Dear Stefano,

I have explained me quite bad. Obviously the rest mass is Lorentz invariant and therefore independent of the state of motion. What is for me dependent of the state of motion is the mass, my analogy is with friction. I don't think that the concept of mass isn't so important, for me gives a clear separation between kinetics and dynamics in Mechanics. And also the conservation of energy and momentum are central for Physics. Lorentz transformations are much less but they help to deduce the spacetime Noether currents, i.e. the conservation laws.

Dear Stefano,

Frankly I don't understand how the concept of mass can be a virus. Mass is a concept related with motion, in fact it is a measure that the bodies has to change its state of motion, thus it seems quite logic its change for different motion observers. What is difficult for my intuition is to keep it invariant for the different observers.

Dear Daniel,

OKUN mentioned "relativistic mass" as being a Virus, and defending the concept of rest mass against that misuse.

What changes and we know it from accelerators is the momentum and energy.

Dear Stefano,

What is really Lorentz covariant is the 4-momentum better than the mass or the 4-velocity. Then I understand that there is a degree of freedom to assign a Lorentz transformation for the mass o the velocity. What is easier to interpret is to put on the 4-velocity the gamma and to forget the velocity transformation. If you do that is not necessary to employ the word rest mass because it is a constant and even the experimental change of mass in a nuclear reaction, for instance, is not easy to be interpreted.

Perhaps I don't understand very well the question and the paper that you recommend.

Dear Daniel,

indeed...not the mass..in circular accelerators the magnetic field of the bending magnets is adapted such as B=K*(mv*1/sqrt(1-v2/c2)), in order to curve the trajectories of the particles.

We cannot attribute the 1/sqrt(1-v2/c2) to the mass, but the momentum is found as mv*1/sqrt(1-v2/c2).

Dear Stefano,

what is more difficult to interpret without assuming a relativistic mass is the 36 equation where there is only energy and mass as possible variables to assign gamma.

Dear Daniel,

I agree !! In this case though it is again the momentum mc is multiplied by gamma and it is variable only according to gamma...

Dear Stefano,

That is only grammar, you are using names. If mc has to be assigned a gamma that only can be done on the m because c is constant and independent of the state of motion.

Dear Daniel,

>

I see, there should be a clash with the constancy of c...

that is a good point!!!

SQ: the momentum is found as mv*1/sqrt(1-v2/c2).

You can write

p=mv*1/sqrt(1-v2/c2)

as

p = m [v/sqrt(1-v2/c2)]

or as

p = [m/sqrt(1-v2/c2)] v

The latter looks like the Newtonian equation but with a speed-dependent mass. That approach allowed science lecturers to teach all the results they were used to simply by assuming mass changed, and the public could understand that "you can't go faster than light because the mass becomes infinite".

It doesn't work so well for energy though. Kinetic energy in Newtonian mechanics was:

KE = 1/2 m v2

In relativity we have

Etotal = m [c2/sqrt(1-v2/c2)]

Erest = mc2

hence

KE = mc2 [ 1/sqrt(1-v2/c2) - 1]

Then there's the whole mess of "transverse mass" versus "longitudinal mass"! This and the following section give some useful history:

https://en.wikipedia.org/wiki/Mass_in_special_relativity#Transverse_and_longitudinal_mass

Understanding mass as the norm of the 4-vector and velocity as a rotation through the angle quantified by the rapidity provides a much more sound basis.

Dear George,

is somebody authorized to make such distinctions other than

mv*1/sqrt(1-v2/c2)

when the measured quantity is neither a mass nor the speed but it is the momentum which is proportional to the magnetic field in the circular accelerator?

These equations apply to any body with mass, we're not just talking about accelerators. They apply to a cricket or tennis ball too. For each, you can measure speed, rest mass and momentum, otherwise Newton could not have noted the conservation law.

Yes George,

but we can measure them is in accelerators only and the only relation available is

B = K* q, hence

B = K* [mv*1/sqrt(1-v2/c2)]

there is nothing which can authorize us to divide it in two parts and attribute the gamma to the mass.

There's nothing to imply it and nothing to prevent it, but using the 4-vector and geometry gives an understanding that works over a much wider range of applicability.

IF you attribute the gamma to the mass you could in principle derive what Yarman and others are claiming!!!

Article On the Abuse and Use of Relativistic Mass

Dear Stefano,

Please see bellow my approach. Mass is constant, the transverse magnetic field that is used to guide the particles decreases for the moving particle with a factor that is inverse of the gamma factor.

Preprint Michelson-Morley experiment, general Doppler's principle, Ma...

Mass is constant, this is why photon can have a finite mass. The mass of a small particle cannot tends to infinity, this is nonsense due to the special theory of relativity.

Voigt's paper about Doppler principle was not presented well. Voigt changed x and t instead of changing k and w in the phase of a wave kx +wt. If we change k and w and repeat Voigt's work we can get the frequency shift for any angle. This is what I did. From this, we can find many results very naturally. Voigt's paper was for frequency shift only, this is why c is constant. When we analyse the propagation the speed of light is relative like any other speed.

Poincare multiplied Voigt's transformation with gamma factor and made a transformation that has no physical meaning (LT).

Lorentz got from LT the transverse relativistic mass and a longitudinal relativistic mass.

Planck's formula for the velocity dependent mass (a unique relativistic mass) is also based on LT with the gamma factor for the mass. Please see Eqs. 1-5 (he multiplied Newton's second law with gamma factor):

https://en.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics

Best regards

Caro Stefano,

The "famous" Okun article has no relevance and does not represent any important contribution.

The SR theory can be well described using the relativistic mass, so its use is more a matter of taste than a requirement.

Mathematicians usually prefer the 4-vectors and proper time approach, avoiding the relativistic mass, because their squares are Lorentz invariant.

τ 2 = t2 − (r/c)2 ,

m2 c4 = E2 − (pc)2 .

However, they do not consider the following disadvantages:

1) Proper time has no physical meaning for radiation (photons), because in SR a reference system fixed to radiation is not admissible.

2) The relativistic mass is not Lorentz invariant, but it is a “parameter of state”. State variables are useful for the study of evolution of any physical system. The importance of state variables can be revealed on the process of creating or annihilating of pairs. Consider a system formed by an electron and a positron that annihilates giving two gamma photons. The energy and the relativistic mass of the system are conserved, while the proper mass disappears. The proper mass is not a state variable. This shows that in dynamic processes the relativistic mass is conceptually more significant than proper mass.

3) In order to avoid the relativistic mass, Okun makes a biased interpretation of Einstein's mass-energy equivalence. The equivalence between mass and energy results a state equation with a single freedom degree if the mass considered is the relativistic one.

Regards

SQ: IF you attribute the gamma to the mass you could in principle derive what Yarman and others are claiming!!!

No, gamma is always > 1 and multiplies rest mass while Tolga seems to be claiming rest mass decreases.

Don't worry though, as I've always said, I think relativistic mass is a very bad idea, it only really occurs in aether theory.

Dear George,

Proper mass could increases or decreases in non-elementary systems.

In systems that are not closed, that is true Hugo, but it isn't the subject of Stefano's question.

The concept of mass in SR is not simple and clear. The question is that there are different forms to associate it to the states of motion (and therefore to the observers). The basic ones are two:

1. Mass as the ratio between linear momentum p to the velocity v

m=p/v

2. Mass as the cocient between a force F to an acceleration a

m=F/a

The first definition seems to be a tautology, over all if think in terms of Mechanics, but it isn't for instance when we introduce a variational definition. For instance having a particle of electric charge within a magnetic field (pc=p+qA), pc is the canonical momentum, p the mechanical and qA the magnetic.

The second definition contains the Newton's second law which is quite different issue and many different subtleties respect to the relativistic observers (or the geometry associated to systems of coordinates). Therefore it is not strange that taking one definition of mass or the other, the results are not the same. This is in summary what happens with this subject and in practice it is avoided without entering in what the mass does. But for me the mass is one dynamical quantity which transforms using the relativistic gamma if the energy and mass want to be properly related.

Caro Stefano,

I knew the paper that you have recently shared (On the Abuse and Use of Relativistic Mass). In my opinion it is worse than Okun's article, because the author’s assertion that "By introducing a concept that is inferred and not primitive, that destroys the Lorentz covariance of the theory, relativistic mass is in direct conflict with the kinematical structure of special relativity" (sic), is a nonsense. Relativistic mass is a definition and definitions cannot destroy anything.

Proper mass is Lorentz invariant, but it is not a state parameter.

Relativistic mass is not Lorentz invariant, but it is a state parameter.

I wonder what the reason will be for not using both.

Best regards

Dear all,

so far at support of the Relativistic mass (RM) there are:

Prof. Baldomir

Prof. Fernandez

not in agreement with the RM concept

Prof. OKUN (unfortunately not any more alive)

Prof. Gary Oas

Dr. Dishman

I think we all agree on the definition of rest mass....which is what is measured in the mass rest frame...

Prof. Baldomir affirms that: "Mass is a concept related with motion,"

I prefer to say that mass is a concept related to work/energy....

Experimental evidences give us information of the extent of the relativistic momentum.

|B| = |K|* [m |v|]*1/sqrt(1-v2/c2)]

speed is mixed with mass to give the relativistic momentum

Is it possible instead to set up an experiment which tests directly the relativistic mass? If no, then there is no way to be able to say that it is something which has an ontology...

LT do not consider the mass at all for example but there is speed as well

t'=gamma*(t-xv/c2) = gamma*t - gamma x*v/c2

x'=gamma*(x-v*t)=gamma*x - gamma v*t

but what comes out is a relation where gamma is linked

to the time which is what occurs in experiments as the ones measuring time dilation....

and to the space (which is supposed to emerge by other experiments for the Lorentz contraction).. gamma is never related directly to speed but always mixed with time or space....

Caro Hugo,

how do you reply to this?

the two viewpoints of kinetic energy lead to a differing ontology of spacetime:

KRM =(1-1/gamma)*mc2 (RELATIVISTIC MASS)

KG = (gamma-1)mc2 (REST MASS ONLY)

for both the first order approximation is

1/2mv2

but at higher orders the result is quite different...

Dear Stefano,

I got the same formula obtained by Planck and which is validated in particle accelerators, except that mass is constant in my approach. I used only a Doppler shift in Maxwell's equations and inverse Fourier transform: this is very natural.

How could the mass of a tiny particle will tend to infinity? This is nonsense.

STR's time dilation is in fact a frequency shift. In the phase of a wave kx+wt, Voigt worked with with changes of x and t instead of working with changes of k and w. His method was justified but not presented well. The multiplication with gamma factor has no physical justification.

Best regards

Dear Halim,

time dilation corresponds to a detected frequency shift of radiation in the transverse Doppler experiment,

time dilation corresponds to a frequency shift of an oscillator (CLOCK) in the twin effect experiment made by Hafele and Keating or in the muon ring experiment.

time dilation corresponds to a frequency shift of oscillators (MASERS) which detect radiation, in the Vessot and Levine experiment...

it is not so simple... there is energy behind...and how energy is handled Conservation laws, mass energy equivalence and the least action principle...

Carissimo amico,

Your puzzle is very difficult to order

KRM =(1-1/gamma)*mc2 (RELATIVISTIC MASS) = (1-1/γ)*mc2

KG = (gamma-1)mc2 (REST MASS ONLY) = (γ-1)* m0 c2

Obviously, KRM= KG

Why do you assert “but at higher orders the result is quite different...”?

Regards

Dear Stefano,

If you follow STR, you cannot avoid relativistic mass. There is no relativistic mass without STR and there is no STR without relativistic mass. Either you accept relativistic mass, either you look for another approach.

Best regards

Dear Hugo,

If m=gamma m0 both expressions coincide for the kinetic energies. And they only give a classical kinetic energy at firts approach term for the gamma. At second approch you have not a coincidence with the usual kinetic energy because you optain an extra term

(3/8) m0 v4 /c2

and so on.

Carissimo Hugo,

Let' consider m0 the rest mass and the relativistic mass m0γ, by admitting that it exists

with this paradigm nobody forbids me to write the following

RKE1 =1/2v2m0γ which at the first order is the classical 1/2v2m0

but with only γm0 c2 , rest mass only, you have to write

RKE2 = Etot-Erest= γm0 c2- m0 c2 =(γ-1)* m0 c2 which also at the first order is 1/2v2m0

both of them give as a first approximation 1/2m0 v2 the classical kinetic energy

but these are not the same at higher orders!!!!

so by admitting the existance of m0γ ,

I get a second version of the kinetic energy of a system which clashes with the first...

Dear Stefano,

If you want to understand what is the relativistic mass (the unique relativistic mass, not longitudinal mass and transverse mass) please see how it was derived the first time:

Planck's formula for the velocity dependent mass (a unique relativistic mass) is based on LT with the gamma factor for the mass. Please see Eqs. 1-5 (he multiplied Newton's second law with gamma factor):

https://en.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics

What is the physical reason to multiply Voigt's transformation with gamma factor? There is no physical reason, this was done the first time by Poincare from a mathematical perspective: in order to form a group. It is not justified physically.

I derived exactly the same formula than Planck, but mass is constant in my approach. This makes more sense than a mass increasing with velocity.

Best regards

Caro Stefano and estimado Daniel,

Stop the world, I want to get off (it's a tango).

You are doing different approximations. Don’t forget that m= γm0

Regards

There are pros and cons. SR is a "theory" which transforms acceleration, and consequently transforms forces. The non-zero forces in two inertial systems are not same. Why then it would be inconceivable to transform mass too? Yes, the Lorentz symmetry is broken--mass is not a scalar, since it transforms, but that's how it just is. We are dealing with non-consistent "theory", so we have non-consistent results.

No, I do not support "relativistic mass increase", because I do not support SR at all.

Caro Hugo,

yes... so you can write KE=1/2γm0v2 which should be the relativistic energy

but this is not in agreement with the well known.... KE =(1-γ)m0c2

although they both at the first order give 1/2m0v2

At the end of the day, one simple question settles the matter for me.

If I hang a bowling ball from a tree on a spring balance, I can stand beside it and read the value of mass. If I walk past it however, the "relativistic mass" is greater, but the balance doesn't change. I'd could assume that gravity has weakened or perhaps something about the spring, but if two people walk past the ball at the same time but at different speeds, the ball has to have two different values of "relativistic mass" at the same time. If mass is a property of the ball, how can it have two different values?

That tells me that the concept of relativistic mass cannot be a simple property of the object, it has to be extrinsic, a value that is dependent on the choice of coordinates or a property of the ball-observer system.

It is far more insightful to realise that "relativistic mass" is just a misleading name for the sum of the mass and the object's kinetic energy.

There is no relativistic mass in my approach. I have found that the transverse magnetic field felt by the particle decreases with velocity. My formula is in agreement with experiments:

Preprint Michelson-Morley experiment, general Doppler's principle, Ma...

In my approach I do not use any postulate. LT cannot give a correct result because the multiplication with the gamma factor was done from a mathematical perspective, not for a physically correct reason.

Dear Halim,

I ve already seen such forumlas somewhere for the RDE..it is quite interesting your derivation

you have to explain me how do you find the 2/3 in the Mossbauer rotor experiments though...

Dear George,

The different observers see different mass, length, time...but all of them obtain the same value when take a rest system for measuring it. The concept that mass is a measure of matter contained in one object is something related with its association to a conserved accelerated motion as it is gravitation, but in fact what really measures is the oposition to change the state of motion: nothing more. Therefore it must be related with the motion! What is the problem?

Dear Stefano,

RDE is not correct. It is correct only for transverse Doppler effect (theta=90degree) for moving source. We cannot obtain correct results by using LT because LT was made from Voigt's transformations with a multiplication by gamma. The multiplication with gamma was done from a mathematical perspective (to form a group), it is not justified physically. Voigt's work was correct but not presented well, he should have modified k and w instead of x and t.

I am willing to explain my result of 2/3 but I would need first that you follow my approach until the formula for particle accelerators. If you follow my approach, the section of Mossbauer will be simple for me to explain.

I derived the correct formula for DE for any angle. It is different from RDE.

I did not use any postulate and do not have any inconsistent concept.

Mass of a tiny particle cannot increase to infinity. This is nonsense.

My approach allows the photon to have a mass.

Best regards

Dear all,

again... if the relativistic mass γm0 is accepted as a concept then KER=1/2γm0v2 is the relativistic energy

on the other side it is well known... KE =(1-γ)m0c2 to be the relativistic energy...

since KER KE , KER cannot be the relativisitc energy of a body in SR, hence the replacement of m with γm0

should be a mistake.

Dear all,

I am sorry, this post will be dedicated to several people. I will try to be concise.

Caro Stefano and estimado Daniel,

Two identical functions cannot have different approximation, whatever be the order. You both are taking a different approximation on each function. I suggest do not continue this simple subject.

Dear Halim,

You wrote: “…,you cannot avoid relativistic mass. There is no relativistic mass without STR and there is no STR without relativistic mass”.

This is not true. Most experts in relativity don't use the relativistic mass. Besides, relativistic mass is a definition, so it can be ignored.

With reference to Max Planck ad-hoc treatment on Newton’s law, I think that it is not recommendable. The relativistic mass is consequence of the space homogeneity, which assures the conservation of momentum. So, if you preserve the momentum definition (p=mv), you must incorporate the relativistic mass as a definition.

Dear Vladimir

. It seems to me that relativity is not your research topic. Allow me to add this: I do not support my mother-in-law, but she exists, as well as SR theory.

Best regards

Dear George,

You wrote: “., but if two people walk past the ball at the same time but at different speeds, the ball has to have two different values of "relativistic mass" at the same time. If mass is a property of the ball, how can it have two different values?

Exactly, that is what happens. Both inertial observers measure different relativistic mass, and different energy, but both conclude that E=mc2, being m the relativistic mass.

Regards

Caro (no so much) Stefano

Wrong, you are mixing Newton with Einstein.

Regards

Hugo Alberto Fernández

I do not support my mother-in-law...

Is it because her mass increases relativistically?

Dear Preston,

yes sure KE =(γ-1)mc2 ,

and there exists only m, there is no room for m0γ otherwise from

1/2mv2 one could write 1/2m0γv2 which clashes with the first...

Caro Hugo

Wrong, you are mixing Newton with Einstein.

Regards>>

if you accept , as Feynman said, that it is sufficient for dynamics to replace m with γm0 to get relativistic dynamics

nothing should prevent you to write that...

I know perfectly that is a concoction of classical dynamics with electrodynamics, but it is what occurs if one is coherent with the expression m=γm0 ....

On the same footing if that is a concontion, the momentum γm0v , should not be considered v*(γm0) but you have to take it as a whole, (mv)γ, since what is experimentally verified is that |B|=k (m|v|)γ

otherwise I see inevitable to fall into the mistake of considering this quantity 1/2γm0v2

When I was at university in the 1970s, some of my lecturers were recent graduates but some had been there for decades and were into their 60's or older so probably born not long after SR was first published. A long time ago, when I was just learning this stuff, a much older chap said that he had a similar experience and that when learning mechanics from one old professor, all the lecture notes he put up on a projector were purely Newtonian with the exception that "m" had clearly had a suffix "r" to indicate "relativistic mass" added long after they were first written. That meant the old lecturer didn't need to learn anything new and could just keep giving the same explanation he had done for decades before.

I think that was the original reason for using the approach, of course you don't learn anything about SR that way, but the equations you get are adequate for many purposes.

that is a possible explanation of the wrong behaviour...which sometimes bring to the wrong quantitiative conclusions not only wrong interpretations.. as 1/2γm0v2

Dear Stefano,

I don't understand your difficulties with the kinetic energy interpretations. The expression that you give for the kinetic energy

KE =(γ-1)m0c2=γm0c2-m0c2=(m-m0)c2 (1)

and at low velocities you can approach

m-m0=1/2m0 (v/c)2

The expression (1) only has an additional term due to have always one rest energy which you must substract and which you can obtain as one of the extremes of the definite integral for finding the kinetic energy. There are not problems at all for working with this mass instead of keeping it constant.

Alternatively you can associate only the gammas to the velocities and you can think that relativity is only "geometrical illution" associated to the state of motion without changing the Physics. This for me is a bad choice because there is a physical limit: the velocity of the light never can be reached by one observer and therefore never can be obtained its kinetic energy too. This degrees of freedom related with the choice on the velocities or the mass can be avoided going to work directly on the momenta, but at the end the judge should be the experiment, which is not so easy because both physical quantities are involved simultaneously on the state of motion.

Apreciado Hugo,

You are right, the expression KER=1/2γm0v2 is wrong because at low velocities you don't obtain the usual kinetic energy. Actually you obtain it but with one extra term due to the rest energy with all the bodies have at zero velocity and which must be added for following the relativistic definition of 4-momentum.

Caro Stefano,

< if you accept , as Feynman said, that it is sufficient for dynamics to replace m with γm0 to get relativistic dynamics >

Did Feynman write such horrible sentence?

Unfortunately, here (vacation at the beach) I do not have access to Feynman's books. If so, I suppose that he added that such replacement is to be only applied on the fundamental law of Mechanics, and that it must be F=dp/dt instead F=ma. Otherwise, you will have serious conflicts, as yours (oops).

Hugo Alberto Fernández

Feynman used relativistic momentum and argued that mass of a body is increasing with speed.

Dear Hugo,

You said:

"This is not true. Most experts in relativity don't use the relativistic mass. Besides, relativistic mass is a definition, so it can be ignored.

With reference to Max Planck ad-hoc treatment on Newton’s law, I think that it is not recommendable. The relativistic mass is consequence of the space homogeneity, which assures the conservation of momentum. So, if you preserve the momentum definition (p=mv), you must incorporate the relativistic mass as a definition."

The mass of a particle that tends to infinity when v is close to c does not make any sense in physics.

The momentum of of a particle that tends to infinity does not make any sense neither.

I derived equations that agree with experiments of particle accelerators. I didn't do it "ad-hoc". In my approach, I have found that the transverse magnetic field felt by the particle is not the same magnetic field that we feel in accelerator frame. In the frame of the particle, the magnetic field felt by the particle decreases with velocity. In my approach, Lorentz magnetic force is not linear anymore:

Preprint Michelson-Morley experiment, general Doppler's principle, Ma...

Best regards

Dear Halim,

Common sense is the least common of all senses.

We are physicists using models which try to explain nature as better as possible. In my opinion, the SR is one of them that works excellent and without verifiable contradictions. It is possible that some aspects must be reworked (for example: twin paradox), but not what you mention about mass and momentum.

On the other hand, the GR showed serious problems and difficulties. I think it is at rest waiting for the successor.

Please Halim, I would like to know the approach you refer (your derived equations). Is it possible?

Regards

Dear Hugo,

my approach is explained in my paper I have cited above. The formulas are in the section about particle accelerators.

I obtained these formulas without using a postulate. I only used an approach based on wave theory. When Poincare decided to multiply Voigt's transformation with gamma factor, he did it from a mathematical point of view (to form a group). There is no physical reason to do that. This is why LT cannot give results with the correct physical insight. The correct equation is not enough. We need the correct physical explanation.

Best regards

HB: When Poincare decided to multiply Voigt's transformation with gamma factor, he did it from a mathematical point of view (to form a group). There is no physical reason to do that.

The physical reason is illustrated by the attached graphic, sometimes known as the "Rod and Tube Paradox". At rest, the outer diameter of the rod is the same as the inner diameter of the tube. If the tube is held at rest and the rod moved towards it, it is length contracted and by Voigt's Transforms its outer diameter should also shrink so the rod passes through the tube with a small clearance. From the point of view of an observer for whom the rod is at rest, the tube is moving, it is length contracted and by Voigt's Transforms its inner diameter should also shrink which means the hole is smaller than the rod which can therefore not pass through.

Symmetry requires that there can be no transverse change, hence the Lorentz Transforms follow.

Caro Stefano,

I would like deeper your question. It is so obvious that a definition can not cause any harm, that one wonders what the real reason is for rejecting the relativistic mass definition, using silly arguments or even "scientific" publications with elementary errors.

Note that to eliminate the relativistic mass it was necessary to redefine magnitudes such as momentum (p = mv) and invent nonsense with the equivalence between mass and energy (E = mc2).

SR theory admits several different treatments. One is with 4-vector, of mathematical "flavor", and another one (3+1-space-time), of taste of physicists.

In my opinion the mathematical formalism (tensors) of GR imposed the 4-space-time description over the 3+1-space-time description, although the latter is superior.

What do you think?

Dear George,

what you show is not a physical reason, it is a graphical representation of this hypothetical and non physical effect. Poincare did it to form a group, this is a very well known fact.

It is better that we return to real science. The momentum of a small particle cannot tend to infinity, this is nonsense.

We learn from Nature, we cannot make Nature to follow our thoughts.

Best regards

Dear Stefano,

I remember that you asked several times that an alternative should be presented to replace LT, if you want to abandon LT.

I have found the same formulas used in particle accelerators only by using wave theory. I don't have the problem of relativistic mass, and the momentum of the particle doesn't tend to infinity in my approach, what would stop you to check my proof?:

Preprint Michelson-Morley experiment, general Doppler's principle, Ma...

You are not forced to follow my approach, but my method resolve the question posed in this thread.

Best regards,

Halim

Dear Halim,

HB: what you show is not a physical reason, it is a graphical representation of this hypothetical and non physical effect.

It is a graphical representation of a thought experiment which would be difficult to do in practice but is valid in principle. What is shows is the physical reason why the Voigt Transforms are ruled out.

When Voigt wrote them, he was applying them only to EM and particularly plane waves so a scaling in the transverse direction was not a problem. Using them more generally to also apply to matter with well defined size does show up this problem. This is a genuine contradiction, not a paradox that can be resolved.

HB: The momentum of a small particle cannot tend to infinity, this is nonsense.

It can "tend to", it cannot reach infinity. That is why massive particles can never reach the speed of light, they can only "tend to" it. Thi is of course what is observed.

HB: We learn from Nature, we cannot make Nature to follow our thoughts.

Exactly.

Dear George,

I think that you have experience with waves. The phase of a wave is written phi=kx+wt. If we consider a medium with refractive index n we write phi=nkx+wt or phi=k'x+wt. With k'=nk.

Voigt's paper about Doppler's principle was well done but unfortunately he changed x and t instead of changing k and w. The multiplication with gamma factor was done after.

If we repeat Voigt's work by changing k and w, we have a very consistent method to derive many formulas such as the formulas used in particle accelerators.

This would not take much of your time to check what I am writting.

Best regards

Dear Halim,

It is well known the Voigt contribution which was devoted to the Doppler effect. It is true that he was the first to realise that Galilean transformations where not good for keeping Maxwell equations invariants and in fact the light velocity. But unfortunatelly he wrote wrongly their transformations. He has written the Lorentz transformation divided by the gamma, which makes depend the transverse component of the spatial components of gamma instead the time and the longitudinal one. This is very well explained by the figures represented by George and that is well explained. One of the places that you can see it very well is in the electromagnetic radiation associated to the Lienard-Wiechert potentials. On the other hand Voigt transformations are not a group because the product of two Voigt transformations do not need to be one Voigt transformation. This is not a mathematical detail, it has enormous high physical implications and nowadays nobody is going to accept these transformations as belonging to Special Relativity. Less if you try to substitute Lorentz's ones by them just considering only the velocity of the light transformation which is obviously the same.

Dear Daniel,

Stefano's concern about relativistic mass is justified. When we consider the momemtum, the problem is still there. According to the equations about mass and momemtum derived from LT, the relativistic mass of a small particle will be larger than the mass of the earth, the same for the momemtum.

I obtained the correct equations verified in particle accelerators without this problem of relativistic mass and extremely large momemtum. I didn't need a postulate. I only used wave theory.

Concerning Lienard-Wiechert potentials, I am presently analysing Heaviside analysis of moving charge. It does not make sense that the field should tend to infinity if the charge moves at speed close to c. Something must be missing in the analysis. I am trying to find a solution of the problem. We all know that electric current is finite, and doesn't have to be very large, whereas the electron moves at speed close to c.

I have experience with the analysis of a Fabry Perot cavity excited from its inside by electromagnetic waves. I have found that the calculated transmission coefficient is greater than one and can tend to infinity. To resolve this unphysical behaviour, I have found that something was missing in the analysis. By adding the missing part, I resolved the problem and got result that agrees with common sense physics.

Nature is not strange, our common senses help us to make a difference between a theory that sounds correct and a theory that sounds unphysical.

This is my paper about Fabry-Perot cavity excited from its inside by electromagnetic waves:

https://www.researchgate.net/publication/242077693_Multi-Layer_Crystals_of_Metallic_Wires_Analysis_of_the_Transmission_Coefficient_for_Outside_and_Inside_Excitation

I resolved the issue by considering the interaction between the waves and the source.

I think that the same problem is in Heaviside moving charge. I am presently working on it.

Best regards

Dear Halim,

Allow me to ask you some things that arise from my first reading of your article "Michelson-Morley Experiment,...".

1 - Why do not you clearly indicate at the beginning which is the theoretical framework where you carry out your treatment?

Is it the absolute space of Galileo-Newton?

2 - In your formula 36 you write:

If it is Galileo's transformation, you are accepting infinite velocities, which does not make sense. Could you clarify the point?

3 - In your formula 106 you write:

Are you matching the tangential force with the normal magnetic force?

If so, I suggest you withdraw or correct the article.

Regards

Dear Hugo,

I will respond to your questions with pleasure:

1- The framework is the wave theory. In the wave theory, a wave does not change the lengths of objects and a wave does not change time. I do not need the concepts of SR. I have read Galileo's work, which I like very much, and Galileo never talked about any transformation. So I don't use terms such as "Galielan transformation", because it doesn't make sense for me. The definition of velocity is very clear and does not need to be called a transformation.

2- The particle cannot exceed c in my approach because the transverse magnetic field felt by the particle tends to zero when v tends to c.

3- You know that Lorentz magnetic force is obtained by doing the cross product of v with B. Actually the formula that I obtained is exactly the formula used in particle accelerators and the formula obtained with SR (please you can check), except that I obtained the correct physical insight. In my approach, there is no relativistic mass, the mass or the momemtum do not tend to infinity (this is nonsense a momentum that tend to infinity).

Best regards

Dear Halim,

HB: I think that you have experience with waves.

I do but wave analysis doesn't show the problem in the Voigt Transforms as easily as the paradox I illustrated.

HB: This would not take much of your time to check what I am writing.

Sure, post the transforms you propose and let's see if they work when applied to the rod and the experiment.

Regards

George

Dear George,

Please see here my approach which is based on wave theory:

Preprint Michelson-Morley experiment, general Doppler's principle, Ma...

In wave theory, a wave does not change the lengths of objects and a wave does not change time.

My approach is confirmed with experimental results.

Best regards

Halim,

You have not been able to answer simple explanations.

Your pseudo article is not serious, is not well founded and has preconceptions of people without training. You should reflect that your behavior is a lack of respect towards all those who are sincerely interested in science.

It is not my intention to waste a second more of my time with your article.

Regards

Dear Hugo,

I am sorry that you believe so. I didn't force you on anything. My work is very serious and scientifically consistent. As you can check, I publish regularly in serious journals, I am regurlarly invited for presentations, I am a reviewer for many journals and I have been involved in steering comittees of many conferences.

I started to work on the subject presented in this paper in 2016, and I consider that I have resolved the main issue related to this problem. However, there are many work to do now in order to re-analyze several phenemena such as blackbody radiation and Compton effect.

I repeat that I don't force anyone, I talk with you and others scientists with respect. I will always use respect.

Best regards

Dear Halim,

HB: This would not take much of your time to check what I am writing.

GD: Sure, post the transforms you propose and let's see if they work when applied to the rod and the experiment.

HB: Please see here my approach which is based on wave theory:

I looked through the paper but I couldn't see any transforms that can be applied to the rod and tube thought experiment, please just post them here, it's only four lines.

George

Dear George,

Let's stop our discussion respectfully. My intend here was to present my recent work related to the question posed by Stefano. There is a wall of misunderstanding between us, because you work in the framework of SR and I work in the framework of wave theory. Like what happened with Hugo, you try to impose your framework and you are not intersted about a different approach.

I respect experience and knowledge of all scientists.

I have given the reference to my appoach for anyone who will follow this thread. Anyone can judge by himself.

I will now focus on other work and I will stop participating to this thread.

Best regards

Stefano> could you please have a look at this thread?

I think the concept of relativistic mass is unnecessary, and may easily lead to confusion. A relevant example is your use (in another thread) of the formula ½mγv2 for relativistic kinetic energy. The correct expression is (γ-1)mc2. As you can easily see, the two expressions differ by a factor 2 when γ becomes very large (which means that v ≈ c). They start to differ at the first relativistic correction.

I am not sure what exactly is meant by relativistic covariance is broken, and how important or dangerous that is. It is certainly possible to perform correct calculations with the use of formulas involving relativistic mass, if one knows what is going on. But I think that manifestly covariant formulations are more elegant and simpler to perform correctly. But such can profitably be supplemented with interpretations in specific coordinate systems, for improved physical insight.

Note however that the kinetic energy of individual species can contribute to the total ("invariant") mass of a composite system, as in the formula

(Mc2)2 = (E1 + E2)2 - (p1+ p2)2 c2,

which explains why one builds colliding beam experiments to produce very massive particles. But in this case, and similar ones, the additional mass cannot be attributed to the individual species in an unambiguous way; it is a collective effect, as can be seen from the formula

M2c4 - m12c4 - m22c4 = 2(E1E2 - p1·p2 c2)

In Newtonian mechanics, the concept of mass is used in several ways. 1. As inertial mass used in the 2nd law, which (urk!) must be replaced by a relativistic mass matrix in relativistic dynamics, if one insists on keeping the 2nd law in its (non-covariant) form. 2. As the proportionality constant relating momentum and velocity, if one insists on keeping this relation in its (non-covariant) form. 3. As the gravitational mass.

Further, 4. with the introduction of the Einstein formula, E = mc2, as an expression for energy.

It is not correct to adapt non-relativistic dynamics to relativistic speeds by just re-interpreting m as "relativistic mass".

Finally, I am not a fan of using the γ-factor at all. A more relevant relation is the formula v = pc2/E, which remain correct for massless particles, if one wants to relate 3-velocity to the covariant concept of 4-momentum pμ = (E/c, p). And, for Lorentz transformations, the parameter η (analogous to rotation angle) such that cosh η = γ.

Dear Halim,

HB: There is a wall of misunderstanding between us, because you work in the framework of SR and I work in the framework of wave theory.

It's not that I'm imposing anything, I just don't see how you would analyse the problem I posed using a wave approach. Turning your approach into the transforms necessary to convert between coordinate systems is neutral way to bridge the gap.

Anyway, I agree that further discussion probably would not make any more progress, I'll leave it with you to resolve what your approach would predict in this case.

Dear Kare,

What you have made is to show how a system of two particles 1 and 2 can be reduced to another in the center of mass (M and P). And you can even generalize it for a higher number of particles, but I don't know what is the relation with the question of relativistic mass that Stefano has made.

Dear Kare,

yes infact, I understand that the correct relation is (γ-1)mc2 at the same time I cannot see how this can be in agreement with

m=m0γ if coherently the direct consequence of the relativistic mass would be ½mγv2 .

Caro Hugo,

Note that to eliminate the relativistic mass it was necessary to redefine magnitudes such as momentum (p = mv) and invent nonsense with the equivalence between mass and energy (E = mc2).>>

The momentum is p=mvγ mainly because it has been verified in several experiments including a day by day experience in the strength of the magnets in accelerators.

The energy is E=mc2 not because of the relativistic mass..

I think that the space-time does not exist at all, does not have an ontology , what I say just follows from the result of the experiments...

The time mixed with space is a convenient tool of calculation which suites very well certain conditions when symmetries are involved but has limitations which I'm currently checking out carefully through performed experiments.

3+1 is certainly better than 4D, from a Physical point of view...but as in Thermodynamics, time should not be involved at all in Dynamics which should describe the evolution of systems and only the comparison of other evolutions give an idea of the time taken...but what exist is the evolution of processes...

The Lorentz Transformations give good results if used within their postulates.. Nature does not follow a "group of transformations", there is no symmetry except if I make a problem symmetric/isolated by building it up with particles.

In accelerators for example the configurations are analyzed in isolation, during the scattering the system is isolated, so that everything is easily expressed in the center of mass of the colliding particles. In this case the transformations work very well.

Stefano> I cannot see how this can be in agreement with m=m0γ if coherently the direct consequence of the relativistic mass would be ½mγv2

What makes you think that the last expression is a direct consequence of relativistic mass? And with which expression for relativistic mass?

Hi Kåre,

I like your presence in this thread. In general, I agree with your initial intervention, except for some aspects that I will discuss.

As I said before, the relativistic mass is a definition, so it can not break anything (... relativistic covariance is broken, is a mistake).

You wrote: .

Elegance and simplicity are subjective notions that have not relation with correct performance.

Elegance

I recognize that 4-vectors formalism using proper time (Tau) is more elegant than 3-vector + 1-scalar (t) approach, but it is not general. Proper time has not physical meaning for radiation and massless particles. By example, the 4-vectors velocity, acceleration and force, are not defined for photons, and its values are magically incorporated to perform correctly. Rigorously, proper time limits this formulation. The same occurs in GR, which is probably connected with radiation difficulties in such theory.

There is another very important reason for which I prefer the 3+1-space-time treatment. The spatial coordinates (x,y,z) and the temporal coordinate (t) have different mathematical domains, because physical time only has nonzero positive increments (arrow of time). This is closely related to causality

Simplicity

So far it seems that the used formalism has nothing to do with the use (or not) of the relativistic mass. The following will clarify the subject.

Allow me to discuss your example (two particles, no radiation):

4- vector approach. The main relation is: E2 = m2 c4 + p2 c2 (without relativistic mass). It is obtained from 4-momentum:

PaPa = E2/c2 - p2 = m2 c2

In our case

(E1 + E2)2 = (M c2)2+(p1 + p2)2 c2

(M c2)2 = (E1 + E2)2 - (p1 + p2)2 c2,

Operating:

M2c4 - m12c4 - m22c4 = 2(E1E2 - p1·p2 c2)

3+1 spacetime approach. The main relation is E=mr c2 (mr = relativistic mass)

E1 + E2 = Mr c2 = m1r c2 + m2r c2

Which approach is simpler?

Of course, they are correct results, but the latter calculation is simpler because the relativistic mass is a property of the system (it is a state variable).

The proper mass is a Lorentz invariant, but it is not a state variable.

The relativistic mass is not Lorentz invariant, but it is a state variable.

Why do not use both?

Besides, there are other reasons to use the relativistic mass.

Regards

Dear Kare,

what prevents me to replace m with m0γ in ½mv2 and obtain ½mγv2 ?

I understand that ½mγv2 is not the right formula. Although the consequence of admitting m0γ as a replacement of m is ½m0γv2 . It is not reasonable that sometimes I can replace m0γ and sometimes I cannot....

it means that the mass cannot be "directly" associated with the gamma factor, but it is the momentum or the rest energy which can be, which is perfectly reasonable.

Dear Stefano,

I obtain the kinetic energy expression below and sorry for my bad processor

Ec= ∫vp*dv= ∫v(γ m v)**dv=( γ-1) m v

The first arterisk on p means its time derivative, while the two asterks mean derivative with respect to the velocity v. The last integral have the extrems in v=0 and v. I think that it is straightforwardly obtained and without any problem for substituting in different form the mass m.

Dear Daniel,

do you mean this?

KE= ∫v (dp/dt) dv= ∫v [d(γ m v)/dv] dv=( γ-1) mc2

Of course, mass is the only. Rest mass.

Dear Stefano,

Your expression is using c as the limit velocity, i.e. the kinetic energy change between a zero velocity and the highest c. But for a given body which is going to reach v you can write my expression. Notice that in fact c is not possible for a particle with rest mass m different than zero.

The idea of the this definition of kinetic energy that explained to my students was very intuitive. If you think in one body at rest and you transform the energy kinetic to another moving at v relative it is immediate to see that the system has an extra energy mc2 which must be substracted..

Dear Daniel,

I see that in term of dimension there is something which does not match...

ENERGY on one side and momentum on the other,

you wanted to mean this Ec= ( γ-1) m v2

Stefano Quattrini :

If one starts from Newton's 2nd law in the form dp/dt = F, and combines it with the energy expression dE = F · ds =F · v dt, one finds

ΔE = ∫ v · (dp/dt) dt

where the integral runs from (say) t0 to t1. For one-dimensional motion one may remove the vector notation, and change integration variable from t to v. This gives the energy expression

ΔE = ∫ v (dp/dv) dv

where the integral now runs from v0 to v1. When p = mγv = mv/(1-v2/c2)1/2 a slightly cumbersome* evaluation of this integral indeed gives

ΔE = (γ1 - γ0)mc2,

as you asked Daniel (for the case that v0=0 or γ0=1).

*) To evaluate the integral one may rewrite the integrand as

v (dp/dv) = d(vp)dv - p = d[vp + mc2(1-v2/c2)1/2]/dv

= d[(mv2 + mc2 - mv2)/(1-v2/c2)1/2]/dv = mc2 dγ/dv,

where the last expression is trivial to integrate. In the above m always denotes the invariant mass (rest mass), which is the only mass I want to recognise.

Stefano> It is not reasonable that sometimes I can replace m0γ and sometimes I cannot....

That is a fact of nature; and it is no help blaming nature for being unreasonable. But this exactly was my main argument for never introducing the concept of relativistic mass. It is bound to lead to confusion and mistakes.

Daniel> The idea of the this definition of kinetic energy that explained to my students was very intuitive.

For proper education of the students at University of Santiago de Compostela, I sincerely hope you bring along a more correct set of notes when you lecture this topic to them.

Dear Kare,

thanks for the clarification ΔE= ∫ v · (dp/dt) dt = ∫ v (dp/dv) dv

in the one dimensional form, it is the integral of the "power" in time in the interval where the power acts.

By using the relativistic momentum p = mγv one finds, after integration

ΔKE=( γ-1) mc2 which is the variation of KINETIC since there is the squared speed in gamma in reference to an IRF.

>

it means that it is not correct to do it...simple as that, it is not a part of a general procedure. In other words m=m0γ it is not a valid substitution, since there is not a general criterion or a principle which discriminates its application.

Since it is not possible in general to use *m0γ* instead of m to enlarge the scope of an expression, hence m=m0γ is something with no general purpose hence meaning.

Hugo> 4-vectors formalism using proper time (Tau) is more elegant than 3-vector + 1-scalar (t) approach, but it is not general.

For massless particles one may still use an affine parameter instead of eigentime τ. I don't see any problems with that; maybe a little extra work at the end, to eliminate the affine parameter in favour of observable quantities only. On the other hand the dynamics of massless particles can have extra (conformal) symmetries, which simplifies calculations.

On may also start with the dynamics of particles with mass m, with fixed 4-momenta as initial conditions, and consider the limiting behaviour as m -> 0.

Regardless of formalism, the restricted dynamics possible for massless particles must be taken into account.

Hugo> because physical time only has nonzero positive increments (arrow of time). This is closely related to causality

With sensible dynamical models, this ought to come out automatically. If not, some deeper study is required. As in the case of potential scattering which allows a trajectory moving forward in time to suddenly turn backward in time. As discussed by Finn Ravndal and Alex Hansen in their beautiful work on the Klein paradox,

http://folk.uio.no/finnr/talks/Klein-Paradox.pdf Article Klein's Paradox and Its Resolution

Hugo> Simplicity

The case you (and I) considered do not really illustrate the enhanced simplicity of a manifest invariant approach. For that one should f.i. consider scattering processes. These are simplest to analyse in the centre-of-mass (or rather centre-of-energy) system, as you also indicate. But what if you must prepare some predictions for a laboratory system experiment, or (horrors) an experiment with two colliding beams of different energy, which are not even collinear? By starting with a formulation in terms of Mandelstam invariants, (s , t, u), which can first be related to calculations in the centre-of-mass system, and afterwards to any other system, one may reasonable simple untangle some of the seemingly insolvable kinematic relations in the other systems.

Relativistic mass, apart a factor of c, is just the 0'th component of the 4-momentum vector. No need to introduce it once more.

Dear Stefano,

I have made the calculation too quickly and I have made mistakes. Sorry. The calculations made by Kare are good but I think that even they can be made easier just solving an immediate integral. There are nothing against the relativistic mass which from my humble point of view is a necessary concept as I told you for the good transformation of the time component of the 4-momentum.

Dear Kare,

Don't be worry I don't need to explain them the motion of two reletivistic particles for explaining the relativistic mass. But in any case I recognice my mistake calculating one integral that just accepted a result.

After my error made in my calculation (really not calculation but only remembering things) I have repeated it

F=d/dt(p)= d/dt(ɣmv)= ɣ3ma

Which is the Newton’s law for SR. Therefore the kinetic energy KE can be found by

KE=∫ ɣ3ma dx=∫ ɣ3mvdv

And we do a change of variable y=1-(v/c)2, for obtaining

KE= -(1/2)mc2∫dy y-3/2=(1/2)mc2 (-2y-1/2)

Substituting the variable y and considering the extremes of the integral between v=0 to v. We have

KE=mc2(1/(1-v2/c2) -1)= ɣmc2-mc2= (ɣ-1)mc2

Which in first approximation is the classical kinetic energy given v

Dear Daniel,

thanks for your derivation which starts from the relativistic momentum as well ɣmv .

In the one dimensional case it is KE=∫ ɣ3ma dx=∫ ɣ3mv dv, hence by integration between 0 to v we obtain

(ɣ-1)mc2 . Yours is faster to derive than Kare's version which is elegant as well though.

While the momentum is as |B|=kɣm|v| and is verified with the strength of the magnets to bend the trajectory in ring accelerators

nothing similar can be said about the relativistic mass ɣm:

a) not directly detectable

b) leads to the wrong formula of the kinetic energy

c) in the LT the gamma factor is coupled either to time ɣt' or to space alone ɣx' and to speed only if with space ɣ(vt) or time ɣ(vx)/c2

we should follow the definition of rest energy E0=mc2 and never consider the rest mass alone.

• relativistic momentum mvɣ=pɣ
• rest energy E0=mc2
• total energy E=ɣE0
• hence the kinetic energy KE= E-E0

in addition the "relativistic mass" is in open disagreement with General Relativity as you can see here

Article Non-linear energy conservation theorem in the framework of S...

Kåre wrote: < For massless particles one may still use an affine parameter instead of eigentime τ. I don't see any problems with that; maybe a little extra work at the end, to eliminate the affine parameter in favor of observable quantities only.>

I agree with your claim, which also matches with what I said earlier about the 4-vector formulation using proper time (., the 4-vectors velocity, acceleration and force, are not defined for photons, and its values are magically incorporated to perform correctly).

There is no mathematical way to obtain the 4-velocity of massless particles. It must be ad-hoc introduced. This means that you must add another condition (postulate) on null-geodesic, which may be a (particular) affine parametrization or impose perpendicularity between acceleration and velocity for massless particles. Without this condition such approach is not general. The reason is that physical time must always have positive increments (dt > 0), fact that is not contemplated in 4-vector approach, since the 4 coordinates are mathematically equivalent.

Regards

Dear Hugo,

>

yes and I think this is one of the limits of the representation.. I don't think though the 3+1 provides such feature. We have to check deeper in the action principle...

Dear Hugo,

Take care when you choose wrong sentences:

For massless particles one may still use an affine parameter instead of eigentime τ.

Time is never an operator and cannot have something as an "eigentime". That is a very wrong concept.

Dear Stefano,

I have not enough time for responding you this morning and I don't want to do it as in my mistaken previous post. But let me to say that what you have concluded

b) leads to the wrong formula of the kinetic energy

is not true and I think that my form of obtaining the KE proves it clearly. The extra term mc2 which could be strange for the kinetic energy is just what is obtained when v=0 and that is for me quite logic. While in classical mechanics the energy of a particle at rest is considered zero (or the energy associated to its potential), in relativity it doesn't happen that. This is very simple and quite straightforward when one has to teach subjects related with these issues. No mistery and no problems with the energies!!!

I'll answer you to the other points in another moment and let me to say that I agree with you and Hugo on the non consideration of non massive particles in the paper preseted by you. But there are also more subtleties to remark.

Daniel> Time is never an operator and cannot have something as an "eigentime".

You misinterpret the meaning of eigentime, which is a synonym for proper time. Both meaning the time ticked out by a travellers own (imagined) perfect clock. I checked the web that I am not the only one to use the word eigentime for that concept. Interestingly, both proper values and eigenvalues have also been used for what is now most commonly denoted eigenvalues of matrices or linear operators.

Daniel> Nature works in one strange way that we need to accept, as Kare said

Can you please point me to exactly where I have said that??? I think I just demonstrated* the relativistic relation for kinetic energy, KE = mc2(γ-1) in a quite straightforward mathematical way; it is not strange that it differs from the ad hoc suggested expression ½ γmv2.

*) By the way, your last derivation, using a · ds = v · dv = ½ d(v · v) = ½ d(v2), is better that mine, because it is completely general (not restricted to motion along a straight line).

[Remark to other readers: If anyone don't immediately see the equality a · ds = v · dv, note that ds = v dt and dv = a dt, so the equality only says that (a · v) dt = (v · a) dt.]

Dear Daniel,

I think you misunderstood me. I said that your derivation is correctly based on the relativistic momentum and it is also general as Kare said.

F=d/dt(p)= d/dt(ɣmv)= ɣ3ma which correctly gives in the general vector form

KE=∫ ɣ3ma dx=∫ ɣ3mv dv = E0(ɣ -1) where E0=mc2

but does not make explicit use of the relativistic mass ɣm....

which in the case of substitution in 1/2mv2 would give something which does not exist.

Dear Kare,

Let me then to copy the whole sentence used by Hugo:

< For massless particles one may still use an affine parameter instead of eigentime τ. I don't see any problems with that; maybe a little extra work at the end, to eliminate the affine parameter in favor of observable quantities only.>

He refers to massless particles where the proper time is even not defined. Less if it associated to an operator as frankly thought that eigentime refered. In any case something seems to be wrong in such sentence.

Respect to the sentence

Daniel> Nature works in one strange way that we need to accept, as Kare said

It was my interpretation that you have finished your post for Stefano

Stefano> It is not reasonable that sometimes I can replace m0γ and sometimes I cannot....

That is a fact of nature; and it is no help blaming nature for being unreasonable. But this exactly was my main argument for never introducing the concept of relativistic mass. It is bound to lead to confusion and mistakes.

In any case it doesn't matter, perhaps it was my bad interpretation of your words. Sorry if I didn't do it properly, but I thought that you said that something was necessary to be accepted besides the pure mathematical calculation for finding the relativistic kinetic energy.

I don't agree with your comments, over all when you used them for remembering me my University, but clearly we don't need to share the same kind of politeness. his doesn't prevent to discuss on Physics in a forum as RG and don't doubt that I not going to use the same kind of arguments for my discussions.

Every good wishes,

Daniel