Light and Doppler effect paradox?

[Boorglar]

Hello, I just thought of something that looks like a paradox to me. Suppose you have a device which can detect a light source's frequency, and will explode if the frequency exceeds a threshold value $f_t$. Place this device stationary wrt the light source. Now, suppose there are two observers, one stationary, and one moving towards the light source fast enough at constant speed. By the Doppler effect, the moving observer will see the frequency larger than $f_t$, and so the device should explode in his frame of reference. But the stationary observer sees a frequency less than $f_t$, so the device does not explode in his reference frame.

I don't even think the effects of special relativity would be relevant here. You could set up the light source to emit at a frequency $f_t - \epsilon$ (for some $\epsilon > 0$), and have the device be sensitive enough to detect the variation of frequency that occurs by moving with some reasonable speed v << c that is sufficient to cause a frequency shift of $2 \epsilon$.

 

[Dale]

Clearly you cannot have something explode in one frame but not in another, so what do you think the resolution of the apparent paradox is. Maybe think of how you would build a detector which could detect a light source's frequency.

 

[Boorglar]

My guess is that obviously the device will not explode in any reference frame. But then this would imply that the device's detection abilities are degraded as the observer moves faster, which also seems strange. A possible detector would be something based on the photoelectric effect, where the threshold frequency corresponds to an electric circuit turning on...

 

[DaveC426913]

What paradox? Just because the observer sees a moving detector and a certain frequency of light, does not mean he gets to determine under what conditions it explodes.

 

[DaveC426913]

My guess is that obviously the device will not explode in any reference frame.

No. The only reference frame that matters to the detector is its own reference frame.
The stationary observer must take into account the relative motion of the detector and the light impinging on it. (Note that the SO sees the detector as time dilated).

 

[Boorglar]

Well, in this setup, the stationary observer is in the same frame as the detector (they're both stationary wrt the light source). The "paradox", as I see it for now, is that in a different frame, the apparent functionality of the device is not the same. Using the photoelectric effect mechanism, for example, a moving observer would see photons hitting the detector with frequency f larger than $f_t$, which should trigger the circuit (but don't).

 

[Dale]

A possible detector would be something based on the photoelectric effect, where the threshold frequency corresponds to an electric circuit turning on...

The threshold frequency of what? Is the threshold frequency the frequency of the source? If not then what frequency is it?

 

[Boorglar]

Sorry, I meant the threshold frequency of the detector. The circuit will turn on if the frequency of the light source photons is greater than the threshold frequency $f_t$.

 

[Dale]

How does it know what the source frequency is?

 

[Boorglar]

Hm I was thinking of using the photoelectric effect. The energy of photons hitting the detector would have to be larger than the work function of the material, and this would produce a current in the circuit. Since the energy of photons is proportional to their frequency, that would be a way to detect frequency.

So one could find a material with the appropriate work function such that $\frac{W_f} {h} = f_t$ is the threshold frequency.

 

[Dale]

hitting the detector

Note that you mentioned "hitting the detector" and not "leaving the source". Think about this a bit.

What actually determines if the threshold is reached? Is it the frequency emitted? Is it the frequency received? How does motion of the source change things? What about motion of the detector?

 

[Boorglar]

It would be the frequency received. Oh I think I understand what you mean. It doesn't actually matter what the moving observer sees. He could just as well see gamma rays hitting the detector, but what matters is what the detector itself sees, which can be calculated knowing its velocity relative to the observer and the light source.

So basically, saying that some object was hit by light of a certain frequency is not enough. One also needs to specify which reference frame was used to measure the frequency.

 

[Dale]

Yes.

If you look into the details the derivation of the photoelectric effect assumes that the metal is at rest. So that is the important frame.

 

[lightarrow]

It would be the frequency received. Oh I think I understand what you mean. It doesn't actually matter what the moving observer sees. He could just as well see gamma rays hitting the detector, but what matters is what the detector itself sees, which can be calculated knowing its velocity relative to the observer and the light source.

So basically, saying that some object was hit by light of a certain frequency is not enough. One also needs to specify which reference frame was used to measure the frequency.

An analogy: in your frame you see (= you "measure") a giant asteroid (let's say m = 10^12 kg) hitting at velocity v = 100,000 m/s in the +x direction a little planet which is moving at 99,999.999 m/s in the same direction. Will the asteroid do a great damage to the litte planet?

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lightarrow