use of transverse and longitudinal mass. In the year that followed, i.e. 1906, Planck showed that the Lorentz force could be written in the form

\bold F = \frac{d\bold p}{dt} = \frac{d(\gamma m_o \bold v)}{dt} = q(\bold E + \bold v \times \bold B)

Or substituting in relativistic mass m = gamma m_o

\bold F = \frac{d(m \bold v)}{dt} = q(\bold E + \bold v \times \bold B)

where m is the relativistic mass of the body. After that paper Planck never tried to prove that mass can in all cases be set equal to m = gamma m_o so he did not get the credit for showing that m = gamma m_o. Hence he does not deserved the credit. Three years later, in 1909, Tolman and Lewis argued that mechanics should be obtained from the conservation laws and the principle of relativity and without reference to electrodynamics. In their famous paper The Principle of Relativity and Non-Newtonian mechanics they demonstrated the feasability of such a notion through the now famous collision thought experiment.Three years later in 1912 Tolman published a more general version in his famous paper Non-Newtonian Mechanics: The Mass of a Moving Body. All relativity texts (at least those which I know of) which derive the momentum equation p = gamma*m_o*v now use Tolman's method as described in that paper. Neither paper had anything to do directly with force. It was due to this work that, in part, was responsible for relativity papers to no longer be restrticted to EM journal references. Hence Tolman and Lewis are given the credit for being the ones to show that mass depends on velocity.

 

[DW]

Hi quart

Don't get me wrong. I never put people on my ignore list except for people who spam threads with the same comments they've posted a thousand times before without end. It is always a good idea to hear somone elses views (in fact you're almost always a better person for it) dw has badly abused that idea by repeating himself to the same person, the same comments a times while ignoring proof under all occasions. Its irritating after the first 100 times.

E.g. to show you what I mean I took a gander at his last one for purposes of illustration. This is the 1,000 th time that he's claimed that the position 4-vector is not 4-vector. I explained to dw why his claim is wrong 1,000 times. He ignores it 1000 times and then stgarts repeating himself

In this case R = (ct, x, y, z) is a Lorentz 4-vector. Its defined as the displacement displacement from a chosen event which is defined as the "origin of coordinates". This is standard stuff found everywhere and in nearly all relativity/em texts (e.g. Ohanian, J.D. Jackson, Thorne and Blanchard etc). Yet dw can't seem to learn it. (sigh)

To be precise, define

\bold X^P \equiv (ct_P, x_P, y_P, z_P) = Event P

\bold X^Q \equiv = (ct_Q, x_Q, y_Q, z_Q) = Event Q

\Delta \bold X \equiv \bold X^P - \bold X^Q = (ct_P, x_P, y_P, z_P) - (ct_Q, x_Q, y_Q, z_Q) = (c\Delta t, \Delta x, \Delta y, \Delta z)

x \equiv x_P - x_Q = \Delta x

y \equiv y_P - y_Q = \Delta y

z \equiv z_P - z_Q = \Delta z

Now define event Q as the "Origin" of the coordinate system. This means, for example, that x is the x-component of a displacement R from something called the "origin" and is written as

\bold R \equiv \Delta \bold X = (ct, x, y, z)

That is the template of all Lorentz 4-vectors.

A previous example was when he claimed that what I was using couldn't be readily used to translate to GR. Thus he took my explanation of what is equivalent of defining and describing the components of 4-vectors and has ignored the numerous times where I've used it in equations in GR wiuth 4-vectors. Here is a perfect example

http://www.geocities.com/physics_world/gr/grav_force.htm

In that derivation you can see how the relativistic mass falls out of a derivation which starts with all 4-vectors. See Eq. (8a) in above link. I assume you'll understand why I'll ignore dw's claims on its correctness when he tries to respond to this right?


Smart move that you took quart.

Pete

Now you are back peddling on what you said. Originally you claimed position was a four-vector. Now you are trying to twist what you said to refer to a displacement. It is obvious you are not being honest particularly because you had considered the displacement OF that position in the numerator of your definition of four-vector velocity. You don't put a displacement of a displacement there! Just admit you were wrong and I corrected you and move on.

 

[DW]

That is only true for the covariant Lagrangian. It is not true for the relativistic, non-covariant Lagranmgian. The (relativistic, non-covariant) Lagrangian for a charged particle in an EM field is given by (Reference: Classical Electrodynamics - 2nd Ed., J.D. Jackson, page 574, Eq. (12.9)}

L = \mu c^2 \sqrt{1 - v^2/c^2} + q\Phi - \frac{q}{c}\bold v \bullet \bold A

where mu is the particle's proper mass. However

m = \gamma \mu

where m is the particle's inertial mass. Solving for mu and substituting into the above equation gives

L = \frac{mc^2}{1 - v^2/c^2} + q\Phi - \frac{q}{c}\bold v \bullet \bold A

Pete

Corrections
The Lagrangian for a charged particle in an EM field is given by (Reference: Classical Electrodynamics - 2nd Ed., J.D. Jackson, page 574, Eq. (12.9)}

L = m c^2 \sqrt{1 - v^2/c^2} + q\Phi - \frac{q}{c}\bold v \bullet \bold A

where m is the particle's mass. However

E = \gamma mc^2

where E is the particle's energy. Solving for m and substituting into the above equation gives

L = \frac{E}{1 - v^2/c^2} + q\Phi - \frac{q}{c}\bold v \bullet \bold A

 

[DW]

In 1905 Einstein attempted to write the equations of a charged particle in an EM field in the form F = ma.

I am the one who has been telling you about this.

That led to his use of transverse and longitudinal mass.

No. He didn't use those. He introduced those concepts. The mass that he actually used in the paper was mass as invariant.

In the year that followed, i.e. 1906, Planck showed that the Lorentz force could be written in the form

\bold F = \frac{d\bold p}{dt} = \frac{d(\gamma m_o \bold v)}{dt} = q(\bold E + \bold v \times \bold B)

Or substituting in relativistic mass m = gamma m_o

\bold F = \frac{d(m \bold v)}{dt} = q(\bold E + \bold v \times \bold B)

where m is the relativistic mass of the body. ...

Which is why I keep referring to "relativistic mass" as Planck's mass concept which is a dead concept having no place in modern relativity.

(snipped some flamming)

Hence dw's (and his alter ego GRCQ} honored place on the ignore list.

Just because every knowledgeable person disagrees with you does not make every knowledgeable person me. I am not GRCQ and such a lie should warrent your dissmissal here. And how is it that you keep responding to someone you are ignoring? You never had me on any such list.

 

[quartodeciman]

Pete,

You may have surmised by now that I'm right on the verge between (what I call) mass-first and energy-first points of view. That is why I posed my earlier question about validating SR momentum fundamentally. Then (I hope) one has (perhaps) a choice of directions for expanding SR dynamics.

SR kinematics --->SR momentum law--->exploitation of mass dynamics--->*
SR kinematics --->SR momentum law--->exploitation of energy dynamics--->*
*--->justification of 4-dimensional dynamic worldview

Quart

 

[DW]

All that has gone before seems to illustrate the point in my last post.
Is it the case that one way to harmonise the two points of view would be to accept that energy and mass are equivalent concepts and 'relativistic mass' can simply be renamed "Total energy"?

Energy and mass are not equivalent. Mass is equivalent to "rest frame energy". Calling relativistic mass energy is NOT renaming it. Calling energy by relativistic mass IS renaming energy with a missnomer.Using Planck's concept of mass harminises nothing. Using mass as invariant is what is correct and as such haminises everything that encorporates it.

Of course particles have other properties too, inertia and other charges, in which case the defining characteristic of 'proper mass' would seem to be its inertia.

Qualifying the word mass with proper is wrong because mass is invariant. The length of the momentum four-vector is the same value for ALL frames.

 

[Garth]

DW - An atom has mass, which we would want to define as invariant, yet that mass may vary by the emission/absorption of a photon.
So we open up the atom and find atomic particles - nucleons and electrons and a total system energy. This energy is the source of that photon's energy. The particles have mass, which we would want to define as invariant, yet they enter into energetic interactions.
So we open up the nucleons and find quarks and strong/weak force energies which supply those interactions with energy. The quarks have mass, which we would want to define as invariant, yet they enter into energetic interactions.
So we open up the quarks??
Where does the process end?
According to one popular theory it ends with strings, which have mass? or is it energy? At this level the strings are vibrations of energy, as the Schrodinger coordinate representation, with its wave packet functions, would have it all along in the first case, so where is mass at the most fundamental level?

 

[pmb_phy]

I recall somewhere someone quoting dw about his claim that the gamma factr is supposed to be associated with the velocity and not the mass. Unforunately for de that's an incorrect assumption. In the first place, any definition of mass should hold in all cases. dw's association of gamma with velocity is invalid for photons since in that case the gamma factor is infinite and the momentum is finite. The relativistic mass is still well defined. Also, as I've explained to dw on countless occassions, the complete description of mass requires a tensor. As an example, consider the case of pressureless dust. In the aforementioned tensor the momentum density, g for matter is given by

\bold g = \rho \bold v

where rho is the mass density given by

\rho = \gamma^2 \rho_0

In this case its not meaningful to associate gamma with v since one is still left with another gamma. And this relation is not valid in general and especially not for radiation.

Pete

 

[DW]

I recall somewhere someone quoting dw about his claim that the gamma factr is supposed to be associated with the velocity and not the mass. Unforunately for de that's an incorrect assumption. In the first place, any definition of mass should hold in all cases. dw's association of gamma with velocity is invalid for photons since in that case the gamma factor is infinite and the momentum is finite. The relativistic mass is still well defined. Also, as I've explained to dw on countless occassions, the complete description of mass requires a tensor. As an example, consider the case of pressureless dust. In the aforementioned tensor the momentum density, g for matter is given by

\bold g = \rho \bold v

where rho is the mass density given by

\rho = \gamma^2 \rho_0

In this case its not meaningful to associate gamma with v since one is still left with another gamma. And this relation is not valid in general and especially not for radiation.

Pete

My definition is not invalid for photons and you know it because you read it and are now misrepresenting it. You know very well that in general I define momentum in terms of a quantum frequency and wavelengths in short in terms of a wavelength k vector which applies for both massive and massless particles. I then define mass in terms of that vector and only afterward demonstrate the relation between momentum and four-vector velocity only as a second hand result for massive particles. How dare you knowingly missrepresent my position?

 

[DW]

DW - An atom has mass, which we would want to define as invariant, yet that mass may vary by the emission/absorption of a photon.
So we open up the atom and find atomic particles - nucleons and electrons and a total system energy. This energy is the source of that photon's energy. The particles have mass, which we would want to define as invariant, yet they enter into energetic interactions.
So we open up the nucleons and find quarks and strong/weak force energies which supply those interactions with energy. The quarks have mass, which we would want to define as invariant, yet they enter into energetic interactions.
So we open up the quarks??
Where does the process end?

It ends at the virtual particles comprising the field energy in the bindings, which are ultimatley what really account for why we can use the potential energy shortcut in the first place.

According to one popular theory it ends with strings, which have mass? or is it energy? At this level the strings are vibrations of energy, as the Schrodinger coordinate representation, with its wave packet functions, would have it all along in the first case, so where is mass at the most fundamental level?

Where it comes to strings, the mass of a string correspond to particular modes. Since I am not a string theorist and relativity was never intended to apply in the string domain, I don't see how bringing up those scales is relevant outside of being interesting if it does. As far as I have gone is to see how general relativity is applicable in electrodynamics all the way down to the subatomic particle quantum domain.

 

[Garth]

DW - The question I am asking is whether mass is inevitably invariant or might it vary to include energies, especially potential energies?

I am not refuting any convention here to cause an argument, I am asking a serious question in order to seek the truth at the most fundamental levels.

I am not particularly happy with string theory either, as I don't like inventing things, like extra dimensions, which then conveniently roll themselves up so you can't see them. - Like the fairies at the bottom of my garden that are really there but you can never see them because they are so shy. Today's 'New Scientist' reports a new string theory that apparently does away with these extra dimensions so I shall be happier with that. We shall see.

However your phrase "all the way down " reminds me of the story of the Flat Earther who gave a lecture on how the Earth was a flat disc sitting on the back of four elephants, which stood on the back of a giant turtle. When he was asked what the turtle was standing on he replied, "Its turtles all the way down".

My point is that if it is not 'turtles all the way down' then we end up at a fundamental level which consists of objects (particles/strings or whatever) that are the final/ultimate repositories of mass. However, because their interactions will have to determine the interactions in the level above, they will have to enter into energetic interactions themselves. In this case as they use/release energy their mass will have to vary to accommodate that exchange of energy. If not there must be a deeper level consisting of particles with mass and an energy bank. And so on...

Perhaps it is "turtles all the way down."

 

[Nereid]

However your phrase "all the way down " reminds me of the story of the Flat Earther who gave a lecture on how the Earth was a flat disc sitting on the back of four elephants, which stood on the back of a giant turtle. When he was asked what the turtle was standing on he replied, "Its turtles all the way down".

OK, I confess; this is OT: Terry Pratchett wrote the Discworld series based on this idea, which itself has a very long history (x000 BC Hindu?). I'm not sure if the first use of the phrase is well determined

 

[pmb_phy]

There are other situations when the mass = proper mass is inadequate. It's been shown that a rod is easier to accelerated when it is pulled rather than when it is pushed. This, of course, implies that a scalar cannot be associated with an object in all cases to describe the inertial properties of an object.

 

[Garth]

OK, I confess; this is OT: Terry Pratchett wrote the Discworld series based on this idea, which itself has a very long history (x000 BC Hindu?). I'm not sure if the first use of the phrase is well determined

I had thought it was ancient Egypt.

 

[selfAdjoint]

I had thought it was ancient Egypt.

India? With the elephants? I think Pratchett has only one level of elephants and one turtle, at least that's all the "disconauts" in one of his books saw.

 

[Haelfix]

I just read this thread, wasting my time more or less.

I agree with Tom Matson btw, I can't see the argument as this is purely a matter of convention and notation. Feel free to rename popular concepts mass, energy, whatever all you want all that matters is the quantity an experiment measures and what a theory predicts is that *number*. I for one see your arguments as more or less equivalent.

All this is soo much easier if you just work in natural units hbar = c = 1. If you want set G = 1 too in the GR context.

Besides if you really want to quibble about semantics, all this stuff is moot game. Field theory and full general relativity is the language proffessional physicists talk in nowdays.

 

[pmb_phy]

I just read this thread, wasting my time more or less.

I agree with Tom Matson btw, I can't see the argument as this is purely a matter of convention and notation. Feel free to rename popular concepts mass, energy, whatever all you want all that matters is the quantity an experiment measures and what a theory predicts is that *number*. I for one see your arguments as more or less equivalent.

Its not always wise to discard any future thought about something like this because you've decided its all a matter of semantics. It took a few years of studying this subject in detail before I realize that it was much more than that. Only then did I start asking myself more fruitful questions on this topic and it was then that it produced fruit.

Pete

 

[Haelfix]

Lets be honest, mass is a completely nebulous concept either way.

Outside of classical mechanics, its somewhat arbitrarily defined depending on the theory.

Already in vanilla quantum mechanics, its hard to say exactly what *is* the mass. In special relativity there is your discussion thread. In GR, there exists metrics where no sensible global notion of what mass-energy is.

In field theory (particularly when talking about 1st order gravity), its just so painful to even think about such things, that no one has bothered muddling their head over what exactly *is* the physical meaning.

What we do have is a bunch of equations, that output a number for a specific situation and experimental setup, and that's that. I think nature has given us a pretty good hint that our intuitions are leading us down a blind alley in this case, and that we should just follow the tried and true equations that match experiment.

And based on those equations, I don't see any mathematical inconsistency between your choice of conventions and DW's. Now if you wish to debate that, please clearly outline the statement and show me that x is not equal to y in say an experiment.

 

[Garth]

I agree with Tom Matson btw, I can't see the argument as this is purely a matter of convention and notation. Feel free to rename popular concepts mass, energy, whatever all you want all that matters is the quantity an experiment measures and what a theory predicts is that *number*. I for one see your arguments as more or less equivalent.

Thank you, a number of us have made the same observation.

All this is soo much easier if you just work in natural units hbar = c = 1. If you want set G = 1 too in the GR context.

So long as you are aware of the limitations that such a convention or 'language' places upon what you are able to say. G = 1 is fine in a strictly GR context but defining it as so would blind you to the possibility that G might vary as in the Brans Dicke theory, likewise defining mass to be invariant.

My intuition is that energy is fundamental and therefore we might take up Feynman's usage and call (rest) mass 'Rest Energy' and relativistic mass 'Total Energy'. This may be a more generalised way of conceiving of the world.

 

[sal]

It's been shown that a rod is easier to accelerate when it is pulled rather than when it is pushed.

It has??

Can you explain that, or perhaps post a reference to a derivation? (BTW I'm happy with references to your website -- unlike DW I have no problem with it it's a good site, IMHO)

Or perhaps I should just ask where the acceleration is measured -- at the front end of the rod, the back end, or the "middle" (for some definition of "middle").

 

[Garth]

The problem with the four-momentum equation P(mu) = mU(mu) is that U defines velocity with respect to proper time tau - dx(mu)/d(tau). But how do we measure proper time? What clocks keep proper time? It is only in the particle's rest frame that its proper time can be measured and therefore the fact that m is the constant rest mass is an observational tautology; it can only be measured in the rest frame. In any other frame of reference the frame dependent time t is measured and the relativistic mass which we may call M or m.gamma according to our convention as discussed above.

 

[DW]

The problem with the four-momentum equation P(mu) = mU(mu) is that U defines velocity with respect to proper time tau - dx(mu)/d(tau). But how do we measure proper time? What clocks keep proper time?

That is not a problem, but is not the definition of four-vector momentum anyway. It is only a result applicable to particles that don't happen to travel at the speed of light.

It is only in the particle's rest frame that its proper time can be measured and therefore the fact that m is the constant rest mass is an observational tautology; it can only be measured in the rest frame.

That simply isn't true. If it were we wouldn't know the mass of any particles, because we never measure it from their rest frames. The dynamics equation of relativity corresponding to Newton's second law is the four-vector equation F^{\lambda } = mA^{\lambda }. It is the m there that is measured in terms of that equation or an equivalent result from it. That m is the only real mass that there is and it is invariant.

In any other frame of reference the frame dependent time t is measured and the relativistic mass which we may call M or m.gamma according to our convention as discussed above.

And relativistic mass is a mistake anyway. If you mean relativistic energy then say relativistic energy because something other than that exact thing called by relativistic mass doesn't even exist in nature at all.

 

[pmb_phy]

It has??

Can you explain that, or perhaps post a reference to a derivation? (BTW I'm happy with references to your website -- unlike DW I have no problem with it it's a good site, IMHO)

Or perhaps I should just ask where the acceleration is measured -- at the front end of the rod, the back end, or the "middle" (for some definition of "middle").

Hi sal

Nice to see you posting here.

I read an article about it in the American Journal of Physics (AJP) several years back. I don't recall the exact reasons but I think it was related to gravitation time dilation. There was a similar article in another journal which I'm trying to get my hands on.


If you'd like I can scan that AJP article in an e-mail it to you?

Pete

 

[sal]

I read an article about it in the American Journal of Physics (AJP) several years back...

If you'd like I can scan that AJP article in an e-mail it to you?

Sure, thanks, I'd like to see it -- off hand I can't imagine how it could work out that way.

This seems like a really pleasant forum. And if I could only figure out how to get threaded message display enabled, I'd be reasonably happy with the user interface, too...

 

[Garth]

That is not a problem, but is not the definition of four-vector momentum anyway. It is only a result applicable to particles that don't happen to travel at the speed of light.

That is obvious of course, what we are talking about in this thread is the definition of mass, i.e. a particle's mass.

That simply isn't true. If it were we wouldn't know the mass of any particles, because we never measure it from their rest frames.

I wasn't talking about the measurement of mass here but proper time, how would you measure it? The measurement of mass is then derived from that, so that is the problem.

And relativistic mass is a mistake anyway. If you mean relativistic energy then say relativistic energy because something other than that exact thing called by relativistic mass doesn't even exist in nature at all.

The phrase "relativistic mass" is used by so many authoritative people that I do not think you can dismiss it that easily, what we are trying to do is ask whether it is a useful concept of not. Of course, as I have said before, we could also use the phrase "Total energy" and use "Rest energy" for "Rest Mass" - or in your convention - mass. If we use the term "Rest energy" we also leave open the question as to whether it is invariant or not under translations and boosts, say within a gravitational field. A postulate that is open to experimental verification/falsification.