Rongqing Dai
Abstract
The intrinsic relationship between mass and energy is undoubtedly one of the most important philosophical enlightenments for the mankind in human history, which has been embodied with the famous Einstein equation E = mc2 for thepast century. Although Einstein derived this famous equation within the framework of special relativity, some others had also arrived at the similar before special relativity was proposed; however, both E = mc2 and other similar ones all exaggerated the contribution of mass to the total energy proportionally. In the meantime, we have also learned that special relativity is wrong because Lorentz length contraction and time dilation are physically unrealistic. Nevertheless, the approach employed by Einstein in the derivation of E = mc2 is more advantageous than others for it paves the path to the final correct form of mass-energy relationship after special relativity is phased out. In this writing, by replacing the relativistic formula of Doppler Effect used by Einstein with the non-relativistic formula of Doppler Effect, the approach employed by Einstein for E = mc2 is applied to work out the correct form of mass-energy relationship which is E = mc2/2.
Keywords: Mass-Energy relationship, Relativistic, Non-relativistic, Doppler Effect, Speed of Light
1. Introduction
One of the greatest groundbreaking knowledge advances around the world at the turn of 20th century was the relationship between mass and energy as revealed by the famous paper “Does the Inertia of a Body Depend Upon Its Energy-content?” of Einstein (1905a [[1]]), which dictates the following famous equation:
E = mc2 (1),
where c is the speed of light in vacuum, m is the mass of an object, and E is the total energy of the object.
When Einstein worked out relation (1), he employed the following formula for accounting the so-called relativistic Doppler Effect for the energy of light:
(2),
where l is the energy possessed by a ray of light in co-ordinates (x, y, z), l* is the energy of the light in co-ordinates (ξ, η, ζ) having its origin of co-ordinates in motion along the axis of x with the velocity v,and Φ is the angle between the ray and the axis.
However, the relativistic formula (2) is wrong because it depends on the postulate of constancy of speed of light in vacuum (Einstein 1905b [[2]]) which is wrong due to the fact that the Lorentz assumptions of length contraction and time dilation are physically invalid (Dai 2022 [[3]]). Therefore, we need to redo the derivation of Einstein by replacing (2) with a non-relativistic formula.
2. Re-deriving the Mass-Energy Relation with Einstein’s Approach
Now let’s repeat the steps of Einstein (1905a) by replacing the relativistic formula of Doppler Effect (2) with the following non-relativistic formula of Doppler Effect:
l* = l {c/(c – v)} (3)
Here angle Φ is taken to be zero for simplicity. Let there be a stationary body in the system (x, y, z) with energy E0 which takes the valueof H0 in the system (ξ, η, ζ). Let this body send out two rays of light along the axis of x in opposite directions simultaneously, each with energy L/2 measured relatively to (x, y, z). If we call the energy of the body in both systems after the emission of light E1 and H1 respectively, then by employing the formula (3) we obtain:
E0 = E1 + L/2 + L/2 = E1 + L (4)
H0 = H1 + {c/(c – v)}L/2 + {c/(c + v)}L/2
= H1 + c2L/(c2 – v2) (5)
Let Ki (i = 0, 1) be the kinetic energy of the body in the system (ξ, η, ζ) before and after the body emitting light. Since velocity v along the x-axis only affects the kinetic energy of the body without impacting its potential energy, the change in the kinetic energy of the body should be:
K0 – K1 = H0 – E0 – (H1 – E1)
= c2L/(c2 – v2) – L
= v2(L/c2)/(1-v2/c2) (6)
Taking Taylor expansion of 1/(1-v2/c2) and omitting higher-order terms, we get:
K0 – K1 = v2(L/c2) (7)
Since the velocity v remains constant throughout the process (according to the conservation of momentum), we have:
K0 – K1 = Δm v2/2 (8)
where Δm is the mass lost by the body after emitting the light rays. From (7) and (8) we get:
Δm = 2L/c2 (9)
Rewriting (9) into more familiar forms, we have:
m = 2E/ c2 (9a)
or
E = mc2/2 (9b)
Equations (9a) and (9b) are the correct forms of the mass-energy relationship, which shows that only half of the familiar mc2 accounts for the total energy E.
2.1. Virtual luminal mass
Equation (9) relates the lost mass of the body and the energy of the emitted light, and thus we might view Δm as the total virtual mass of the emitted photons. This would not only get us a virtually balanced account sheet for total mass, but more importantly would help us to better comprehend the physical significance of the mass-energy relationship E = mc2/2.
3. Physical Meaning of E = mc2/2
Literally, the relation E = mc2/2 tells that the total energy of an object is its kinetic energy when all its mass imaginarily moves at the speed of light. In a more realistic sense, we might imagine a perfect nuclear chain reaction that leads an object completely turned into pure energy, i.e. pure massless photons, and let’s visualize in our mind the scene that the object would gradually disintegrate into smaller and smaller parts, and finally completely “evaporated” or sublime into pure photons. During that subliming process, the potential energies contained within that object would gradually turn into kinetic energies of those small parts, and thus the masses of all small pieces would gradually diminish while their kinetic energies gradually increment.
During this whole process, energy is conserved, and thus the total energy remains as E = mc2/2, and the total mass remains as m before some part of it begins to turn into pure energy. Finally, when all the mass turns into pure energy of massless photons, the total energy of those photons would still be E = mc2/2 and the total virtual mass of the photons would still be m.
3.1. The Upper and Lower Limits of Light Frequency and the Virtual Number of Photons
From the famous Planck-Einstein formula we have:
E = hf (10),
where E is the photon energy, h is the Planck constant and f is the frequency of light. From (9b) and (10) we have:
Nhf = mc2/2 (11),
where f is the (average) frequency of the photons when all the mass sublimes into pure photons, and N is the total number of the photons.
Now since mc2/2 is a finite number and N cannot be infinity, we know that there must be an upper limit and a lower limit of the frequency f.
Suppose all the photons during the subliming process are of the same frequency f, then we have:
fmax = mc2/2Nmin (11a)
fmin = mc2/2Nmax (11b)
where fmax / fmin are the upper and lower limits of the final frequencies and Nmin / Nmax are the corresponding lower and upper limits of the number of photons in that virtual subliming process.
3.2. Zero luminal potential
The relationship E = mc2/2 confirms the century long knowledge that the potential energy of light would be nil.
4. Speed Limit
In the special theory of relativity, the Lorentz factor γ = 1/(1-v2/c2)½ determines that the speed of light in vacuum c would be the upper limit of all speeds in nature. This would no longer be considered valid after we phase out special relativity. From the Maxwell formula of speed of light (e.g. Wikipedia (a) [[4]]):
c = 1/( ε₀μ₀) ½ (12)
we learn that the speed of light in vacuum is limited by the values of the permittivity of free space ε₀ and the permeability of free space μ₀, instead of any intrinsic energy barrier or space and time hurdle.
While the mass-energy relationship E = mc2/2 renders the upper limit of the kinetic energy when a body of mass m completely turning into pure photons, it does not dictate how fast a body of mass m can move when accelerated by external power. Further, the speed of light in vacuum c is not constant to all observers either (Dai 2022).
4.1. Superluminal Doppler Effect
Although it is practically difficult to observe anything from a superluminally moving coordinate system, it should not stop us from doing so in imagination. However, with the relativistic Doppler Effect formula (2) even our imagination would be limited because of the physical meaninglessness of a mathematically imaginary frequency of light.
Now with the non-relativistic Doppler Effect formula (3), we can see that when v approaches c the observed energy become infinite, and when v surpasses c the light wave would apparently propagate in the opposite direction. The “observed infinite energy” is forbidden in the special theory of relativity because of its first postulate (i.e. the principle of relativity), but now we would not have this problem after phasing out the special theory of relativity.
5. Paradoxical Level Inconsistency and Its Resolution
The discussion in this section does not depend on the specific form of the mass-energy relation, and thus we might conceptually put it in a general form of function:
E = f(mc2) (13)
Now since E represents the total energy of an object, it would encompass both kinetic and potential energies; accordingly, we would suppose the kinetic energies of all the particles composing a piece of rock down to quarks and leptons would contribute to the total mass of that piece of rock. If this logic holds sound, we might expect that the mass of particles in the accelerators would increase tremendously as assumed by early relativistic scholars. However, in recent decades most relativistic scholars seem to have abandoned the idea of mass created by kinetic motions (or the relativistic mass) for reasons (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 the object is, the more deviate it will off the original horizontal position. Hence, even though particles in accelerators are all of very little mass and all fly extremely fast, if the tremendous increment of mass is real, it would not be a mission impossible to observe its effect after nearly a century’s of countless runs in thousands of accelerators around the world.
Therefore, the main 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.
Hence, we are now facing such a logical level inconsistency: we assume that the kinetic energies of subatomic particles within an object would contribute to the total mass of the object, but we deny the contribution of the kinetic energy of a macroscopically moving object to its mass.
We know that a universal scientific law about nature should be applicable at all levels of nature. For example, we cannot assume that the Newton’s law is only applicable to cars but not to the parts of cars. Therefore, we need to resolve the abovementioned level inconsistency in order to prove the theory valid.
One simple way out is to assume that only potential energy contributes to mass. But then even without a perfect nuclear chain reaction, when the efficiency of the nuclear chain reaction increases, as demonstrated in the above virtual subliming thought experiment, because of the conservation of energy, we should expect that the final released energy would approach the total energy expressed in (13) which encompasses both kinetic and potential energies.
Another simple way to avoid the difficulty might be to name E of (13) as the binding energy (e.g. Wikipedia (b) [[7]]), which refers to the smallest amount of energy required to disassemble a system of particles into individual parts, without distinguishing between kinetic energy and potential energy. But the drawback of this trick is obvious: the notion of binding energy is closely bound to the notion of binding force which is not directly related to kinetic energy.
One more way out seems to be to assume that the existing quantum theory has overestimated the kinetic energy of the subatomic particles in an object. But if we accept this view, then we need to radically revise how potential energy acts within a macroscopic body. This is because based on the existing microscopic (quantum) model, the subatomic particles need to have enough kinetic energies to balance the potential energies; otherwise, all the subatomic particles would collapse to a big chunk and no more potential energy could exist either.
Nevertheless, we might resolve the above paradoxical level inconsistency with this verdict: the total energy E in equation (13) only accounts for the kinetic energy and potential energy that participate in binding all the ingredients together into an object.
The essential difference between the subatomic particles in a macroscopic body and the particles accelerated in accelerators is that the kinetic energy of those component particles is in dynamic balance with the potential energy of the binding forces, while the kinetic energy of the particles in accelerators do not contribute to the dynamic balance for maintaining the integrity of the macro system. Accordingly, only those kinetic energies that are involved in maintaining the potential energies of the binding forces contribute to the mass of the system. For example, if we manage to have a high precision measurement of the mass of the Sun-Earth system, we would find that the kinetic energy of the Earth moving around the Sun does contribute to the total mass of the system.
6. Final Remark
The idea of converting mass into energy could be traced back to decades before Einstein (1905a) published his landmark paper “Does the Inertia of a Body Depend Upon Its Energy-content?” (Ricker 2015 [[8]]); nevertheless, similar to Einstein’s work, they all exaggerated the contribution of mass to the total energy proportionally. The approach employed by Einstein to reach E = mc2 is more advantageous than the others because it paves the path to the final correct form of the mass-energy relationship E = mc2/2 as demonstrated in this writing.
[[1]] Einstein, A. (1905a). “Does the Inertia of a Body Depend Upon Its Energy-content?”. Retrieved from: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf
[[2]] Einstein, A. (1905b). “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/
[[3]] Dai, R. (2022) “The Fall of Special Relativity and The Absoluteness of Space and Time”. https://www.researchgate.net/publication/363582341_The_Fall_of_Special_Relativity_and_The_Absoluteness_of_Space_and_Time
[[4]] Wikipedia (a). “Maxwell’s equations”. Retrieved from: https://en.wikipedia.org/wiki/Maxwell%27s_equations. Last edited on 21 April 2023, at 19:36 (UTC).
[[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 (b). “Binding energy”. Retrieved from: https://en.wikipedia.org/wiki/Binding_energy. Last edited on 15 October 2022, at 18:17 (UTC).
[[8]] Ricker, HH. (2015). “The Origin of the Equation E = mc2”. Retrieved from: http://www.dankalia.com/delloro/gravity-cone/The%20Origin%20of%20the%20Equation%20E%20=%20mc2.htm
The World of Science and Science fiction of Ron Dai 
An updated version of this article has been posted at:
The reason of update is to add some more detailed discussion on the mass of photon.
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