Why doesn't time-dilation impact the threshold frequency of the photoelectric effect?

Question

Consider the following hypothetical: 

An atom is traveling with a velocity of v to the left, directly towards an incoming photon, as crudely depicted below.

p----> <----A

Question (1): What is the resonance absorption frequency of the atom?

Question (2): What is the threshold frequency of the atom?

Question (1) requires that we assume that time-dilation is occurring within the atom, in order to be consistent with experimental evidence that shows that the resonance absorption frequency of an atom does in fact decrease due to time-dilation.

Kundig Experiment

Note that the Kundig experiment shows that the threshold frequency of the atom itself is reduced due to time-dilation. This is distinct from any Doppler shifting that occurs with respect to light incident upon the atom.

Question (2) requires that we assume that time-dilation is NOT occurring within the atom, in order to be consistent with the equations for the work function of an electron, which, as far as I can tell, assume that time-dilation is not occurring within the atom.

Electron Work Function

This seems to imply that there is no single answer to "how much time has elapsed" as measured within the atom, since (1) and (2), by definition, require different measurements of time.

What is the correct measure of time in this set of facts?

Here is my question, presented as an experiment:

If I rotate a metallic plate with a threshold frequency of $f_0$ around a light source with a frequency of $f_0$, in the same manner that was done in the Kundig experiment, would the threshold frequency of the plate be reduced below $f_0$ due to time-dilation?

For those that are interested, I came across this in connection with my research applying information theory to time-dilation (the working paper is here: https://www.researchgate.net/publication/323684258_A_Computational_Model_of_Time-Dilation)

Comments

The binding energies $E_n$ are impacted by the atom velocity since  the energy is Lorentz-transformed together with the atomic momentum $\vec{P}$. Voting to close.

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Correct me if I'm wrong, but this would imply that the binding energy of an electron would decrease as a function of the velocity of the atom which contains the electron. This seems to imply that electrons in faster moving atoms will require less work to extract. Are you implying that this is indeed the case?

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Why are you suprised by this? A fast moving atom has instead an additional kinetic energy, which can be used for extracting the electron.

Consider also the following situation: you have an EMF generator with the theshold frequency in your laboratory. If a moving atom approaches your generator, the electron can be extracted. If the atom is moving away, one-photon ionisation is no longer possible (but multiphoton ionisation is still possible, with much smaller probability). Are you familiar with the Doppler effect?

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Yes I understand that the kinetic energy of the atom can be used in that manner, but that is a distinct point from the question of whether time-dilation itself, independently impacts the work function of the electron. So you are saying that the work function of an electron is actually given by $hf_0\frac{1}{\gamma}$, where $f_0$ is the threshold frequency of the material?

If so, then are there experiments analogous to the Kundig experiment where the threshold frequency of a rotating plate is reduced?

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No, it is not a distinct point at all. The product $E_n\cdot t - \vec{P}\cdot\vec{x}$ is a relativistic invariant.

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Apologies for being unclear, but perhaps it would best if you could just answer the one question I posed:

Is the actual work function of an electron given by $hf_0\frac{1}{\gamma}$, where $\gamma$ is the Lorentz factor?

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It depends on the experimental situation described above. There is no notion of "actual work function" alone, but "an effective work function" depending on the velocity vector with respect to your lab generator.

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OK, but is the work function adjusted by the Lorentz factor γ? I believe your statement that it is a relativistic invariant implies that it is not, but please correct me if I'm mistaken.

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OK, but is the work function adjusted by the Lorentz factor $\gamma$? I believe your statement that it is a relativistic invariant implies that it is not, but please correct me if I'm mistaken.

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You must consider a problem of ionisation of an atom with an incoming EMF. It is a Dirac-like equation with an external EMF. Then you can consider it in a reference frame where the atom is at rest. The incoming photon frequency will be modified depending on the relative motion of the atom and the source. There is no other option for considering the question about "the actual work function". Study the relativistic Doppler effect.

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I understand the incoming photon frequency will be affected by the Doppler effect. Let's agree that question is closed.

There is a second order question of whether the threshold frequency of the atom is adjusted by the Lorentz factor, which I don't think you've yet answered.

For example, the resonance absorption frequency of an atom (not the frequency of the light) is reduced due to time-dilation. This has nothing to do with the frequency of the incoming photons, but is instead a property of the atom itself that is changed due to time-dilation.

My questions are:

(1) Does the threshold frequency of an atom get adjusted in the same manner that the resonance absorption frequency of an atom does? 

(a) If not, why?

(b) If yes, what experiments show this to be the case (i.e., an analog of Kundig et all re Mossbauer)

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Property of the atom "itself" is meaningless. You need some interaction. Thus the Doppler effect gives you the answer.

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With all due respect, you are avoiding a very simple question through obfuscation. 

The resonance absorption frequency of an atom decreases due to time-dilation when it is rotated, as is demonstrated by the experiments of Kundig et all.

Here is my question, presented in a yes / no format:

If I rotate a metallic plate with a threshold frequency of $f_0$ around a light source with a frequency of $f_0$, in the same manner that was done in the Kundig experiment, would the plate eject electrons?

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The answer is given with the relativistic Doppler effect.

Note, the Mössbauer experiment deals with a very narrow resonance, thus very sensitive to the frequency shifts.

Atomic or metal "levels" have much wider resonance curves (spectral line widths) and thus require much higher relative velocities to be observed/non-observed.

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I'm not sure how you turned a yes / no question into an opportunity for more confusion, but here goes:

"The answer is given with the relativistic Doppler effect." 

what does that mean?

"Atomic or metal "levels" have much wider resonance curves (spectral line widths) and thus require much higher relative velocities to be observed/non-observed." 

are you saying that the threshold frequency of an atom would be impacted, but that a lot of velocity is required to achieve a point where it would be observable? If so, could you provide a reference that supports this claim? 

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The relativistic Doppler effect (formula) demonstrates how the rest-frame frequency is modified as a function of the relative velocity direction and the velocity absolute value. The original frequency is implied exactly defined, without any width $\Delta\omega$. Atoms emit lines with some widths $\Delta\omega$, so the resonance absorption may take place for somewhat different frequencies.

Kundig experiment demonstrates the effect and answers your question; the effect, which may be difficult to realize with the usual light source and a metal plate due to important $\Delta\omega$ and necessity of much higher velocities. I am not familiar with metal threshold uncertainties $\Delta\omega$ so I cannot (and do not want to) estimate the feasibility of your experiment.

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"The relativistic Doppler effect (formula) demonstrates how the rest-frame frequency is modified as a function of the relative velocity direction and the velocity absolute value. The original frequency is implied exactly defined, without any width ΔωΔω. Atoms emit lines with some widths ΔωΔω, so the resonance absorption may take place for somewhat different frequencies."

The relativistic Doppler effect formula tells us how the frequency of the incident LIGHT changes, it doesn't tell us anything at all about the threshold frequency of the ATOM.

"Kundig experiment demonstrates the effect and answers your question; the effect, which may be difficult to realize with the usual light source and a metal plate due to important ΔωΔω and necessity of much higher velocities. I am not familiar with metal threshold uncertainties ΔωΔω so I cannot (and do not want to) estimate the feasibility of your experiment."

It most certainly does not. The Kundig experiment demonstrates that resonance absorption (in the nucleus of an atom) is affected by time-dilation. I'm asking about the photoelectric effect, which deals with the excitation of electrons, and is a completely distinct phenomenon.

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Photoelectric effect is due to absorption. I regret that I was engaged in this useless discussion.

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So you're now saying the photoelectric effect is due to light being absorbed in the nucleus of an atom? I'm almost certain that is incorrect. 

I'm admittedly not an expert on this subject, which is why I posted the question. But, with all due respect, I'm getting the sense you have no idea what you're talking about. I asked you a simple, yes / no question, and you responded with a bunch of cryptic nonsense.

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I am not saying the photoelectric effect is due to light being absorbed in the nucleus of an atom! I impied absorption by an atomic electron or by an electron from a metal plate.

I am not in position to give you some lectures on atomic physics here. Learn from other sources, please.