Going crazy: can someone spot the error in this simple derivation?

[nonequilibrium]

I've been staring at this for hours now.

Can someone spot an error in the following derivation of the relativistic doppler shift?

Imagine a flashlight (not on) and an eye passing each other in space. Call the frame where the flashlight is fixed S for Source, and that of the eye O for observer. Call v the relative velocity between both. In S the flashlight sends out a flash after a time $$\tau$$ (after passing). If t is the time in S for that flash to reach the eye, then $$ct = v \tau + vt$$, because the lhs is the total distance covered by the flash to get to the eye and the first term on the rhs is the distance the eye is at at the moment of the flash; the second term is there since the eye is also moving away while the flash is going toward it.

In O the flashlight sends out a flash after a time $$\tau' = \gamma \tau$$ after the passing. If t' is the time in O for the flash to reach the eye, then $$ct' = v\tau' (= \gamma v \tau)$$ because as soon as the flash has "left" the flashlight, all the light has to do is travel the distance crossed by the flashlight during $$\tau'$$, which is exactly the rhs.

Rewriting the first equation as $$(c-v)t = v \tau$$ and dividing this equation by the latter, we get $$\frac{c-v}{c} \frac{t}{t'} = \frac{v}{\gamma v}$$ or $$t = \frac{\sqrt{1-(v/c)^2}}{1-v/c} t'$$. We can easily rewrite the latter equation as $$t = \sqrt{ \frac{c+v}{c-v} } t'$$.

If we regard t and t' as the period of the EM-wave in resp. S and O, we get the wrong formula: the minus and plus in front of the v has switched places...

A BIG THANK YOU TO ANYONE WHO CAN FIND IT!

 

[JustinLevy]

The math and reasoning looks okay (besides the typo of accidentally dropping a $$\tau$$ at one point) up to:
$$t = \sqrt{ \frac{c+v}{c-v} } t'$$

I think there is an error of interpretation in your next line:
"If we regard t and t' as the period of the EM-wave in resp. S and O"

The period of an EM wave for a frame should be the time period between crests at the same spatial point.

However, instead t refers to the time between two events spatially separated in S. So does t' in O.

The time you want in S is $$\tau$$.
The time you want in O is $$t' + \tau'$$ which I'll call T.

If you plug your relations in for those two times, I think it will work out.

$$T = \tau' + t' = \gamma \tau + \gamma \frac{v}{c} \tau = \gamma(1 + v/c) \tau = \sqrt{\frac{c+v}{c-v}} \tau$$

or for comparison to your original equation

$$T_s = \tau = \sqrt{\frac{c-v}{c+v}} T_o$$

Since the observer is moving away, he should measure a lower frequency (longer time period). So the signs appear correct.

 

[nonequilibrium]

Hm, I think you're right about the wrong interpretation of the period, but doesn't that mean I can't derive it using just "one flash"? Because then at any spatial point the wave passes only once so the period is effectively infinite? But I wanted to ask: can you clarify your choice for tau and tau' + t' for the period in resp. S and O? Thank you!

 

[grav-universe]

What you found there is not the relation between the frequencies or the periods, but just between what an observer in each frame measures for t and t', the time that each says passes for the photon emitted at S to reach the observer O. For the frequency, you will need two or more consecutive photons, although two is enough for the calculations.

So all found from the frame of S, after a time τ, S emits a photon. After a time of τ + T, where the frequency according to S is f = 1/T, S emits another photon. The first photon travels to O in a time of t = v τ / (c - v), as you had, so reaches O in a total time of τ + t = τ [1 + v / (c - v)] = τ / (1 - v/c), so when O's clock reads τ / (γ (1 - v/c)), which is the actual reading upon O's clock upon receiving the first photon since the clock and photon coincide in the same place and all obervers must agree upon the reading of the clock when these events occur in the same place. The second photon leaves S at a time of τ + T and reaches O at a time of (τ + T) + v (τ + T) / (c - v) = (τ + T) [1 + v / (c - v)] = (τ + T) / (1 - v/c), so when O's clock reads (τ + T) / (γ (1 + v/c)). The period observed by O, then, is T' = (τ + T) / (γ (1 - v/c)) - τ / (γ (1 - v/c)) = T / (γ (1 - v/c)). Since the frequency O observes is f' = 1/T', the ratio of the frequency observed to the frequency emitted becomes f' / f = (1/T') / (1/T) = T / T' = γ (1 - v/c) = (1 - v/c) / sqrt[1 - (v/c)^2] = sqrt[(1 - v/c) / (1 + v/c)].

All found from the frame of O, S emits a photon when S's clock reads τ, as all observers must agree upon that since the photon and S' clock reading coincide in the same place, so when O's own clock reads τ' = γ τ according to what O measures for both clocks, as you had. The photon travels to O in a time of t' = v τ' / c, as you had for that also, so reaches O when O's clock reading is τ' + v τ' / c = τ' (1 + v/c) = γ τ (1 + v/c). The second photon leaves S when S's clock reads τ + T, so when O's clock reads γ (τ + T) according to O. The second photon reaches O in a time of v γ (τ + T) / c, so the reading upon O's clock when receiving the second pulse is γ (τ + T) + γ (τ + T) (v/c) = γ (τ + T) (1 + v/c). The time between the two pulses as O measures them, then, is T' = γ (τ + T) (1 + v/c) - γ τ (1 + v/c) = γ T (1 + v/c), where f' = 1 / T'. The ratio of the frequency observed to the frequency emitted according to O, therefore, is f' / f = (1/T') / (1/T) = T / T' = T / (γ T (1 + v/c)) = 1 / (γ (1 + v/c)) = sqrt[1 - (v/c)^2] / (1 + v/c) = sqrt[(1 - v/c) / (1 + v/c)], the same as with S.

 

[grav-universe]

Perhaps a much easier way of performing this, which is similar to the way you might be trying to consider things in the OP by the looks of it, is to have S simply emit the first photon immediately upon passing O, and then emit a second photon a time of τ later according to S's clock, so T = τ between pulses according to S, and the frequency according to S is therefore f = 1 / T = 1 / τ.

According to S, then, the second photon reaches O a time of t = v τ / (c - v) later, so when a total time has passed for S of τ + t = τ + v τ / (c - v) = τ / (1 - v/c), and therefore a time of T' = τ / (γ (1 - v/c)) has passed for O between the readings upon O's clock between receiving the pulses, so O says the observed frequency is f' = 1 / T' = 1 / [τ / (γ (1 - v/c))] = γ (1 - v/c) / τ, whereas the ratio of the observed frequency to the emitted frequency becomes f' / f = [y (1 - v/c) / τ] / [1 / τ] = γ (1 - v/c) = (1 - v/c) / sqrt[1 - (v/c)^2] = sqrt[(1 - v/c) / (1 + v/c)].

According to O, S emits the second photon when S's clock reads τ, so when O's own clock reads γ τ according to O. The photon takes a time of t' = v (γ τ) / c to reach O according to O, so the time that passes between receiving the photons according to O is T' = γ τ + v (γ τ) / c = (γ τ) (1 + v/c), and the observed frequency is therefore f' = 1 / T' = 1 / ((γ τ) (1 + v/c)). The ratio of the observed frequency to the emitted frequency is f' / f = T / T' = τ / [(γ τ) (1 + v/c)] = 1 / (y (1 + v/c)) = sqrt[1 - (v/c)^2] / (1 + v/c) = sqrt[(1 - v/c) / (1 + v/c)].

 

[jason12345]

It's not as simple as this since all you seem to be doing is relating the difference in time between events in one frame to another. Where's the plane wave?

You need to start with a plane wave in one frame with a frequency w and wave number K. Next, you need to prove that this will also transform as a plane wave with frequency x' and k' in another frame S'.

It isn't that obvious and involves some head scratching and hair pulling. It's worth doing because you will discover that for a plane wave traveling with velocity v < c in a medium, there is a relativistically induced optical anisotropy which quite a few people are ignorant of.

 

[JustinLevy]

Hm, I think you're right about the wrong interpretation of the period, but doesn't that mean I can't derive it using just "one flash"? Because then at any spatial point the wave passes only once so the period is effectively infinite? But I wanted to ask: can you clarify your choice for tau and tau' + t' for the period in resp. S and O? Thank you!

The two "crests" or "pings" I imagined (and I thought what you were trying to calculate) were:

Source frame:
crest/ping #1: when the source and observer were at the same location / passed each other
crest/ping #2: when the source emitted the next crest/ping

Observer frame:
crest/ping #1: when the source and observer were at the same location / passes each other
crest/ping #2: when the next crest/ping from the source reaches the observer

Notice the time between crests is measured at a single spatial location in each frame.

I didn't read the other posts, since this hopefully clarifies.

EDIT:
Okay, I read one

It's not as simple as this since all you seem to be doing is relating the difference in time between events in one frame to another. Where's the plane wave?

You need to start with a plane wave in one frame with a frequency w and wave number K. Next, you need to prove that this will also transform as a plane wave with frequency x' and k' in another frame S'.

All you need for these is to track the location of two crests or "pulses", not every single crest / pulse / entire wave. Since the coordinate transformations are linear, you don't need to worry whether a plane wave transforms to a plane wave. You know this is true from the start.

It isn't that obvious and involves some head scratching and hair pulling. It's worth doing because you will discover that for a plane wave traveling with velocity v < c in a medium, there is a relativistically induced optical anisotropy which quite a few people are ignorant of.

Yes, the relation starts to look uglier once the velocity of the wave front isn't colinear with the velocity of the observer. But to anyone that has heard the doppler shift of a train going by, I think the "direction dependence" should be fairly intuitive. So I'd hope there aren't too many people ignorant of it.

It's been awhile since I've taught a class involving SR, but I remember the most common problem being students wanting to use length contraction and time dilation for everything. No matter how many times I showed in office hours where these relations came from, and that they were just specific examples of relating coordinates of multiple events between two coordinate systems using the Lorentz transformations ... many kept "feeling" that the length contraction and time dilation were more "fundamental" than the Lorentz transformations.

It's tough now. Students have grown up hearing these concepts in vague terms for so long before they see them precisely, that there is a lot of "rearranging" of intuition that needs to be done. Even worse is teaching quantum mechanics. They parrot these phrases that they've heard so many times, and feel they have intuition for, but that they haven't even learned what they mean yet. It's a really strange phenomena when you start getting classes of students thinking they already understand the basics of QM just because they've seen so much reference to it in sci fi or popular sources.

 

[nonequilibrium]

Thank you guys a lot for the help on this matter! :)

PS:
"It's tough now. Students have grown up hearing these concepts in vague terms for so long before they see them precisely, that there is a lot of "rearranging" of intuition that needs to be done. Even worse is teaching quantum mechanics. They parrot these phrases that they've heard so many times, and feel they have intuition for, but that they haven't even learned what they mean yet. It's a really strange phenomena when you start getting classes of students thinking they already understand the basics of QM just because they've seen so much reference to it in sci fi or popular sources."
Very true. I had read many popular books and always read about length contraction and time dilation, so I was very surprised to suddenly see the variable x in the lorent transformation for time. On the other hand, this became more logical once I realized that without this that other popular phrase "no simultaneity" would be impossible.

And as for QM: I've only had an introduction at college, but I must say that introduction was as vague as the popular books I've read. Sure we've done some math like time-independent Schrodinger equation, but I mean, in terms of insight, it was as useless as any popular book on the matter (sadly...). If you happen to know a fine book for someone in my position on that matter, it is very welcome! (I'm also very interested in the historical development of the theory; at this moment I'm just looking for anything that will pay due respect to the possible interpretation(s) of what is written down because the way I've seen it till now: it's all very mingled... At one time the UP is statistical, at the other time it's physical; at one time the wave-nature of particles breaks down upon measurement, at the other time it is perennial, etc...; books I have on the matter include: In Search of Schrodinger's Cat (John Gribbin), Quantum (this is more like a novel about the Einstein-Bohr debate), The Physical Principles of Quantum Theory (Heisenberg), Physics and Beyond/Der Teil und Das Ganze (Heisenberg), Feynman Lectures, so all in all a pretty meaningless collection)