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As will be shown in this short easy, the Einstein-Planck formula—the opening formula of quantum mechanics—and the Law of Conservation of Energy is incompatible when we examine motions in both macroscopic and microscopic levels. We might even not be able to resolve the incompatibility in question by extending Planck’s blackbody experiment or some photoelectric effect experiment to involve the red/blueshift caused by the change of the speed of source or receiver. This is because the fundamental physical or philosophical difficulty encountered here is that the kinematic motion of the observer is not supposed to be able to change the real physics of another unrelated object.

Keywords: Einstein-Planck formula, photon, redshift, blueshift, random relative motion, energy conservation

1. Planck’s Blackbody Experiment and Einstein-Planck Formula

In 1900, based on his blackbody radiation experiments, Max Planck proposed that the radiant energy of light is proportional to its frequency (Wikipedia, Planck postulate)^[[1]]:

E = nhν (1)

where E is the light energy, n is a positive integer, h is Planck’s constant, and ν is the frequency of light.

In 1905, based on Equation (1), Albert Einstein proposed the concept of light quanta which was later known as photons, and stated that the energy of each photon is:

E = hν (2)

In a sense, Einstein’s theory of light quanta officially launched the study of micro world by treating physical quantities as quanta, which is pretty much what quantum mechanics does; accordingly Einstein was awarded the 1921 Nobel Prize in Physics for his explanation of the photoelectric effect based on his interpretation of light as quantum particles.

Equation (2) is then known as the Einstein-Planck formula.

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To better appreciate the discussion of this writing, we need first to clarify some fundamentals concerning the speed of light.

1.1. The revised postulate of speed of light in vacuum

In order to correct the faulty first postulate of special relativity, Dai (2022c)^[ [2]^] proposed the revised postulate of speed of light in vacuum as follows:

[The speed of light c in vacuum is constant with respect to the source of the light if the source is moving at a constant velocity. In case the source is not an inertial system, c will be constant as calculated with respect to the source at the moment it is emitted. The value of the constant speed of light is given by the Maxwell formula c = 1/(ε˳µ_o)^1/2where ε_o is permittivity in vacuum and μ_o is permeability in vacuum. Here the term vacuum refers to the complete emptiness and free of any externally imposed energy, and thus the vacuum itself would not have any impact on the speed of light.]

The original form of the revised postulate of speed of light in vacuum when Dai first proposed it in 2022, it took the following form:

[The speed of light is constant in vacuum that is not attached to any specific material object and is always the same to the source of the light in vacuum. The value of this constant speed of light is given by the Maxwell formula c = 1/(ε˳µ_o)^1/2where ε˳ is permittivity in vacuum and µ_o is permeability in vacuum.]

Obviously, a defect in the 2022 postulate is that it lacks a necessary restraint to the source of the light which is the requirement of being inertial. Therefore, the revised postulate of speed of light in vacuum is modified as shown in this writing by adding the requirement that the source needs to be an inertial object, which is what we see at the beginning of this section.

1.2. The impact of gravity upon the speed of light

The influence of gravity (such as the Sun's gravity) on the speed of light is generally considered negligible (unless in extreme condition like blackhole) with all theoretical schools. Relativity merely generalizes and absolutizes the constancy of the speed of light, assuming it is not only unaffected by any gravity but also absolutely constant for any observer (though Einstein himself provided a formula for calculating the change in light speed with gravity in his 1912 paper).

2. The Einstein-Planck Formula and Random Red/Blue Shifts

Suppose there are two objects, A and B, moving randomly in a vacuum. Object A emits a beam of light that hits object B. Let the frequency of the light observed by an observer fixed on A be ν₁, and the frequency observed by an observer fixed on B be ν_2.Because A and B move randomly in a vacuum, redshift or blueshift will generally occur so that ν₁ ≠ ν₂.

In the open universe, there are different scenarios for two random objects A and B doing relative movements, but all fall into the following three categories:

(1) Both A and B are inertial. When both A and B are inertial, despite the frequency of the light shot by A looks different in A and B due to the red/blueshift, when looking only in A or only in B, the frequency does not change, i.e. it stays as ν₁for A and ν₂for B until it hits B.

(2) The receiver B is non-inertial. In case the receiver B is non-inertial, suppose the source A is doing inertial motion and B is moving towards A with an acceleration a. According to the above modified revised postulate of speed of light in vacuum, the speed of light with respect to A is a constant c, but varies with respect to B and thus the calculated the red/blue shifted frequency will constantly change as long as a is not zero. This will also apply to the case when both A and B are doing accelerated motion, even when the relative acceleration between A and B is zero.

(3) Only the receiver B is inertial. If B is inertial, since the speed of light is constant with respect to B after leaving A according to the above modified revised postulate of speed of light in vacuum, the observer on B will not see any change of the color of light; however, since A is now not doing inertial motion, the energy of light beam will change after it leaves A.

3. Energy Imbalance between Macroscopic and Microscopic Levels

In section 2 we went through different scenarios of Doppler Effect red/blueshifts happening in the universe due to the perpetual relative random movements of all objects. In all those scenarios, the frequency of the same beam changes depending on whether observed on A or observed on B, i.e. ν₁ ≠ ν₂, no matter if any of A and B is inertial or not. For the ease of discussion, let’s examine the least violent scenario in which both A and B are in inertial motion because the violation of the known physical law would be more serious in other scenarios.

Now let’s extend our investigation into microscopic level to involve things like atomic electron transition (Wikipedia, Atomic electron transition) ^[[3]]. Suppose the energy released at the microscopic level for emitting the light beam toward B is Ea, then according to formula (2) we have:

Ea = βhν₁ (3)

where β is the number of photons in the beam. Now suppose the energy of the beam is Eb when examined from B, then we have:

Eb = βhν₂ (4)

Since we have ν₁ ≠ ν₂, we must have:

Ea≠Eb (5)

This means that the energy of the light beam measured by the observer B cannot balance the energy released at the microscopic level for emitting the light beam.

4. The Root Cause for the Incompatibility

The incompatibility between Einstein-Planck formula (2) and the law of energy conservation is profoundly rooted both mathematically and philosophically (physically).

The existing human theoretical framework concerning energy is constructed on the balance between the changes of the velocity-determined kinetic energy and the force-determined potential energy, i.e. the conservation of energy, at both macroscopic and microscopic levels, while Einstein-Planck formula (2) does not directly involve the expression of either kinetic energy or potential energy but only indirectly related to the kinetic energy through the red/blueshift caused by the kinematic movements of the source and the receiver of the light beam. But on the other hand, the kinematic relative movement between A and B cannot change the energy released by A at the microscopic level.

5. Discussion

The Einstein-Planck formula (2) is incompatible with the law of conservation of energy, which is not accidental but rather substantial and thus cannot be remedied by any pure mathematical tricks without changing the fundamental structure of the system that involves both items.

Furthermore, we might even not be able to resolve the incompatibility in question by extending Planck’s blackbody experiment or some photoelectric effect experiment to involve the red/blueshift caused by the change of the speed of the source or the receiver. This is because the fundamental physical or philosophical difficulty encountered here is that the kinematic motion of the observer is not supposed to be able to change the real physics of another unrelated object.

The so-called Galilean relativity has been wrongly extended by the modern community of physics as “the same laws of physics apply in all inertial frames of reference, regardless of one frame's straight-line, constant-speed motion relative to another” (LibreTexts)^[[4]], while Galileo was only interested in the kinematic motion. This misinterpretation of ancient knowledge became part of the source for special and general relativities in early 20^th century.

6. Final Remark

A few years ago, Dai (2021) ^[[5]^]pointed out that suppose there are two objects A and B moving in random relative motion within a vacuum. If object A emits a beam of light that hits object B, without the intervention of any external force, the energy of light traveling in the vacuum is not conserved in general. However, that analysis overlooked one thing: as mentioned earlier, the conclusion of that 2021 analysis does not hold at the macroscopic level for motion within purely inertial systems. Nevertheless, it is not completely meaningless because it was the first time that the incompatibility between the quantum energy system and the general law of energy conservation was brought to light.