Abstract: So-Called RELATIVISTIC MASS IS A GRAVE ERROR. MASS IS AN INVARIANT. WHAT VARIES IS FAST TIME GOING SLOW. We develop more natural and fundamental axiomatics for mechanics, in the hope of elucidating the Foundations better.
We propose the following physically meaningful conceptual chain ordered by evidence:
Buridan principle
→ momentum as primitive
→ time dilation from light clocks
→ relativistic momentum
That provides a dynamical path into relativity from first principles and physical observations instead of a geometric, top down one, physically unjustified.
[This is a matter of conceptology, MSF, Most Significant Facts, not just pedagogy… It’s also a sketch, the more formal and complete Axiomatics of Relativity will appear in a separate essay.]
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Momentum is a vector more fundamental than energy. Indeed, energy is the integral of the derivative of momentum over a spatial path, it’s a scalar (a scalar which gets replaced by a stress-energy tensor in general gravitation theory)… Momentum is also how Matter Waves are defined and there it’s derived directly from Matter Waves’ amplitude (p = hk/2π) (the resulting Feynman’s Path Integral and notation simplify Quantum Field Theory)..
Momentum P = mV (vectorial). was invented/discovered by Buridan as a definition expressing Motus Quantitas, Quantity of Motion, just as mass defines Quantity of Matter:. (Buridan called Momentum “Impetus”).
Buridan also discovered an ultra-major law, Buridan’ Principle, tying up force, time and momentum, F(dt) = dP (vectorial)… That law is more general than F = ma. From the fact that FAST TIME IS SLOW, and the classical Buridan Principle, one can demonstrate a generalization which makes the Buridan Principle RELATIVISTIC (who would have thought Buridan → Relativity?)
Defining P = m dX/dτ (using proper time τ) naturally yields P = m Vγ without ever needing to touch the definition of mass (a clean and clear proof).
This may have consequences for Matter Waves as those were discovered through relativistic considerations.
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We must acknowledge the 300-year “DARK AGE” between the Oxford/Merton Calculators/Parisian “Nominalist” doctors of the 14th century and the Scientific Revolution of the 17th century imposed by the theocrats and the plutocrats (“nobles’, monarchs) they were entangled with. This explains in part why Newton is often credited with “discovering” laws that were actually “recovered.”
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BURIDAN FOUND THE TWO MAIN LAWS OF MECHANICS Three Centuries Before Newton exploited them decisively:
Momentum was initially called impetus by Buridan who invented it. Later it was called “Quantity of Motion” (and is still called that way in French), a better terminology generalizing “quantity of matter” which is mass.
Buridan wanted to associate just one vector to the motion of an object of mass m and speed V where V is directional magnitude, that is a vector.
P = mV was the simplest way to do so.
Key Insight: Impetus is not self-dissipating; it persists unless diminished by external resistance. In a vacuum, an object would move indefinitely —a notion called Newton’s first law of motion by Newton’s fans, although it’s in Descartes’ Principia published in 1644, when Newton was 2..[1]
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The concept of founding mechanics on mV turned out to work splendidly… until Gabrielle Émilie Le Tonnelier de Breteuil, marquise du Châtelet, translating Newton from Latin to French noticed that Isaac had confused mV and ½ mVV, energy, which she introduced, experimented upon, and elucidated (Lagrange and Hamilton introduced functions that energy trajectories minimized using Maupertuis’ Action Principle… He was an amused collaborator of Émilie who died from childbirth at 42…) Two centuries after Émilie, still another Parisian, Louis De Broglie, replaced P by the gradient of the amplitude of matter waves (long story; Feynman’s career was all about that).
[The meta-reason behind the discovery of the foundations of mechanics was the Principle Of Least Reason combined with the MSFT, Most Significant Fact Theory.].
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Buridan asserted: F (dt) = dP.
(BURIDAN PRINCIPLE; BP; it generalizes to Relativity, see below!)
Here dt is a small time interval, dP is a small variation of momentum P and F is an applied force (and also a vector).
Buridan’s Principle is more general than F = ma, in particular because it allows for varying mass.
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Buridan did not use the equal sign, because that was invented, with the equation notation, in 1557, two centuries later. Instead Buridan wrote sentences in Latin which said exactly the same thing that an equation says in a more abbreviated form.
In particular, Buridan uses what we now call differentials, also defined verbatim as what they are, rates of change.
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If P = mV, Buridan Principle implies F = m dV/dt = mA, where A is the acceleration vector. This is generally called the “Second Law of Mechanics”, written F = ma, and is named after Newton (Newton was not even born three centuries later). Moreover the F = mA Law was perfectly known to Descartes and Galileo whom scholars were able to demonstrate that Newton picked it from (by looking in Newton’s “Waste Book”in 1965).
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Buridan was hyper famous in his lifetime, rector of the university of Paris, adviser to four kings, and had as pupils thinkers such as (bishop) Oresme. Oresme also became a celebrity, rector of the university and tutor to the Dauphin.
Oresme published in 1350 CE the proof of the fundamental theorem of calculus, complete with a drawing (the theorem that differentiation and integration are inverse to each other, up to constants). Oresme proceeded to develop the geometric differential calculus which Newton would use to demonstrate that the gravitational attraction (discovered by Bouillaut) 1/dd plus F = mA implied Kepler’s laws.
Above we used the infinitesimal notation invented by Leibnitz which was proven logically coherent by Model Theory only around 1950… But the equation, as a Latin sentence, is entirely in Buridan’s work. Moreover he uses it to prove that a falling body constantly accelerates (a discovery often attributed to Galileo who published nearly three centuries later…)
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Work-energy theorem: ΔE = ∫F·dx, and F = dP/dt:
As F = dP/dt Force is the derivative of Momentum. Now Energy is the sum (=integral) of Force along a path (the famous path integral which generalizes to Quantum Mechanics). So Energy is the composition of a derivation from momentum, followed by an integration along a path. In other words, momentum is more fundamental. It’s worth pointing out that modern physics, especially Quantum Field Theory, rests on Lagrangians or Hamiltonians, which are expressions depicting Energy. By studying Momentum itself, one goes back to the source. De Broglie’s Matter Waves depend upon Momentum…,
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F = mA plus F = dP/dt gives m dV/dt = dP/dt. Integrating, we get: P = mV + C, thus P = mV if we reasonably assume P = 0 when V =0.
So the usual second law plus Buridan Principle implies the expression for momentum in classical mechanics.
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MAKING BURIDAN PRINCIPLE COVARIANT: SUBSTITUTE PROPER TIME!
It turned out that classical mechanics was not correct at HIGH SPEED. By 1904, Henri Poincaré had announced a new mechanic which he called “Relativity” and Poincaré toured the USA to instruct the cowboys. By June 1905 he published the exact rotations of the four dimensional spacetime which he called “Lorentz Transformations”.
In a September 1904 lecture in St. Louis named The Principles of Mathematical Physics, Poincaré drew some consequences from Lorentz’s theory and defined (in modification of Galileo’s Relativity Principle) “The Principle of Relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion.” .
Applying the Principle of Relativity, we should get:
F γ dτ = dP
This way Buridan’s Principle looks the same in the moving frame.(That’s called “covariance”)
Well, OK, we are actually not just going to state it, but to prove it!
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Here τ is so-called “proper time”, the time of the moving mass m at speed v.
Indeed, considering light clocks, one can readily see that FAST TIME IS SLOW (it turns out all of Relativity can be derived from fast time being slow, as I will show in a companion essay; some will scoff and evoke “supplementary hypotheses” such as the constancy of the speed of light, but those hypotheses are all contained in the concept of light clock, which is used to demonstrate that fast time is slow… And some like the delay term -xv/cc came actually from telegraphy, a subject Poincaré was an expert at). Pure Euclidean geometry shows that:
. τ = t/γ … So dτ = dt/γ
= 1/1 – v2 /c2
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:PROVING RELATIVISTIC BURIDAN PRINCIPLE FROM PROPER TIME:
Time on the moving mass m moving at speed V is: τ = t/γ … As above.
We can apply F on m. For example F could be gravitation, which depends only upon m, independently of the state of motion of m (Buridan was already aware of that and applied gravity to demonstrate it caused universal acceleration).
Let’s consider dP/dτ. We have: . dP/dτ = dP/dt dt/dτ = F γ
Or: dP = F γ (dτ)
So from Buridan’s law in the rest, lab frame, we get Buridan’s law in the moving frame according to the Relativity Principle, once we have adjusted the time for the “proper time” of m, the real time in the moving frame.
When I learned Relativity, for years I was confused whether that “proper time” was “real” or “apparent”… Although the so-called “Twin Paradox” shows clearly that “proper time” is the real time around m, moving at v, and yes it goes slower, for REAL.
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MASS IS AN INVARIANT:
Buridan put together P = mV = dX/dt.
… m has a universal meaning: it’s the mass of m, the quantity of matter as measured by weight in the lab/rest frame.
Notice in passing that a serious mistake which was made by our hero Henri Poincaré… Poincaré talked about “inertial mass increasing” in 1904. That’s an HORRENDOUS error which was parroted by many celenbrity physicits, all the way to Feynman and beyond. With all dues respect for those tremendous geniuses, that mean they had NO understood Relativity (Einstein did better than most, as he came to reject relativistic inertial mass in his later years…)
It is an horrendous error, because it shows a lack of understanding about the nature of Relativity. Relativity: Relativity is all about slow time, NOT about changing mass! Poincaré in 1904 did not understand what we are trying to explain in the present essay, namely that inertial mass, the quantity of matter, is NOT increasing.
Instead what Poincaré, and generations of parrots who squawked after him, did NOT understand, was that it is FAST TIME GOING SLOW which caused the augmentation of momentum.
(Why the following generations of super hyper geniuses didn’t get it, and insisted upon changing mass, is a testimony to the Parrot Principle: why not repeat what sounds good?)
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PROOF OF RELATIVISTIC MOMENTUM FROM SLOW TIME: P = mVγ
P = mV (vectorial) is Buridan’s momentum. More exactly: P = m dX/dt (Vectorial)
The constructing idea of P is to find the simplest combination of mass and speed, which are the most significant facts of motion, and define the quantity of motion.
P = mV is not compatible with Relativity because speed, V = dX/dt is limited in norm to c. So the norm of P is limited by mc, and the quantity of motion would be bounded, even as more force is applied..Thus, to make P compatible with Relativity, we must change its definition.
Moreover, as v → 0, we want the Relativistic generalization of Buridan’s definition of Momentum to converge to mV.
We can’t touch m, it’s an invariant, independent of speed, mass (we are not repeating Poincaré’s 1904 mistake). That leaves us with having to modify dX/dt…
Now we found above that how P is changed by dP during that application depends upon dτ, “proper time”, time of m going at speed V. If V is gigantic, close to c, time in the moving frame comes to a near standstill, applying force F to will have nearly no observable effect… Because time is so slow in the moving frame (m, V) that F is applied for a very small time. THIS is the essence of the increase of Momentum (and the illusion of greater inertial mass which confused Poincaré initially).
Thus, all we can change is dt. Whereas in the lab/rest frame X and V are observed, and m is invariant, one observes that time proper to m moving a t V is τ.
So: P = m dX/dτ = m Vγ.
Or: P = mV/1 – v2 /c2
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Conclusion: Modern physics rests on Lagrangians and Hamiltonians, which are functions of a scalar, energy, and most stable around trajectories. However, Momentum is more general, multidimensional, fundamental and directly associated with the main characteristic of Matter Waves.
Lagrangians and Hamiltonians give no intuition (that is, no simple math) about how to derive Relativity… But Momentum, which has direct cognitive meaning, does, as we will show in a companion essay.
We just gave, from Most Significant Facts, why proper time (“fast time”) matters, and a proof of the formula for relativistic momentum.
Physics ought to be natural, because physics is from the Greek ta physika, literally “the natural things,” title of Aristotle’s treatise on nature. So physics ought to be deduced from Most Significant Facts. For example, Quantum Physics can be discovered laying on a sunny beach and nearly closing one’s eyelids: the eyelashes, acting like multi-slits, form interference patterns with the incoming sunlight. Hence light is a wave.
We will try to pursue our simplification program… One of the reason to scrutinize momentum is that the De Broglie-Matter Wave relation is obtained from inside the conceptual universe above. So the SPQR, Sub Quantum Physical Reality, fundamentally rests on Relativistic considerations (a conclusion whom Prince Louis De Broglie himself drew, I would not say in person lest it makes the present author look antediluvian)…
Patrice Ayme
P/S: Why was the stupendous Fourteenth Century physics subsequently hidden? In 1474 CE, Louis XI of France, the most powerful king in Europe, decided that most advanced thinking was not his friend, because he was keen to seduce the Pope and his Catholic Inquisition, which was itself determined to foster irrationality. So Louis XI, and the Vatican, made reading Buridan a capital crime.(The first one to launch that method of executing families when the wrong book was found at home, was that great innovator, Roman Emperor Constantine. Neither Caligula, nor Nero or Commodus had sunk that low…)
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