During a course of mine on light & time, a student asked me during the break an excellent question to which I did not have a satisfactory answer in my own opinion at that moment. The student seemed satisfied, but the question kept buzzing around in my mind like a pesky fly in the room that couldn’t find the exit.

## The Cosmic Microwave Background Temperature

Before I start to describe the question, let me say the following. It is my strong opinion that we can imagine an expanding EM wave as a cloud of photons expanding at the speed of light from the light source whose density decreases with distance from the source squared. But the energy per photon, I say, does not decrease with distance as the energy of each individual photon is determined by its frequency. Viewed in this way, Maxwell’s EM wave is a phenomenon emerging from the behavior of this photonic quantum cloud. It is a cloud of photons materializing according to their quantum probability waves , not the electromagnetic wave that is usually used to explain the wave behavior of light. For more explanation, see elsewhere on my website under ‘What is light?‘.

The student question was in response to my statement that the energy of a photon does not change with traveled distance because it would mean that its frequency would have to change accordingly. The universe background radiation, discovered in 1964 by Penzias and Wilson, with a wavelength of 7.35 cm and a temperature of 2.7 K, is today seen as a residue of the original radiation from the Big Bang. Due to the expansion of the universe, the original energy of the photons has decreased enormously. That the energy of the Big Bang photons has obviously decreased, contradicts my above statement that with the expansion of the EM wave, the energy of the individual photons does not decrease. So a very good question.

## The redshift and the expanding universe

We encounter the same problem with the so-called redshift of light from galaxies moving away from us at great speed. The photons we receive from it here have gotten a lower frequency because of the so-called Doppler effect and that’s how Edwin Hubble discovered that the universe seemed to be expanding, as the light from galaxies showed a greater redshift on average the further away they were.

## The Doppler effect stretches the wave

As a physicist, I like to approach such a question with a thought experiment that contains the basic ingredients of the question. Think therefore of a rocket whizzing away from you at half the speed of light. From the rocket a beam of light, i.e. photons, is sent back by a laser to be received by you. The laser emits violet light of a wavelength of 400 nm (750 Thz). When the laser has sent 1 wave, that’s 1 wavelength of course, then the wave must stretch, the wavelength as received by you becomes longer. The end of the wave is emitted when the rocket has covered half a wavelength since emitting the beginning of the wave because it’s going with half light speed. It must have covered a distance of 200 nm between these two moments . The wavelength, received by you, thus becomes 600 nm (yellow-orange). That indeed means that the photon you receive must correspond to a wavelength that is increased by af factor 1.5. The corresponding frequency of the foton becomes 2/3 of the emitted frequency – 500 Thz. The energy of a photon is directly proportional to its frequency (Planck’s law). The photon that you receive has therefore only 2/3 of the energy of the photon that was emitted. Energy is missing! Where does that energy go then? Isn’t that contrary to energy conservation?

## The effect of relativity on the Doppler effect

Above that, we should not forget relativistic effects. If we choose to view the source, the rocket, as static and the receiver speeding away at half the speed of light, the outcome of the above Doppler calculation will be different. But because the speed of light is the same for all moving systems, we have to take that into account as well. We have to look therefore at the behavior of the clocks. You will ‘see’ the clock In the flying rocket ticking slower, because of the time dilation. The formula for that time dilation is: T=T_{0}√(1 – v^{2}/c^{2}) where T represents the time in the rocket as seen by the stationary observer, you, and T_{0} the time for the stationary observer himself, so your own time. The ratio ^{v}/_{c}=^{1}/_{2}, which gives in the time dilation formula: T = 0.866 x T_{0}.

So in the time that 1 second elapses for you, 0.866 second elapses in the rocket. Not only the clock in the rocket will tick slower from your point of view, the laser will also be slowed in the same way and will emit light of 400/0.866 nm = 462 nm (indigo), from your point of view . The already lower energy of the photon received by you, due to the Doppler effect described above, is therefore extra reduced by an additional factor of 0.866. The wavelength of each photon that you will receive is thus, now including both Doppler effect and relativity, λ = (400 x 3/2)/ 0.866 = 693 nm (red). You will obtain the same result when you switch roles and assume a moving receiver (yourself) and a stationary source (the rocket). For a comprehensive explanation of the relativistic Doppler effect, see Wikipedia.

## Energy conservation law and relativistic effects

Now the question is whether this relativistic decrease in frequency also violates energy conservation. But a closer inspection reveals that no energy is lost by relativistic effects. The relativistic decrease in frequency is the result of the slower time in the rocket as ‘seen’ by you as a stationary observer, so that less energy is sent to you per unit of your time. From your point of view, an ‘outstrechted’ photon of 462 nm (649 Thz) is emitted and also received (if we forget the doppler effect for a moment). The fact that this is experienced differently in the rocket, 400 nm (750 Thz), is only a consequence of the fact that the slowing down of the clock in the rocket is not experienced, since everything runs slower there in the same way. So the photon does not lose energy along its way to you due to the relativistic effect.

So back to the energy of the individual photon that seems to leak away through the non-relativistic Doppler effect. My first guess tells me that energy is not an absolute given in systems that move at different speeds relative to each other. Two bullets rushing next to each other at the same speed have no kinetic energy relative to each other, but they do have kinetic energy relative to their target. The same kind of thinking could therefore probably apply to photons. We should search for a solution in that direction.

## Is a photon a closed system in terms of energy conservation?

After thinking about the question for a some time, I did come to the following conclusion. Conservation of energy only applies to closed systems. Apparently a photon cannot be considered as a closed system, even not as a system in the usual sense. That image of a closed system seems to come from considering the photon as a real particle speeding through space. But in my opinion a photon is really nothing more than the our material interpretation of an event in which an energy exchange takes place between two systems. As long as those systems do not move relative to each other, the energy change in one system will be equal and opposite to the energy change in the other. The registration of received energy is what we interpret as the flight of the photon. It seems to us as if there is a flying photon carrying energy over, but there is no particle flying through space. It is our interpretation of an amount of energy that disappears from its source to appear elsewhere. Should such a non-existing particle be regarded as a closed system in which the total energy remains the same?

I assume it has to do with the fact that in order for the photon to arrive at me apparently at the speed of light, it has to compensate for the speed at which the rocket speeds away from me. And this compensation costs energy. Compare it with shooting a bullet from a speeding train to a target standing next to the railway track. The kinetic energy of the bullet that it received from the gun is not equal to the kinetic energy with which it hits the target. That’s less in the case the train speeds away from the target. Part of the kinetic energy of the bullet is then spent on the difference in speed between the train and the target. But this is not in contradiction with the preservation of energy. I hope you can understand that. The bullet is not losing energy on its way to the target, provided that air resistance can be ignored.

As soon as the two systems move relative to each other, an imbalance in energy exchange exists. It takes extra energy to send energy to a system moving away from you. But this extra energy is not added to the photon. So for energy conservation you have to look at the total system, photon, sender and receiver, not only at the photon. The greater the speed difference between the systems, the greater the imbalance between spent and received energy. And that explains the low energy of the cosmic microwave background radiation. Its source, the remains of the Big Bang explosion, is moving away from us with a speed which must be very close to the speed of light. Very little energy is left for the photons coming our way. They do not lose their energy on the way. From our perspective they started 13.7 billion years ago with that small amount of energy because most energy was spent on compensating for the speed with which their source was receding from us.

## Conclusion: the imaginairy existence of the photon

Considering the photon as a material traveling particle that carries and transfers a specific amount of energy through space leads to these kinds of difficult contradictions. Elsewhere on this website I’ve argued that the photon, while most of the time a useful, however abstract, concept, doesn’t exist along the way and is probably completely imaginary. There is energy that disappears in one system and energy that appears in another. Something in which the quantum state wave plays a major role. And between disappearing and appearing there is a time difference that, divided by the distance between the systems, always turns out to lead to the observation of the speed of light.

My statement that the EM wave is the result of a cloud of photons expanding with the speed of light, whose energy per individual photon does not decrease, must therefore be extended with the condition that the source does not move relative to the observer. No problem at all.