Discord between Doppler Effect of Light and Photon Energy

Rongqing Dai

Abstract

It has been noticed by the author since 2021 that the existing theoretical system of calculating photon energy (and thus energy of light beams) is not compatible with the general theoretical framework of energy conservation when Doppler Effect is involved in the macroscopic motion of light beams. This anomaly in the energy-related theoretical system is not accidental but rather substantial and thus cannot be remedied by any pure mathematical tricks or by extending the known theoretical framework or experimental approaches in a simple way. This writing summarizes the typical scenarios that would manifest the anomaly in definitive ways. Within the existing theoretical framework, it is not clear at all how we can resolve the said anomaly, and thus we might need a so-called new physics to handle the issue.

Keywords: Photon, Energy, Doppler Effect, Light Beam, Violation of Conservation

1. Introduction

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 = nhf                                                            (1)

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

In 1905, based on equation (1), Albert Einstein (1905)^[[2]^] proposed the concept of light quanta which was later known as photons, and stated that the energy of each photon is:

E = hf                                                  (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 (Wikipedia, Photoelectric effect)^[[3]] based on his interpretation of light as quantum particles (quanta). Equation (2) is then known as the Einstein-Planck formula.

1.1. Discord between Doppler Effect and photon energy

From the above brief introduction we can see without any doubt that equation (2) was established through a logically sound empirical process with careful and diligent efforts, which offers a convincing explanation for its success in extensive applications over the past more than a century’s time. Similarly, we also have another two well established and fully verified foundational theoretical systems in the domain of physics, which are the general framework of energy conservation and the knowledge of the Doppler Effect.

However, as Dai has demonstrated since 2021, in real life scenarios for which we need to apply both equation (2) and Doppler Effect formula, the energy conservation law will be violated (Dai 2021, 2025, 2026) ^[[4]^,[5]^,[6],[7]].

In this writing we will look into the violation of the energy conservation caused by the discord between the Doppler Effect and equation (2) from different angles.

2. Augmentation and Reduction of Energy of Photon Bouncing between Two Moving Mirrors

Suppose we have a pair of mirrors A and B of extremely high quality surfaces apart from each other at a distance L; mirror A is stationary and mirror B moves with respect to A at a speed V << c. Now let’s shoot a pulse of monochromatic laser beam from A to B at time t₀ for a short interval of ∆t so that the beam will bounce between A and B for as long as its photons are not absorbed by the surfaces completely.

Let’s denote the frequency of the light beam from A to B as f_AB and the frequency of the light beam from B to A as f_BA, the frequency of the light beam received by A from B as f_AR and the frequency of the light beam received by B from A as f_BR.

2.1. Three relevant laws

Given that physical laws are those assumed to be deterministic rules that people summarize from commonly observed phenomena, some of them being explained to a certain extent (such as Boltzmann's explanation of entropy increase), but many others being just descriptions of patterns without further explanation, we can also call them postulates or hypotheses or principles.

The discussion in this paper is all based on three basic laws (or postulates, principles) of light propagation, some of which are often ignored by researchers.

2.1.1. The revised postulate of speed of light in vacuum

The first of these three laws (or postulates or principles) is based on the well-known Maxwell’s law of the speed of light, which states that the speed of light in vacuum is constant. However, although Maxwell’s law of the speed of light is mathematically based on a complete set of Maxwell’s equations, its philosophical presentation is not rigorous. This is why it became the second postulate of special relativity.

A few years ago, Dai (2022)^[[8]] restated the Maxwell’s speed law in a philosophically more rigorous way and called it the revised postulate of speed of light in vacuum, stating that the so-called constant speed of light in vacuum refers to its constancy with respect to the light source, with the implication that the light source in the revised postulate should be inertial. In 2026, Dai further made the “inertial” requirement of the source to be explicit in the revised postulate (Dai 2026)^[6].

2.1.2. Frequency law of reflection at the interface of two media

When monochromatic light is reflected from an interface, its frequency remains unchanged. For visible light, this means that the color of monochromatic light remains unchanged.

2.1.3. Huygens’ principle of reflection

When a light wave encounters a reflecting surface, such as a mirror, each point on the wavefront behaves as a source of secondary wavelets, which makes the surface of the mirror as the source of the reflected beam.

2.2. The Math

When calculating the Doppler Effect we need to separately account for the scenario of redshift and the scenario of blueshift.

2.2.1. For redshift

Assume that mirror B moves away from mirror A with a constant speed V.

According to the Doppler Effect formula we have:

When the beam is moving from A to B: f_BR = ((c-V)/c) f_AB                      (3)

When the beam is moving back from B to A: f_AR = ((c-V)/c) f_BA                           (4)

Based on (3) and (4) we know, for a single round of light moving from A to B and then bouncing back to A, we have

f_AE = ((c-V)/c)²f_AI                                                                     (5)

where f_AE is the end frequency of the round of bouncing movement, and f_AI is the initial freuqncy of the round of bouncing movement, when observed by A. And at the end of the n^th bouncing round, we have

f_AE = ((c-V)/c)²ⁿf_AI                                                                   (6)

2.2.2. For blueshift

Assume that mirror B moves towards from mirror A with a constant speed V.

According to the Doppler Effect formula we have:

When the beam is moving from A to B: f_BR = ((c+V)/c) f_AB                     (7)

When the beam is moving back from B to A: f_AR = ((c+V)/c) f_BA                          (8)

Based on (7) and (8) we know, for a single round of light from A to B and then bounced back to A, we have

f_AE = ((c+V)/c)²f_AI                                                                    (9)

At the end of the n^th bouncing round, we have

f_AE = ((c+V)/c)²ⁿf_AI                                                                   (10)

Because c>>V, for a not very big distance L, within a very short time period, n will quickly grow to a big number to make the color of the beam to change significantly.

Now let’s denote the initial energy of one photon as E_i, from equation (2) we have:

E_i = hf_AI                                                                                   (11)

Let’s denote its energy of one photon at the end of the n^th round of bouncing as E_n. Then for redshift we have according to (2) and (6):

E_n = hf_AE = h((c-V)/c)²ⁿf_AI                                                          (12)

For blueshift we have according to (2) and (10):

E_n = hf_AE = h((c+V)/c)²ⁿf_AI                                                       (13)

From equations (11) and (12) we can see that for redshift we have:

E_n = ((c-V)/c)²ⁿ E_i                                                                     (14)

From equations (11) and (13) we can see that for blueshift we have

E_n = ((c+V)/c)²ⁿ E_i                                                                    (15)

Therefore, within a very short time period, E_n will be significantly different from E_i for both redshift and blueshift.

Suppose the surface quality of both mirrors is superbly great and the initial distance between mirrors A and B is 10 m, and the speed of B is 1 m/s, and the initial color of redshift is blue (6.66 × 10¹⁴Hz) and the initial color of blueshift is red (4.62 × 10¹⁴ Hz), then within 3 seconds, the color of the beam in both redshift and blueshift tests will turn to green (5.45 × 10¹⁴ Hz).

2.3. Conclusion

Equations (14) and (15) tell us that the conservation of energy is categorically violated for the bouncing movement of the light beam between a pair of relatively moving mirrors, which cannot be blotted out with any mathematical tricks like transformation of frame of reference.

2.3.1. Cause

The ostensible reason for the violation of energy conservation in the above analysis is the combination of four factors: 1) the energy of photon is calculated only with the frequency of the light beam as shown in equation (2); 2) the postulate that the speed of light is constant in vacuum with respect to the source of light; 3) the frequency of light remains unchanged after the reflection; 4) the Huygens principle of reflection.

The combination of the above four factors determines that the energy conservation will be violated for the bouncing movement of a light beam between a pair of relatively moving mirrors.

2.3.2. Discussion

A straightforward remedy for the above problem seems to be to change the way of calculating the photon energy, which is currently done only with equation (2), by taking into consideration of the change of the speed of light. The reasons for doing this are quite simple:

1) By replacing the second postulate of the special relativity with the revised postulate of speed of light in vacuum for the calculation, the speed of light is no longer constant to all the reference frames as claimed by special relativity;

2) As mentioned earlier, the empirical bases of equation (2) are the blackbody experiment and the photoelectric effect, while both of them do not involve the kinematic motions of the source and receiver of light. Obviously, in the bouncing light case that we are dealing with at the moment, the kinematic motion of mirror B matters a lot and thus it makes sense for us to suggest taking into account the impact of that motion upon the energy of the photon. In some sense, this is like treating light more of a duality of wave and particle as we call it by taking into consideration of both its wavy nature (frequency) and its particle nature (speed based kinetic energy), instead of treating it as wave or particle only.

However, if we decide to do so, we will soon find that it might make the matter worse, not better. This is because we have no idea how to mix these two things together properly. Historically, people have calculated the energy of light either by equation (2) or by its kinetic energy when light was treated as particles, not by both at the same time. Now suppose we count the Doppler Effect for the propagation in vacuum and count the kinematic variation for the reflection on the mirror surface, then we just simply double the imbalance of energy rather than blot it out.

Even if we can somehow manage to balance the sheet of energy change by taking into account the kinetic energy of the photon particle, we will still see (with bare eyes) the color change of the light beam, as determined by equations (6) and (10), as long as the surface quality of mirrors A and B are good enough. This may cause energy imbalance somewhere down the road.

Another intuitive solution is to extend the Planck’s blackbody experiment and some photoelectric effect experiment to involve the red/blueshift caused by the speed of the source or the receiver of the light beam, so that we might empirically obtain something similar to equations (1) and (2) with Doppler Effect being taken care of to resolve the energy imbalance issue in the open universe.

But similar to the challenge mentioned earlier with the attempt of mixing the wavy account and the particle account theoretically, with the above experimental idea we still cannot solve the problem that the change of color of the beam when bouncing between a pair of relatively moving mirrors are only tied to the Doppler Effect which is not directly related to the mechanism of the blackbody emission or the mechanism of photoelectric effect.

3. The Random Energy Loss and Creation in a Nonexpanding Universe

Now let’s look into the issue of the random energy loss and creation in the universe even if it is not expanding, as discussed Dai (2021)^[4].

Let’s say John shoots out a beam of blue laser from the rear window of a spaceship A, he does not know there is another spaceship B at the same speed following them far away, and Jack on spaceship B receives that blue light. Then Jack slows down spaceship B, and John shoots another beam of blue light, which is received by Jack again; but this time, that beam of light is not so blue due to the redshift effect. Therefore, for the second beam of light, some energy is lost when it travels from John to Jack, and that loss of energy is not transferred to anything, but just purely lost!

The above statement from my 2021 paper was criticized by relativistic critics who claimed that after Jack slowed down spaceship B, he was no longer in the same inertial system with John, and within the frame of reference of spaceship B, the second beam of light never experienced any energy loss as long as the speed of spaceship B did not change during the movement of the beam from A to B.

However, the reality is not so simple..

To rebuke the above criticism, we only need to put a mirror on spaceship B and assume that the second beam hits on the mirror and reflected. Then from the discussion of last section we know that the color change of the light beam, and thus the change of total energy in the universe, cannot be cancelled out by switching the frame of reference from A to B.

Therefore, based on Dai’s discussion of 2021, we know that the Doppler Effect of light will cause random energy loss and creation in the universe even if it is not expanding.

4. Incompatibility of Energy Changes in Macroscopic and Microscopic Domains

Now let’s extend our investigation into microscopic level to involve things like atomic electron transition (Wikipedia, Atomic electron transition) ^[[9]]. In the above example, let’s suppose when John shoots a beam of light towards spaceship B the energy released at the microscopic level for emitting the beam is E_a, and then according to equation (2) we have:

E_a = βhf₁                                                          (16)

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

E_b = βhf₂                                                          (17)

Since we have f₁ ≠ f₂ due to the Doppler Effect, we must have:

E_a ≠ E_b                                                              (18)

This means that the energy of the light beam measured by Jack on spaceship B cannot balance the energy released at the microscopic level for emitting the light beam. Similarly, if the beam is absorbed by an object on spaceship B, then the energy transferred into the microscopic structure of that object will be E_b, which is different from the energy released on system A.

From this example we see that we will have energy creation through blueshift or energy loss through redshift in the universe when we compare the energy released on A with the energy absorbed on B.

Here an important point that often gets people (especially relativistic scholars) confused is that the macroscopic motion of an object cannot change the microscopic quantum process of another unrelated object. Consequently, the motion of B relative to A cannot change how much energy will be released for emitting a light beam on A, and likewise the motion of A relative to B cannot change the energy variation at the atomic level of any object on B. Accordingly, on the atomic level in the open universe, we have E_a lost from A and E_b gained on B, and the difference ∆E = E_b – E_a is the energy created or destroyed by the Doppler Effect alone.

5. Discussion

Behind all other causes mentioned earlier, there are two types of root causes that we can say for sure for the incompatibility problem of energy calculations as discussed above:

1) The existing human theoretical framework of energy conservation is constructed based on the balance between the changes of the velocity-determined kinetic energy and the force-determined potential 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 Doppler Effect induced by the kinematic movement of the source and the receiver of the light beam.

2) The kinematic motion of an observer cannot change the real physics of another unrelated object.

6. Final Remarks

Obviously, the anomaly in the energy-related theoretical system as discussed in this writing is not accidental but rather substantial and thus cannot be remedied by any pure mathematical tricks or by somehow integrating the wavy energy expression in equation (2) with the classical kinetic energy of particles of light in a simple manner, without changing the fundamental structure of the system. Furthermore, we might not even be able to resolve the incompatibility in question by extending Planck’s blackbody experiment or some photoelectric effect experiment.

As a conclusion, within the existing theoretical framework, it is not clear at all how we can solve the above problems, and thus we might need a so-called new physics to handle the issue.

References

[[1]]Wikipedia, Planck postulate. Retrieved from https://en.wikipedia.org/wiki/Planck_postulate. Last edited on 4 August 202 at 12:15 (UTC).

[[2]] Einstein, A. (1905). On a Heuristic Point of View about the Creation and Conversion of Light. translated from German by Wikisource. Retrieved from https://en.wikisource.org/?curid=59468. Last edited on 6 January 2021, at 13:46.

[[3]]Wikipedia, Photoelectric effect. Retrieved from https://en.wikipedia.org/wiki/Photoelectric_effect. Last edited on 10 April 2026, at 16:03 (UTC).

[[4]]Dai, R. (2021). The Random Energy Loss and Creation in a Nonexpanding Universe. Retrieved from: https://www.researchgate.net/publication/350086785_The_Random_Energy_Loss_and_Creation_in_a_Nonexpanding_Universe

[[5]]Dai, R. (2025). An Energy Dilemma Created When Quantum Physics Started. Retrieved from: https://www.researchgate.net/publication/399235529_An_Energy_Dilemma_Created_When_Quantum_Physics_Started

[[6]]Dai, R. (2026). Metaphysical Symphony, Book One, The Cracking Scientific Foundation. Amazon. Paperback ISBN: 979-8249046958, ASIN: B0GPPD2ZHM. eBook ASIN: B0GQX9G6TV.

[[7]]Dai, R. (2026). Testing Doppler Effect of Light with a Pair of Mirrors. Retrieved from: https://www.researchgate.net/publication/403473842_Testing_Doppler_Effect_of_Light_with_a_Pair_of_Mirrors

[[8]]Dai, R. (2022). The Fall of Special Relativity and the Absoluteness of Space and Time. Retrieved from: https://www.researchgate.net/publication/363582341_The_Fall_of_Special_Relativity_and_The_Absoluteness_of_Space_and_Time

[[9]]Wikipedia, Atomic electron transition. Retrieved from: https://en.wikipedia.org/wiki/Atomic_electron_transition. Last edited on 3 January 2026, at 09:57 (UTC).

 ====

附录： 在academia.edu的交锋

D Langstaff
6 hrs ago

Dear Rongqing,

Thank you for sharing your paper and for the careful thought you’ve put into this setup. It’s an interesting thought experiment that highlights an important subtlety in how we apply the photon energy formula \(E = hf\) together with the Doppler effect.

The apparent violation of energy conservation arises from treating the moving mirror (or spaceship) as a perfectly passive object whose velocity never reacts to the light. In reality, each photon reflection imparts a tiny momentum kick \( \Delta p = 2(hf/c) \) (in the mirror’s instantaneous rest frame). For a mirror moving at velocity \(V\), this momentum transfer does mechanical work on the mirror—exactly equal to the energy gained or lost by the photons due to the Doppler shift. If you hold the mirror at constant speed with an external force, that force supplies (or absorbs) the precise energy difference. When you include the mirror’s kinetic-energy change (or the external work), the total energy of the closed system is conserved in any inertial frame. The frequency shift is not an independent “energy creation/destruction” event; it is the bookkeeping that *preserves* conservation.

This same principle is used routinely in working technology and observations:

- **Synthetic Aperture Radar (SAR)** on aircraft and satellites relies on the Doppler shift from the platform’s motion to form high-resolution images. The radar energy budget is calculated and verified to high precision with no unexplained loss or gain.

- **GPS** corrects for both special- and general-relativistic Doppler shifts (plus gravitational redshift) in satellite signals; the system works because photon energy and momentum are conserved across frames.

- **Astronomy** routinely measures cosmological redshift, stellar radial velocities, and binary-star Doppler shifts. Energy conservation holds throughout; the observed frequency change is fully accounted for by the relative motion of emitter and observer.

No new physics is required—the existing relativistic treatment of photon momentum and radiation pressure resolves the apparent paradox cleanly. The setups you describe have been analyzed in textbooks and papers since the early days of quantum optics and relativity, and experiments confirm the accounting works exactly as expected.

Best regards,

Rongqing Dai
< 1 min ago

Dear Langstaff,

Thanks for sharing your view.

I guess probably you have not noticed that I have tried best to use "imbalance", "incompatibility" where I could just simply use "violation" despite it is the violation of energy conservation......The reason I did that is because I know people might come up with various imaginations as you just did in your comment to suggest that the energy conservation is actually not violated.

If you read my post more carefully you might find that what I emphasized in the post is NOT how energy conservation is violated, but instead is the fact that even if you can imagine how the energy conservation is not violated, you are NOT able to balance the sheet of energy change within the existing theoretical system....YOU JUST SIMPLY CANNOT DO IT with the existing theoretical system.

Why is that? I give the explanation in the writing which you did seem to have paid attention to:

[The ostensible reason for the violation of energy conservation in the above analysis is the combination of four factors: 1) the energy of photon is calculated only with the frequency of the light beam as shown in equation (2); 2) the postulate that the speed of light is constant in vacuum with respect to the source of light; 3) the frequency of light remains unchanged after the reflection; 4) the Huygens principle of reflection.

The combination of the above four factors determines that the energy conservation will be violated for the bouncing movement of a light beam between a pair of relatively moving mirrors.]

If you really want to prove that energy conservation is not violated here, you should not resort to your imagination, you should try to deny the above reason that I give….But I am sure you won’t be able to. As long as we apply those 4 things together, we just cannot balance the energy change here!

You were imagining that during the interaction of photon and the mirror, momentum exchange just balance the energy change, but the existing system does not offer you the tool to balance it mathematically! You need to play as a fiction writer to make your argument through. I have written fiction book and I know the difference between fiction and science!

SO which one is worse? The energy conservation is violated or it is not violated but you are just UNABLE to balance it with your imagination? Which one is worse???

The community of physics has been selectively choose their so-called positivist way of thinking to serve their PERSONAL NEEDS. They claim what people cannot see did not happen (which is terribly wrong), but here you even cannot balance the energy change, but you just assume it is conserved.

Well, I have to say: IMAGINATION is not science. If you think that energy is conserved here, please balance the change mathematically!!!

Cheers,

Ron