Doppler shifts appear to violate conservation of energy

Question

If the relative distance between an infrared source and a spectrometer is shortening at a such a rate that the spectrometer detects that radiation at a Doppler- (or blue-) shifted wavelength in the UV band, then the photonic energy the spectrometer detects is by definition a shorter wavelength than the wavelength emitted from the infrared source.

This seems to violate the thermodynamic law of conservation -- We can't say that the photonic energy was increased, because no additional energy was added. Yet, UV has more photonic energy than infrared.

Moreover, given the propagation of the EM radiation is always C in a vacuum, the relative motion does not add any Newtonian inertia either (you can't add the relative velocity to the propagation velocity -- besides, photons are massless, hence able to propagate at C, no faster).

Likewise, if the distance were growing instead of shrinking (the red-shifted case) there (seemingly) is an equal violation, in that there (seemingly) is a loss of received photonic energy.

Keeping in mind that given an ideal laser beam or a point source doesn't matter, because this has nothing to do with the energy density of the emitter versus the energy density at the spectrometer; the effective change in photonic energy is still the same.

How is this photonic energy delta reconciled?

Answer 1

Let's consider a simpler situation. I throw a ball at a car that's not moving. The ball hits the windshield can causes some damage. Now I throw a ball at the same speed, but at a car that's coming towards me at high speed. The ball will do more damage to the windshield, right? From the moving car's perspective, the ball is moving faster. The amount of damage depends on the relative motion of the ball and car. If the car were moving away from me, less damage would be done to the windshield. There's no problem with conservation of energy because it took energy to get the car moving in the first place.

It's the same with the photon and the detector. Because the detector is moving towards the photon source, it sees a higher energy photon. You can also think of it as the energy that the detector absorbs comes from the photon and the work needed to accelerate the detector in the direction of the photon.

Just like different observers will disagree on the kinetic energy of a thrown ball, different observers will disagree on the energy of a photon. They will all agree that energy is conserved in any interaction. Similarly, they will disagree on the momentum of the photon. Even though a photon does not have mass, it does have momentum equal to $h/\lambda$, where $h$ is Planck's constant (about 6.6 × 10$^{-34}$m$^2$kg/s) and $\lambda$ is the wavelength of the photon. The Doppler shift affects the observed wavelength of the photon, so the observed momentum will change.

Answer 2

This is actually a very common question, and at heart it's identical to the questions we get about the Oberth effect every few days. The Oberth effect is the fact that when a rocket is already moving, firing the exact same fuel will lead to a greater increase in the rocket's kinetic energy, seemingly in contradiction with energy conservation. The resolution is that you have to account carefully for all sources of energy, and the fuel had kinetic energy before being fired out of the rocket, which is effectively being harvested.

A similar thing is going on here. Suppose, for example, that an atom begins at rest. It then emits one photon and starts moving a bit, and then emits what would have been second photon, in its direction of motion. By the Doppler effect, this second photon is instead blueshifted, so where does the energy come from? It ultimately comes from the kinetic energy $mv^2/2$ of the atom. Some of this is harvested and given to the photon, because the act of emitting the photon decreases the atom's rest mass, since $E = mc^2$. In fact, this was one of Einstein's original arguments for $E = mc^2$.

Answer 3

then the photonic energy the spectrometer detects is by definition a shorter wavelength than the wavelength emitted from the infrared source.

Photonic if you are using it to describe an ensemble of photons belongs to the quantum mechanical framework, where classical electromagnetic light emerges from a large ensemble of photons, elementary point particles in the table.

This seems to violate the thermodynamic law of conservation --

Thermodynamics is an emergent framework from statistical mechanics of a large number of particles.

Mixing mathematical frameworks produces apparent inconsistencies that are resolved once the correct framework is chosen.

We can't say that the photonic energy was increased, because no additional energy was added.

But you have been adding energy , as your initial sentence says explicitly.

If the relative distance between an infrared source and a spectrometer is shortening at a such a rate that the spectrometer detects that radiation at a Doppler- (or blue-) shifted

Italics mine.

Answer 4

No the energy is conserved. Energy is a quantitive quantity so when you compare red light to blue light you're comparing the energydensity not the whole energy in the system. And yes the energy density decreases, but since in this process the wave gets larger due to the "stretching" the equal amount of energy is stored. It's just like if you have a sin wave across say 100m and a sin wave across 1000m the wavelength is the same and the energy density is the same but there is more energy in the system where the wave is across 1000m.