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

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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)

edited Apr 20, 2018

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.

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?

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.

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.

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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