This is the formal derivation of the relativistic mass formula from the force equation and the assumption E=mc² and P=mv with v=ds/dt. The derivation is based on elementary physics and elementary mathematics (calculus) only, without referring to higher domains of mathematics or physics. But identifying the constant c in the derived mass formula m=m₀/√(1-v²/c²) is not part of this derivation. In the context of this derivation c only is an arbitrary constant which has the physical dimension of a velocity.
We begin this derivation with momentum=force*time, P=F*t, infinitesimal dP=F*dt, or F=dP/dt. Followed by energy=force*distantance,E=F*s, infinitesimal dE=F*ds, F=dE/ds. ds is a Vector therefore dE/ds=(∂E/∂sx, ∂E/∂sy, ∂E/∂sz).
Physics must be consistent, therefore the force F in both equations must be the same. F=dP/dt=dE/ds.
We now consider that P with P=mv is proportional to the mass m and conclude that this proportionality also must apply to E. We therefor set E=mc² but with an arbitrary parameter c². We select this special form because the parameter which relates mass and energy must be a velocity squared.
Inserting E=mc² and P=mv leads to d/dt(mv)=d/ds(mc²). We now consider the chain rule of calculus.
(1) d/ds=d/dv*dv/ds, dv/ds=dv/dt*dt/ds, dt/ds=1/(ds/dt)=1/v
(2) d/dt=d/dv*dv/dt
(2) leads to (3) dP/dt=dP/dv*dv/dt
(1) leads to (4) dE/ds=dE/dv*dv/dt*(1/v)
We can cancel dv/dt in (3)=(4) an bring 1/v on the other side and get
(5) v*dP/dv=dE/dv. This equation now only depends on v.
Now we consider the left side of (5) with the product rule and get
(6) v*dP/dv=v*d/dv(mv)=v²(dm/dv)+vmdv/dv=v²dm/dv +vm and
(7) dE/dv=d/dv(mc²)=c²dm/dv
Now we consider (6)=(7):
v²dm/dv+vm=c²dm/dv which leads us to dm/dv*(c²-v²)=vm. Or
(8) dm/dv=vm/(c²-v²). By division with m we get:
(9) m‘/m=vm/(c²-v²) in mathematical notation: y'/y=x/(c²-x²)
We now can formally solve the equation y'/y=x/(c²-x²). This is a differential equation from type Bernoulli. Its solution derived by integration on both sides is:
(10) ln(y)=-ln((c²-x²)/2)+C. Exponentiation leads to
(11) y=C/√(c² - v²).
Selecting the constant C as C=m₀c then formally leads to
(12) m=m₀/√(1-v²/c²).
c still is only the constant in E=mc². Within this derivation from pure mechanics, c still is an arbitrary velocity and is not yet assigned to the speed of light.
This derivation, entirely based on fundamental physics and mathematics, is devoted to people who generally doubt the concept of relativistic mass m=m₀/√(1-v²/c²).
Are there any objections?
Are there any ideas how the assignment of c to the speed of light can be proven to people who generally refuse to acknowledge relativity theory?
Can we claim that people who refuse relativity theory also would refuse this basic derivation?
Derivation of relativistic mass formula is based on the principle of mass energy equivalence where in the c is the speed of light as derived from permittivity and permeability constants of vacuum or free space.
Gurcharn Singh Sandhu "Derivation of relativistic mass formula is based on the principle of mass energy equivalence where in the c is the speed of light as derived from permittivity and permeability constants of vacuum or free space."
Where is the link from permittivity and permeability constants of vacuum or free space to the mechanic equations, which lead to the relativistic mass formula, but with an arbitrary speed?
Do you know a textbook reference where the derivation of the relativistic mass formula from basic mechanics is described? You obviouly had this idea long before I had a similar idea.
The origin of the concept of mass-energy equivalence is generally attributed to Albert Einstein. He integrated this concept with SR in such a way that now it seems impossible to think of it as an independent stand-alone concept. Yet it is a documented fact that the concept of mass-energy equivalence, in one form or the other, was already in existence prior to Einstein’s 1905 paper. Nikolay Umov, in his ether-based studies of ‘Energy in Moving Bodies’, had alluded to the inertial property of the energy as dE/dm = c2 in 1873. In 1900, Henri Poincaré had deduced that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious fluid with a mass density of E/c2. Olinto De Pretto, a native of the Veneto region of Italy, studied nuclear physics and the prevailing ether theory from 1899 to 1903. As a result of his research, on November 29, 1903 De Pretto published a 62-page paper in the Procedings of the Royal Veneto Institute of Science, Letters and Arts, vol LXIII , entitled "Hypothesis of Aether in the Life of the Universe". He wrote, “Matter uses and stores energy as inertia, just like a steam engine that uses the energy in steam and stores energy in inertia as potential energy ... All components of a body are animated by infinitesimal but rapid movements equal to perhaps the vibration of the ether”. De Pretto used the expression mv2 for the "vis viva" and the energy store within matter, where he identified v with the speed of light.
Essential physical concept behind this derivation is that the kinetic energy of a body gets stored in its deformed field and this stored energy displays inertia like ordinary matter.
Dear all,
I wanted to inform you that ResearchGate will be removing comments on papers as of May 1st, 2025.
I will be requesting a copy of the 5,000 comments on a paper by Wolfgang Engelhardt, titled Free Fall in Gravitational Theory. I encourage others to do the same, as this might increase the chances of receiving a response.
Below is the message I have sent to ResearchGate:
Subject: Request for Comments on Paper
Dear Colleagues,
I noticed that you will be removing comments on papers, and I would like to request a copy of the comments on the following paper: https://www.researchgate.net/publication/312118218_Free_Fall_in_Gravitational_Theory/comments
There are nearly 5,000 comments on this paper. Could you please compile them into a PDF and send them to me?
Thank you very much for your assistance.
Best regards, Prof. Halim Boutayeb
You can't just look at the mathematical derivation, not that the mathematical derivation seems fine, then the theory is fine, because the mathematics itself may also have problems. We can't prove the math problem first, it would be very difficult.
Dear all,
I have proposed a new technique to derive Planck's law without the concept of light quanta. It is in the end of this video:
https://youtu.be/oudaulTkuJg?si=2xVoYOjnvXTN_TTa
It is not published yet but I have published a paper about Compton effect, where I have replicated the results without relativity or quantization of energy but only with classical electromagnetism.
Best regards
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