Rongqing Dai
Abstract
Opposite to the common perception, the relativistic system of momentum and energy is fundamentally different from its classic counterpart because: 1) the relativistic momentum is an independently defined notion that does not share the same empirical foundation as the classic notion of momentum; 2) the relativistic energy is constructed by multiplying the Lorentz factor and the famous E = mc2 while both of them are not physically real; 3) the notions of both relativistic momentum and relativistic energy were constructed based on the unreal notion of relativistic mass. We will explore why it is so in this writing by delving into how the relationship of relativistic energy with relativistic momentum and mass is established.
Keywords: Momentum, Energy, Conservation, Mass, Relativity, Velocity
1. Introduction
Contrary to the common perception that the relativistic system of momentum and energy is a logical extension of the classic notions of momentum and energy that were developed in Newtonian mechanics, the relativistic system of momentum and energy is fundamentally different from its classic counterpart. This is not only because both the relativistic energy and relativistic momentum involve the physically unreal Lorentz factor, but more importantly because the relativistic momentum is a new definition that does not have the same empirical root as of the classic notion of momentum, and the relativistic energy is constructed on top of the exaggerated mass-energy relation.
In relativistic mechanics, the famous Einstein mass-energy relationship E = mc2 is normally replaced by the relationship of relativistic energy with relativistic momentum and mass as follows:
E2 = p2c2 + m2c4, (1)
where m is mass, c is speed of light in vacuum, E is total energy given by
E = γmc2, (2)
p is the magnitude of momentum p given by
p = γmv, (3)
v is the velocity of the object with its magnitude being v, and accordingly, the magnitude of momentum is:
p = γmv, (3a)
where the Lorentz factor γ is given by:
γ = 1/(1-v2/c2)½. (4)
Despite its popularity no one can find a rigorous derivation of (1) in any context, because it is not rigorously derived from the fundamental classic notions of mass, velocity, energy, and momentum, but is rather partly derived and partly extended from the relevant classic notions through redefining the notion of momentum and total energy. Consequently, the notions of relativistic total energy and relativistic momentum as expressed in (2) and (3) are not empirically established as people have generally believed so and consequently they are indeed defective, as will be discussed in this writing. This means that the conservation laws of relativistic momentum and relativistic energy, two of the pillars of quantum theory, are not solidly established empirical laws.
1.1. Relativistic Mass
Neither the relativistic energy nor the relativistic momentum is empirical logical extension of the classic counterparts, but instead both of them are defined based on the concept of relativistic mass. While it is said that Lorentz first proposed the concept of relativistic mass, Einstein formalized it in his “On the Electrodynamics of Moving Bodies” (Einstein 1905a [[1]]), in which the relativistic mass is formally conceptualized with the following mathematical formulas:
(5)
(6)
Einstein needed to differentiate between the longitudinal mass and transverse mass in order to make the relativistic dynamic equations of motion resemble the classic ones; but obviously this need of differentiation is an intrinsic defect of the notion of relativistic mass, which will make it impossible to be reconciled with the gravitational mass general relativity. For this reason, relativistic scholars redefined the relativistic mass as follows:
M = γm (7)
1.2. Relativistic Energy and Relativistic Momentum
With equation (7) being assumed to be the legitimate relativistic extension of the classic notion of mass, relativistic scholars went on to define the relativistic counterparts of the classic total energy and momentum as:
E = Mc2 (8)
p = Mv (9)
Given the definition of γ in the form expressed in (4), it is obvious that E and p in (8) and (9) are identical to E and p in (2) and (3).
1.2.1. Momentum for massless particles
From (8) and (9), when v = c, the following relation can be reached:
p=E/c (10)
Equation (10) has been used by relativistic scholars as the definition of momentum for massless particles since it does not involve mass, despite this relation was derived from definitions for massive particles.
2. Derivation of the Relationship of Relativistic Energy with Relativistic Momentum and Mass
Once we have definitions (2) and (3), it is quite straightforward to work out equation (1) as follows:
Taking square of the terms on both sides of (2), we have:
E2 = m2c4 + c2(m2v2c2/(c2 – v2)) (11)
Taking square of the terms on both sides of (3) (i.e. making both sides dotted with themselves), we have:
p2 = m2v2c2/(c2 – v2) (12)
From (11) and (12), we have E2 = p2c2 + m2c4, which is equation (1). When m = 0, equation (1) reduces to E = pc, which is the same as equation (10) for the definition of momentum of massless particles.
3. The Faultiness of the Relationship of Relativistic Energy with Relativistic Momentum and Mass
The relationship of relativistic energy with relativistic momentum and mass as expressed in (1) is merely a mathematical equation without the true physical meaning as it is supposed to have, and the foremost reason of this is certainly because special relativity is wrong due to its problematic postulates (Dai 2022 [[2]]). Nevertheless, even if we could tentatively assume special relativity is valid, we might still find a few logical defects which would determine that the relationship of relativistic energy with relativistic momentum and mass is faulty.
3.1. Relativistic momentum is a sheer mathematical expression without empirical physical root
We might find many examples of people trying to derive the expression (3) of relativistic momentum from the classic conservation law of momentum, but none of them is correct no matter how much the authors claim so. We might have a better idea about why this is the case by imitating a seemingly most rigorous approach of its kind.
Let’s start from the classic Newton’s law. Let p and p’ be the momentum of the object in (t, x, y, z) and (τ, ξ, η, ζ) systems respectively, F and F’ be the only force on the object in (t, x, y, z) along x direction and in (τ, ξ, η, ζ) along ξ direction respectively, we have:
dp/dt = F (13)
dp’/dτ = F’ (14)
Since F and F’ are the only force on the object, they must be the same, and thus we have:
dp/dt = dp’/dτ (15)
By assuming the Lorentz transformation to be valid, we would have:
dt/dτ = γ (16)
and γ is given by equation (4). This would lead to:
p = p’dt/dτ = γp’ = γmv (17)
Equation (17) looks the same as (3); therefore, it seems that we have worked out equation (3) step by step from the classic notion of momentum p = mv through the classic Newton’s Law.
Nevertheless, the above equation (17) is false because v of equation (4) and the magnitude of v of (3) are the same, which is invalid due to the fact that when an object moving at v in (t, x, y, z), it will be at rest in (τ, ξ, η, ζ) since the latter system is moving at v with regard to the former system. Hence, the seemingly most “rigorous” derivation of the expression (3) for the relativistic momentum from the basic notion of momentum is wrong.
Now if we let the object move at v1 in (t, x, y, z) and let coordinate system (τ, ξ, η, ζ) move at v with regard to (t, x, y, z), then we will have the momentum p’ in (τ, ξ, η, ζ) as
p’ = mv2 (17a)
where v2 is the velocity of the object in (τ, ξ, η, ζ) which needs to be calculated with v and v1 using Lorentz transformation for velocity when we assume the special theory of relativity to be valid, and v2 as the magnitude of v2 in general is different from v. So equation (17) will become
p = p’dt/dτ = γp’ = γmv2 (17b)
Therefore, we will not be able to get the famous relativistic momentum as defined by equation (3) from the basic notion of momentum if we actually try to follow a strict logic even within the framework of special relativity. This is not an accident but an inevitable consequence of the fact that equation (3) could never hold true according to the classic notion of momentum, despite that equation (3) is one of the foundational blocks in modern science, especially for quantum field theory.
In fact, one important consequence of defining relativistic momentum with equation (3) by requiring that the same speed v would be used for both γ and the magnitude of v in “mv” is that as long as the object’s “mv” is not zero, γ would not be 1, so equation (3) can never reduce back to the classic momentum at all.
The lack of rigorous logical connection between the relativistic momentum and classic momentum makes the relativistic momentum is a literal extension instead of a physical (logical) extension of the classic notion of momentum, which means the relativistic momentum is a new concept defined by equation (3) instead of a subset of the classic notion of momentum. Further, equation (13) or equation (14) should also be considered as new definition if p or p’ is defined by equation (3) instead of the classic “mv” because in that case they would not have the natural legitimacy endorsed by Newton’s law.
3.2. Relativistic energy is not the total energy even if v=0
The need of phasing out special relativity due to its faultiness determines that we need to redo the derivation of the mass-energy relationship. By repeating the steps of Einstein (1905b [[3]]) and replacing the relativistic Doppler Effect applied by Einstein with the non-relativistic Doppler Effect, Dai (2023 [[4]]) worked out the corrected form of mass-energy relationship as:
E = mc2/2, (18)
which tells that the total energy deviates from the famous Einstein mass-energy equation E = mc2 by a factor of ½. Therefore, even if we could accept the relativistic mass to be physically meaningful, the relativistic definition (8) (and thus (2)) for the total energy would not reduce to the correct result (18) when v=0 as it is supposed to do.
Accordingly we might have the virtual momentum of photons as:
m = 2E/c2 (19)
and the virtual momentum of photons as:
p = mc = 2E/c (20)
which deviates from equation (10) by a factor of 2.
3.3. Relativistic mass can only be calculated instead of measured
Both relativistic momentum and relativistic total energy are defined by replacing the classic mass in the relevant classic relations with the relativistic mass as shown in equations (8) and (9) while the concept of relativistic mass itself is indeed problematic.
As mentioned earlier, in order to imitate the classic dynamic equations of motion, people need to define anisotropic relativistic mass as Einstein did in 1905. Most probably due to the mismatch of the anisotropic relativistic mass and the isotropic gravitational mass, relativistic scholars finally chose the isotropic form of relativistic mass as defined in equation (7). Nevertheless, in recent decades, the majority of relativistic scholars seem to have already completely given up the idea of relativistic mass (e.g. Freeman 2019 [[5]]; Muller 2014 [[6]]).
As we know, in a gravitational field, no matter how faster it is, a horizontally free flying massive object would take a parabolic path, and the heavier is the object, the more will it deviate off the original horizontal position. Therefore, the most plausible reason for relativistic scholars to forsake the notion of relativistic mass should be the missing evidence of ever observed significant increment of mass after the particles were accelerated to around the speed of light during the countless runs in thousands of accelerators around the world for the past century.
4. Unsatisfactory Experimental Validation
Despite that the relativistic mechanics and its subfields (e.g. the quantum field theory) have been portrayed by the community of modern physics as much more advanced than Newtonian mechanics in reflecting or modeling nature, we do not have the same degree of logical confidence in the former as in the latter for the following two reasons:
1) Newtonian mechanics was empirically established while the relativistic mechanics was pretty much a literal extension of Newtonian mechanics by redefining the notions of total energy and momentum without empirical tracks during its establishment.
2) Newtonian mechanics has been extensively verified in all relevant levels and areas of engineering and technological practices in empirically direct manners while the so-called verifications of relativistic mechanics always involve the calculations using relativistic formulas.
5. Final Remarks
Despite the fundamental difference between the relativistic mechanics and Newtonian mechanics as discussed above, we often see the erroneous demonstration showing that when the speed of particle v becomes small the relativistic mechanics would reduce to Newtonian mechanics through the Taylor expansions of relativistic energy and relativistic momentum as follows [[7]]:
E = mc2 + mv2/2 + 3mv4/(8c2) + 5mv6/(16c4) + … (21)
p = mv + mv2v/c2 + 3mv4v/(8c4) + 5mv6v/(16c6) + … (22)
This is incorrect since equation (3) or (22) can never reduce to p = mv except for the single point p = 0 when v = 0. Therefore equation (22) only acts as a pure misleading mathematical illusion. As for equation (2) or (21), although they do reduce to E = mc2 when v = 0, they would exaggerate the total amount of E (which is mc2/2) by a factor of 2.
However, despite that equation (1) is a sheer mathematical formula without much real physical significance as demonstrated in this writing, surprisingly physicists did reach some meaningful conclusions starting from that relationship as Dirac did in 1928 (Brown, 21st Century [[8]]; Dirac 1928[[9]]). There are many reasons for this apparently lucky coincidence; for example, for the case of single electron orbiting the nuclear in the Hydrogen atom which Dirac was interested in his 1928 writing, the Lorentz factor γ would be (almost) a constant since the magnitude of the speed of the particle in question stays constant, and thus the relativistic momentum is indeed conserved as long as the real momentum in classic sense is conserved. This is one example of the kind showing that physics could be full of lucky coincidences. Nevertheless, while we should take the advantage of those lucky coincidences, we should not assume the haphazard approaches leading to those coincidental results to be universally valid without careful scrutiny.
References
Brown, K.S. (21st Century) “The Dirac Equation”. Retrieved from: https://www.mathpages.com/home/kmath654/kmath654.htm
Dai, R. (2022). “The Fall of Special Relativity and The Absoluteness of Space and Time”. Retrieved from: https://wp.me/p9pbU7-dX
Dai, R. (2023). “Modifying Mass-Energy Relationship”. Retrieved from: https://wp.me/p9pbU7-gk
Dirac, P. A. M. (1928). “The Quantum Theory of the Electron”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624. Retrieved from: https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.1928.0023
Einstein A. (1905a). “On the Electrodynamics of Moving Bodies”. Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891, 1905, translations by W. Perrett and G.B. Jeffery. Retrieved from: https://www.fourmilab.ch/etexts/einstein/specrel/www/
Einstein, A. (1905b). “Does the Inertia of a Body Depend Upon Its Energy-content?”. Retrieved from: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf
Freeman, P. (2019). “Relativistic Mass or Rest Mass?”. OAPT Newsletter. Retrieved from: http://newsletter.oapt.ca/files/relativistic-mass-or-rest-mass.html
Muller, A. Derek. [Veritasium]. (2014) “Demystifying Mass ft. Sean Carroll” [Video]. YouTube. https://www.youtube.com/watch?v=n_yx_BrdRF8
Wikipedia (2023). “Relativistic mechanics”. Retrieved from: https://en.wikipedia.org/wiki/Relativistic_mechanics. Last edited on 18 January 2023, at 02:51 (UTC).
[[1]] Einstein A. (1905a) “On the Electrodynamics of Moving Bodies”. Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891, 1905, translations by W. Perrett and G.B. Jeffery. Retrieved from: https://www.fourmilab.ch/etexts/einstein/specrel/www/
[[2]] Dai, R. (2022) “The Fall of Special Relativity and The Absoluteness of Space and Time”. Retrieved from: https://wp.me/p9pbU7-dX
[[3]] Einstein, A. (1905b). “Does the Inertia of a Body Depend Upon Its Energy-content?”. Retrieved from: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf
[[4]] Dai, R. (2023). “Modifying Mass-Energy Relationship”. Retrieved from: https://wp.me/p9pbU7-gk
[[5]] Freeman, P. (2019). “Relativistic Mass or Rest Mass?”. OAPT Newsletter. Retrieved from: http://newsletter.oapt.ca/files/relativistic-mass-or-rest-mass.html
[[6]] Muller, A. Derek. [Veritasium]. (2014) “Demystifying Mass ft. Sean Carroll” [Video]. YouTube. https://www.youtube.com/watch?v=n_yx_BrdRF8
[[7]] Wikipedia (2023) “Relativistic mechanics”. Retrieved from: https://en.wikipedia.org/wiki/Relativistic_mechanics. Last edited on 18 January 2023, at 02:51 (UTC).
[[8]] Brown, K.S. (21st Century) “The Dirac Equation”. Retrieved from: https://www.mathpages.com/home/kmath654/kmath654.htm
[[9]] Dirac, P. A. M. (1928). “The Quantum Theory of the Electron”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624. Retrieved from: https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.1928.0023
The World of Science and Science fiction of Ron Dai 
The current version of this post is the upgraded version of the original “The Faulty System of Relativistic Momentum and Energy”. This upgrade is made mainly because my new finding that the real mass-energy relationship should be E = mc2/2 instead of the famous E = mc2, as presented in “Modifying Mass-Energy Relationship” (https://wp.me/p9pbU7-gk).
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