Relativistic mass is a velocity dependent mass, and if we use it in Newton's second law of motion in relation to a given force, it means that as the velocity increases due to acceleration, the mass will increase too, and so the acceleration will decrease, and so the rate of increase of velocity will decrease. The velocity will then tend towards the speed of light, but never actually reach the speed of light.
How did Einstein derive the relativistic mass formula? Where can we see the derivation? It doesn't appear to be in either the June 1905 paper or the September 1905 addendum.
Mass variable with speed predates relativity and derives from Lorentz transverse mass and longitudinal mass. Einstein used it in his 1905 papers, later endorsing the concept of relativistic mass for books by Max Born in 1922 and Peter Bergmann in 1942.
By 1948 in a letter to Lincoln Barnett, Einstein denounced the use of relativistic mass as it was being used in Special Relativity.. The hand written letter is published in part.
https://web.archive.org/web/20110607061318/https://www.worldscientific.com/phy_etextbook/6833/6833_02.pdf
Realistic mass was used as the quotient of momentum divided by velocity which is no longer done. The research community did experiments with particle accelerators testing differential equations in the 1950s and 1960s that concluded the concept of relativistic mass should be abandoned in favor of non linear momentum.
In special relativity with constant c it can be proven in the conventional equations of energy and momentum that rest mass is constant.
To have variable mass the local speed of light must also be become variable, decreasing as gravity potential increases. In 1921 Einstein presented in Princeton a derivation of his equation for variable light speed in a gravity field, the well known equation 107 that was published in 1922 reprint of the lecture notes.
A traveler moving at high speed will measure time passing more slowly and distances apparently being shortened. This gives the traveler an apparent local light speed faster than standard c.
For a traveler measuring a variable light speed locally, the same traveler will measure a variable mass using the conventional equations of energy and momentum. Using Equivalence and Lorentz Functions a traveler far from a gravity source will have an opposite effect from gravity, with kinetic energy causing measured light speed to increase with acceleration. Also the effective rest mass will be measured to decrease locally, approaching zero near light speed. It agrees with accepted research that says anything moving at light speed has no effective rest mass.
You can still find researchers using the concept of relativistic mass in Special Relativity, but seldom in General Relativity.
Frederick David Tombe > " Concept of Relativistic Mass"?
As far as I am concerned it cannot be real. Some quasars are ejected at a considerable fraction of the speed of light; from where such energy would be available? If at all, it can only be an apparent phenomenon (illusion) to an observer in IRF.
I think this idea of “mass increase with velocity" originated from Max Planck’s publication at the link below:
https://en.m.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics
Planck probably borrowed it (as he mentions Lorenz and Einstein in the first line of his paper) from electrodynamics and applied it to quantum dynamics of massive particles. I had a long argument with my friend Wolfgang W. Engelhardt in both his and in my forum as well as through e-mails, on this issue.
In electrodynamics the term used is m = m0 x gamma
Planck used it for massive particles. But as I argued with Wolfgang, if m0 = 0 for photons, the whole expression above for mass m becomes meaningless. This is my whole point against the mathematical idealism of Einstein and his thought experiments. You hide a mathematical fallacy by replacing numerical expressions with alphabetic expressions of your fantasy idea; like m0, t0 and so on!
And what is more: what is the nature of "increased (new) mass" at high velocity? Is it of the same qualitative type as the old ones (in terms of individual atoms); or is it different? What happens to the "increased mass" when the the particle or body comes back to regular/normal velocity?
Dr. Malek, Thank you very much for the link to the 1906 Planck paper. This may well be the paper that I was looking for, and equation (6) may well be the relevant equation.
I still haven't worked out the implications yet, but I'll take a closer look at Planck's argument.
Meanwhile, does the large X on the right-hand-side of equation (6) signify "Force"?
Hi Stefano, Thanks. But in the last section of the 1905 Einstein paper "On the Electrodynamics of Moving Bodies", there are indeed formulae for velocity dependent mass. But they are not the familiar formula that we see at equation (6) in this paper by Planck, supplied earlier by Dr. Malek,
https://en.m.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics
Hi Fredric,
look at page 22
http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf
Hi Stefano, Yes, thanks. I see it now. I was thrown off first time because of the minus 1 term, but now I see that he was just subtracting the rest mass so as to leave the kinetic energy remaining in isolation.
Hi Stefano,
I should just add that on page 21 of Einstein's 1905 paper, I'm not clear about how he arrived at equation (A). I know he used the Lorentz transformations, but he didn't show the derivation line by line.
Have you ever seen how it is done? If so, I'd be grateful if you could elaborate on it please.
Dr. Malek, Regarding your question about the actual physical nature of the relativistic mass increase, it could be one of two possibilities. (1) The Lorentz factor may simply be an asymptotic factor for the dissipation of the input energy to the surrounding luminiferous medium, as like by analogy to air resistance. Or it could be (2) a genuine increase in mass, forming due to the input energy.
I don't want to commit myself as to which, just at the moment. First I want to see how exactly Einstein and Planck derived it. The details are skipped in both Einstein's and Planck's papers.
Hi Fredrick,
it is explained here how to arrive at those relations
Preprint FROM THE RELATIVISTIC MOMENTUM TO THE MASS ENERGY-EQUIVALENC...
Frederick David Tombe> "The Lorentz factor may simply be an asymptotic factor for the dissipation of the input energy to the surrounding luminiferous medium, as like by analogy to air resistance".
I gave a similar explanation in my discussion in the two forums I mentioned before and also with Wolfgang. For me (depending on Dirac's QED and Hegel's dialectics of space and time, please see my RG profile) ) the "surrounding" medium consists mainly of the virtual particle pairs of the quantum vacuum and the accelerating particles increasingly lose energy to this medium with accelerating velocity and at very high velocity to matter/antimatter particle production. The inner linings of the pipes of the accelerators are known to be damaged due to the production of such highly energetic stray particles (including their annihilation energy). In other words, the accelerating particles indirectly lose their energy to the inner walls of the pipes. This seems to be a rational explanation for me rather that a fictitious "Velocity dependent increase of mass". The Lorenz "gamma factor" is the primary Fairy Monsters that gave rise other Monsters in relativistic physics and cosmology!
Dr. Malek, For many years, I would have agreed with you. But now I'm not so sure. I'm still investigating the matter.
Hi Stefano, I've just read your paper. But how was the gamma factor connected to mass in the first place? I'm not questioning the experimental evidence, or how we establish a kinetic energy theorem from the relativistic mass. I just want to know where the relativistic mass formula originated. I can see it on page 22 of Einstein's 1905 paper, but I can't see how he derived it.
It all comes down to how he derived equation A on page 21 from the Lorentz transformations.
I can see a connection between section (c) in your paper and equation A in Einstein's paper. You've both cubed the Lorentz factor in the force equation. Can you please elaborate on that specific bit.
Abdul Malek
Frederick David Tombe
Dear Sirs,
Allow me to show the physical model of the relativistic mass. Any material body consists of two components: visible and invisible. The visible material component is determined by the mass m. An invisible component is a shell of a material body in the form of an attached vortex of an electromagnetic field medium. The invisible field component is determined by the electromagnetic (relativistic) mass (gamma* m).
In such questions, we must define the mass. Otherwise, everyone understands this term in their own way. If mass is "the amount of matter in the body," then it cannot increase with speed. Relativistic (electromagnetic) mass as a shell of a body (boundary layer of the medium) increases with increasing speed.
I have detailed this idea in a preprint "Electromagnetic mass and Gravity". If you find time, please take a look at my profile.
Yours
Valeriy Pakulin
Dr. Pakulin, At the moment, I'm just trying to establish how the Lorentz factor gets involved with mass mathematically. After that, I'll take a closer look at the physical interpretation.
But the mass of a body can increase if more mass is injected into it during the acceleration process.
@ Stefano Quattrini
I've now studied you paper in more detail and I understand your derivation. It begins with an expression for relativistic momentum that has been established from experiment. I can then follow the mathematical manipulations from then on. It leads us to where Einstein was at equation A on page 21 of his 1905 paper.
But the question still remains "How did Einstein himself get to equation A?".
Fredrick,
what I wrote gives the equation at page 22. Presumably Einstein arrived there in the same way but with very little justifications.
Stefano Quattrini and Frederick David Tombe > “Presumably Einstein arrived there in the same way but with very little justifications.”
The only justification of Einstein was based on LT – the bugbear of Einsteinian physics, which is accepted as a universal truth and like c is just a mathematical axiom, without any rational justification, through practice.
In electrodynamics the term used is m =gamma x m0. Planck used it in quantum dynamics for massive particles, and from then on, this mathematical fantasy of LT became the foundation of the virtual edifice of Einstein’s “Castle in the Air”; where modern physics lives happily ever after and carry on infantile scholasticism.
But as I said in my first comment, if m0 = 0 for photons, the whole expression above for mass m becomes meaningless. You hide a mathematical fallacy by replacing numerical expressions with alphabetic expressions of your fantasy idea; like m0, t0 and so on as Einstein did!
In modern physics, all the theories, starting from Newton’s deliberate mutilation of Kepler’s observational Laws (and the re-establishment of the essence of the Epicyles of feudalism, after the Copernican, Galilean revolution)), including Maxwell’s electromagnetism, SR, GR, QFT, QCD etc., ad nauseum, are posteriori mathematical (and theological) scholasticism that subverted physics all along and continues till today. Only Dirac’s QED is an exception. These theories are imposed on physics as requirements of class rule and to keep the priests (trained in the universities) busy in infantile scholasticism; detached from reality.
So, as I argued with my friend Wolfgang W. Engelhardt, his claim that modern accelerators are based on Planck’s theory is a false claim. In fact, all the mathematical idealism-based theories of physics in astrophysics & cosmology in one end and quantum physics (except QED) on the other end; have nothing to do with technological practice in modern times in these realms and are a hindrance to the further progress of physics!
Thus, all the claims that Space exploration technologies, GPS, electronics technology (except photo-electron and QED), nuclear technology, etc., are the fruits of these esoteric theories are complete lies and all these centuries long “proofs” (including the "Big Bang", the “God Particle” etc.) of these theories are contrived make-beliefs, to pacify and intimidate the working people of the world – the real force behind the development of these technologies and all progress in natural science! Only an overthrow of this parasitic ruling class and its false theories can lead to further unhindered progress of natural science!
@ Stefano Quattrini
Hi Stefano, Einstein didn't have the benefit of the experimental result which was your starting point. Therefore I am naturally curious as to how Einstein did it.
I'm not questioning the result.
The same happens up in section 3. Einstein arrives at the standard Lorentz transformation results, but without using four-vectors. It's a bit of a mystery as to how he could have done that.
It makes one wonder whether or not he had access to other material which he copied out incompletely. And I note that he did work in a patents office.
Dr. Malek, I'm not willing to criticize anything until the investigation is complete. Stefano's paper is very convincing, but I'd still like to know more about the origins in 1905. Einstein doesn't appear to have fully justified something, which nevertheless appears to be correct. The mathematics works out too neatly for it to be "all wrong".
Frederick David Tombe > "The mathematics works out too neatly for it to be "all wrong".
If you consider "the consistency of mathematics” alone as the criteria and the validity of positive knowledge, then you are following the very Mantra on which Einsteinian physics is based and I doubt you will discover any mathematical fault in these esoteric theories of modern physics, because these theories are the results of not only of brilliant Einstein, but some other best mathematical minds in history. I think you fail to realize that the expression m = gamma x m0, if m0 = 0 for photons; is a mathematical fallacy and have no meaning in objective reality. This discussion unfortunately, also has no meaning for me! I joined in only to respect your repeated private note to me.
Dr. Malek, I would never apply the relativistic mass formula to a photon, because a photon is a wave. A photon is not interacting with the luminiferous medium in the same way that an ordinary particle does.
Frederick David Tombe > "I would never apply the relativistic mass formula to a photon, because a photon is a wave".
This is an axiom you are using, which has lots of evidence both for and against. So, how much scientific is your approach? And if you deny m = gamma x m0, then you deny the validity of the most theories of modern physics, including SR, GR, SM etc., and the official claim behind the workings of particle accelerators, may be particle physics in general!
As a matter of fact Tangherlini Transformations which keep the simultaneity invariant are much more general. LT are just a particular and unfortunate case...
https://www.researchgate.net/post/Tangherlini_Transformations_have_less_limitations_than_Lorentz_Transformations_why_arent_they_used_instead
Dr. Malek, When maths checks out so perfectly, as in the case of the relativistic mass derivation, it's usually an indication that the physics is correct, at least in a particular context. The objective is then to identify that physical context. I think that Larmor was perhaps the one who was getting closest to identifying the context in question. And that context is certainly not the photon.
The context will likely be that of a particle moving through the aether, where that particle has a dipolar vortex structure. The gamma factor is closely related to both transverse Doppler effect and 'return-path' longitudinal Doppler effect. This would tend to point to a circular energy flow mechanism as a rotating-dipole moves through a sea of rotating dipoles.
What I don't agree with is the notion that the relativistic mass formula has general application, at least without introducing a scaling factor.
Frederick David Tombe > “When maths checks out so perfectly, as in the case of the relativistic mass derivation, it's usually an indication that the physics is correct..”
But how do you get the “maths (that) checks out so perfectly”? You get it from some initial and crude impression of the system under consideration; then immediately you fall back on the thought and fantasy derived maths of the past and you set yourself to build a perfect “castle in the air” on the shaky initial foundation and hope that no new facts or impressions pop up to disturb your castle. When they do (they always do), you have to add another section to your castle and so on until the load is too high on the weak foundation, to save your castle from biting the dust. But this saving is possible through very expensive and contrived “proofs”; sponsored by the State and theology, only if that castle is useful for them!
Dr. Malek, You're not listening to what I have said. You asked me how I get the maths to check out properly. That's the essence of the title question of the thread. I'm still investigating the matter. I'm trying to get to the starting point. But the end result is pretty neat, and pretty plausible, providing that we can identify a physical context where it applies.
I do not believe that the relativistic mass formula has general application.
Your response above suggests that you are responding to someone else. It doesn't correspond to anything that I have said.
Mass variable with speed predates relativity and derives from Lorentz transverse mass and longitudinal mass. Einstein used it in his 1905 papers, later endorsing the concept of relativistic mass for books by Max Born in 1922 and Peter Bergmann in 1942.
By 1948 in a letter to Lincoln Barnett, Einstein denounced the use of relativistic mass as it was being used in Special Relativity.. The hand written letter is published in part.
https://web.archive.org/web/20110607061318/https://www.worldscientific.com/phy_etextbook/6833/6833_02.pdf
Realistic mass was used as the quotient of momentum divided by velocity which is no longer done. The research community did experiments with particle accelerators testing differential equations in the 1950s and 1960s that concluded the concept of relativistic mass should be abandoned in favor of non linear momentum.
In special relativity with constant c it can be proven in the conventional equations of energy and momentum that rest mass is constant.
To have variable mass the local speed of light must also be become variable, decreasing as gravity potential increases. In 1921 Einstein presented in Princeton a derivation of his equation for variable light speed in a gravity field, the well known equation 107 that was published in 1922 reprint of the lecture notes.
A traveler moving at high speed will measure time passing more slowly and distances apparently being shortened. This gives the traveler an apparent local light speed faster than standard c.
For a traveler measuring a variable light speed locally, the same traveler will measure a variable mass using the conventional equations of energy and momentum. Using Equivalence and Lorentz Functions a traveler far from a gravity source will have an opposite effect from gravity, with kinetic energy causing measured light speed to increase with acceleration. Also the effective rest mass will be measured to decrease locally, approaching zero near light speed. It agrees with accepted research that says anything moving at light speed has no effective rest mass.
You can still find researchers using the concept of relativistic mass in Special Relativity, but seldom in General Relativity.
Dr. Decker, Thank you for this interesting response. I had noticed that Einstein used longitudinal and transverse mass in his 1905 paper, so thanks for back-dating that idea to Lorentz.
Reading between the lines of your response, it would certainly seem that doubts prevail over the general applicability of the standard relativistic mass formula. I don't think there has ever been a perfect theoretical derivation of that formula from scratch. Stefano provides a good derivation, but it is experimentally rooted. But I suspect that the formula will be correct for a particular ideal context, that likely being for a rotating electron-positron dipole.
Hi Stefano, I'm now looking at the energy-momentum relationship and how to derive it using the norm of the four-momentum. The inner product leads to E^2 - p^2.c^2. But I'm having difficulty with the reasoning behind why this is equal to (mc^2)^2. It's something to do with mc^2 being invariant under Lorentz transformation.
Can you please clarify the argument. I'd be most grateful. If mc^2 is invariant, then so are many variations using those to parameters, c and m.
Dr. Decker, I've just sent this same message to Stefano. I'm now looking at the energy-momentum relationship and how to derive it using the norm of the four-momentum. The inner product leads to E^2 - p^2.c^2. But I'm having difficulty with the reasoning behind why this is equal to (mc^2)^2. It's something to do with mc^2 being invariant under Lorentz transformation.
Dear Frederick David.
I will quote part of our correspondence. Maybe this will facilitate our discussion.
Simpler and clearer is the equation proposed by Heisenberg for the total energy E.
E ^ 2 = P ^ 2 * c ^ 2 + (m * c ^ 2) ^ 2
The conclusion of this equality is as follows:
Let the energy of a particle with invariant mass m0 at rest be equal to m0 * c ^ 2. This energy corresponds to an impulse m0 *c. Obviously, it is not directed along any of the spatial coordinates. Let's assume that it is directed along the time axis. The kinetic momentum of the particle is equal to m0 * v, where in is the modulus of the velocity vector. Therefore, the square of the total momentum of a particle is the sum of the squares of the kinetic and momentum at rest: (m * c) ^ 2 = (m0 * c) ^ 2 + (m0 * v) ^ 2. From this, the connection of the total (observed) mass m with the invariant (rest) m0 is derived. And there are no transverse and longitudinal masses.
Sincerely,
Yuriy.
Of course, these relations are valid only in the absence of fields (potential energy). If necessary, I can give a more accurate ratio. It was derived based on the considerations of Lanczos.
Dr. Zevatskiy, OK, thanks. I think I'm getting closer to the answer now. The energy-momentum relationship can be proven on the assumption that the four-momentum is Lorentz invariant. Hence the inner product of the four-momentum can be equated to −m0^2c^2.
The question then remains as to how we know the four-momentum is Lorentz invariant. So far my investigations are pointing towards the fact that it is Lorentz invariant by definition.
This in turn all starts to suggest what Dr. Decker and Dr. Malek have been suggesting above, which is that there is only one mass, i.e. the rest mass, and that the Lorentz factor is part of the equations but has no bearing on the actual mass.
Dear Frederick David.
Quite true. This rest mass in the Russian-language scientific literature is called invariant mass. It is unchanged under any Lorentz transformations.
Sincerely,
Yuriy.
Dr. Zevatskiy, Yes, and what I'm beginning to think now is that the 'rest mass' is the only mass, at least the only inertial mass.
As regards the concept of relativistic mass, this is basically a definition based on multiplying the rest mass with the gamma factor. But as Dr. Malek asked earlier in the thread, "Does this mean that the mass actually increases?".
I'm beginning to think that it doesn't, and that the gamma factor is an asymptotic factor based on shear interaction with the luminiferous medium, and which leads to a terminal velocity, 'c'.
The reason that I say this is because when we apply the so-called relativistic mass to Newton's second law of motion, we have to add yet more gamma factors to the equation. So why would only one of the gamma factors be owned by the mass?
A similar conclusion was reached in a paper supplied to me above by Dr. Decker.
Dear Frederick David.
One of the formulations of Newton's second law says that one hundred changes in the momentum of a body is equal to a change in the momentum of a force. The momentum of a body, following a late (1948?) Remark by Einstein, can be calculated without relativistic mass. This, it seems to me, corresponds to your ideas. The force impulse can be represented using the gradient of the field potential.
One of the options for solving this problem is presented in the work:
Article Derivation of Quantum Conditions out of Provisions of the STR
Formulas (7) and (8).
Sincerely,
Yuriy.
Dr. Zevatskiy, Thanks for providing your paper. I think the issue in question is simply that we shouldn't be blending rest mass with any gamma-type factors, so as to imply that a relativistic mass exists as a concept in its own right.
The very fact that Einstein had a separate longitudinal mass and transverse mass should have been evidence that he was using the wrong terminology. He was looking at longitudinal and transverse effects, but that didn't imply that there were separate masses.
I wonder did Lorentz also use that misleading terminology.
Conventional equations of energy and momentum have been confirmed countless times in accelerators for Special Relativity. Gravity differences were not tested in the accelerators. So this is not for General Relativity.
E2 = (mc2)2 + (pc)2
pc = E(v/c)
dE = vdp
Special relativity is inherent in these equations.
E2 = (mc2)2/(1-v2/c2)
p2 = (mv)2/(1-v2/c2)
dE2 = c2dp2 = d(pc)2 - p2dc2
d(mc2)2 = - p2dc2
dLn(mc)2 = - (dLnc2)/(1-v2/c2)
From this last equation it is shown that (mc) is constant when c is constant, meaning that mass is constant in Special Relativity where c is also constant.
Einstein required locally measured c to decrease in a gravity field (his equation 107) meaning that mass increases when locally measured by a traveler.
With Special Relativity constant c makes the above equations a complete set allowing exact solutions with constant mass. When local c and m become variable in gravity fields then another equation is required which is covered in other threads.
There is no support for relativistic mass in Special Relativity, which was discarded decades ago.
As commented before Einstein denounced relativistic mass as early as 1948, but allowed it in 1905, endorsed it in letters around1920 for Max Born's book, and again in a forward he wrote in 1942 for Peter Bergmann's book.
The concept comes from Lorentz but was not much used until Special Relativity. Then it was tested in accelerators and discarded
Dr. Decker, Thank you very much for this historical background. Yes, it does seem that the idea of relativistic mass was premature. Nevertheless, they were still teaching it at universities in the late 70s.
The fact that Einstein based his definition of 'longitudinal mass' on Newton's second law, whereas Planck based his definition on momentum and energy, means that it should have been obvious right from the outset that we should not be directly blending the gamma factors with the mass in order to create a new entity in its own right.
Einstein, unfortunately bears the ultimate responsibility for “relativistic mass" and other Fairy Tales in modern physics that are being ladled out to students and the public in general, in his name; whether he later agreed with these or not. The genie for the fairy tales lies in the mathematical mysticism he initiated, starting with SR. The following comment about SR reveals his mysticism and the source of all infantile scholasticism, which is now known as modern official theoretical physics: “Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties?” A. Einstein, in "Relativity, The Special and General Theory" (Three Rivers Press, NewYork, 1961). Einstein’s later negation (“denunciation”) had little impact on many of the wild derivatives of his theories, conveniently used (in his name) by official physics, monopoly capitalism and theology.
“Black Hole” for example got benediction this year from the “Nobel Church”; in spite of Einstein’s published negation of its possibility even in theory: "The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that mote general cases will have analogous results. The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.
This investigation arose out of discussions the author conducted with Professor H. P. Robertson and with Drs. V. Bargmann and P. Bergmann on the mathematical and physical significance of the Schwarzschild singularity. The problem quite naturally leads to the question, answered by this paper in the negative, as to whether physical models are capable of exhibiting such a singularity." Albert Einstein. A. Einstein, The Annals of Mathematics, Second Series, Vol. 40, No. 4 (Oct., 1939), pp. 922-936
https://pdfs.semanticscholar.org/8dd0/dfafef0c53c428fdc3b58f8099aafcf7d089.pdf
And Einstein’s final refutation of GR in a letter (1954) to his friend Michele Besso, just about a year before his death: “I consider it quite possible that physics cannot be based on the field concept, i.e., continuous structure. In that case, nothing remains of my entire castle in the air, gravitation theory included, (and of) the rest of modern physics” A. Pais, Subtle is the Lord …” The Science and the Life of Albert Einstein”, Oxford University Press, (1982) 467,
So, I wonder why any "conscientious" physicists should still be preoccupied with Einstein’s theories of relativity, “relativistic mass” included!
Dear Colleagues.
How can we answer the question about the relativistic formulation of Newton's second law.
Sincerely,
Yuriy.
Dr. Zevatskiy, I'm still working on it. It seems that we have to add gamma factors to take account of the fact that the luminiferous medium has retarding asymptotic effect on the acceleration.
Dear Frederick David.
If impulse P = mv vector. The potential field U is a scalar. dU = -Fdr, where F and dr are vectors. Then in the classical limit Newton's second law looks like this:
dP × v = Fdr
dP = Fdt.
In the relativistic case:
dP × v = Fdr (1 + U / mc ^ 2)
dP = Fdt (1 + U / mc ^ 2).
Sincerely,
Yuriy.
Most surprisingly, for a charge in a magnetic field (U = 0), Newton's second law coincides in the classical and relativistic limits.
Now it remains to substitute these equations in the expression for the angle and calculate the displacement of the perihelion of Mercury.
Dr. Zevatskiy, I have to confess that I have no idea where the expression dP × v = Fdr (1 + U / mc ^ 2) came from.
I never take any expressions from relativity for granted. I'm still working on the justification for the relativistic mass formula.
According to SR, when a mass (m) moves very close to the speed of light, an external observer will see a contraction of the space that (m) occupies, while its travel time experiences a dilation. If the volume of space containing (m) contracts, i.e., decreases, then the mass density increases, therefore the physical effect at that travel speed is an increase in mass. The apparent relativistic mass (m ') can be written as:
m '= m * Z
Z = 1 / (1-v ^ 2 / c ^ 2) ^ 0.5
Dr. Echaurren, Increase in density does not mean increase in mass. It has already been considered above that the mass doesn't change at all, but rather that Newton's second law has to be amended by a velocity dependent asymptotic factor in order to take account of the interaction with the surrounding luminiferous medium.
The increase in mass is not real, it is only apparent, that is why it is called relative, it is relative to the external observer, the equations only quantify that apparent increase. This is what many fail to understand about relativity, we speak of reference systems independent of each other with their own observers. The observer on the object moving at the speed of light will not see any increase in mass, but all external observers will, that is what you have to understand, that is the beauty of relativity. This apparent increase is only seen in dynamic masses in contrast to masses at rest, it all depends on the reference system, this is how the laws are designed in our universe.
Dr. Echaurren, There was a time in earlier days, and still taught at universities into the 80s, where they believed that the mass actually does increase. But the more I look into it, I now simply see an additional velocity dependent asymptotic factor added into Newton's laws of motion, with the actual mass remaining unchanged.
Well, it's the same thing that I tell you, to teach that mass really increases at very high speeds is not to understand relativity, all the changes are apparent to external observers. To look for a real measurable increase in mass is to look for something that does not happen, at least in this universe. Changes can only be estimated but always within the framework of relativity.
Dear Frederick David.
I express my gratitude to you for your work and deep interest in this discussion. Thanks to our communication on the topic of relativistic mass, I turned to my old work Article Derivation of Quantum Conditions out of Provisions of the STR
and found there an error in formula (8). Correction of this error led to the creation of a new workPreprint Relativistic Dynamics and Hamiltonian
I hope you and many of our colleagues will find in it the answers to the questions that you raise in your discussion.Sincerely,
Yuriy.
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