Testing some SR assumptions

[DiracPool]

Thought experiment: Alice, Bob, and the planet Zolan.

Alice is floating in space 10 light seconds from the planet Zolan (Z) with a radio emitter and detector. She and planet Z are in the same rest frame. Bob has the same devices as Alice but is on planet Z. Bob then accelerates from planet Z toward Alice and then stops accelerating once he reaches a velocity of 0.5c relative to planet Z (and Alice, of course).

At the instant Bob reaches Alice's position, they both simultaneously shine their radio emitters at a reflector on planet Z and leave the emitters running. Say the frequency of the radiation is 1000 kiloHertz.

Here are my assumptions. 1) At the end of 20 seconds, Alice's detector will detect both her's and Bob's emitted signal, and read that the signal reflected by her emitter is still 1000 kH, but Bob's is less than 1000 kH.

2) Bob, on the other hand, from within his reference frame, will detect both his and Alice's signal at the same time, but the time elapsed will be well after 20 seconds according to his clock. Furthermore, both his and Alice's signal will read less than 1000 kH on his detector, although his emitted signal's frequency will read even less than Alice's, the reason being that Bob's reading of his own signal has a doppler shift going in both directions, while Bob's reading of Alice's signal only has a doppler shift after being reflected from planet Z.

Are these assumptions correct?

 

[PeterDonis]

DiracPool, a general comment: when specifying scenarios, it really helps to be painfully explicit about exactly what frame everything is relative to. That helps to avoid a lot of potential misunderstandings. You'll see this theme repeated in a number of my comments below.

Bob then accelerates from planet Z toward Alice and then stops accelerating once he reaches a velocity of 0.5c relative to planet Z (and Alice, of course).

I assume you intend that the distance it takes Bob to accelerate (in the rest frame of Alice and Z) is much less than 10 light-seconds, correct? In other words, the acceleration itself plays no role in the problem; we can idealize it away by saying that Bob accelerates basically instantaneously, without affecting anything of interest to you, correct?

Say the frequency of the radiation is 1000 kiloHertz.

I assume you mean here that Alice measures the frequency of her emitted radiation to be 1000 kHz, and Bob measures the frequency of his emitted radiation to be 1000 kHz, correct? In other words, these two frequencies are relative to two different frames (the Alice/Z rest frame, and Bob's rest frame).

1) At the end of 20 seconds, Alice's detector will detect both her's and Bob's emitted signal, and read that the signal reflected by her emitter is still 1000 kH, but Bob's is less than 1000 kH.

I assume you mean, at the end of 20 seconds by Alice's clock; and that the signals being detected are the *reflected* signals. (Strictly speaking, the emitted signals are the ones going out towards the reflector, not coming back from it.)

We also have to assume that the reflection of each signal does not change its frequency. (Actually, this assumption is frame-dependent, as we'll see in a moment; but it holds in the Alice/Z frame, so it applies at this point.)

With those assumptions, yes, since Bob is moving away from Alice while he emits his signal, its frequency relative to Alice will be redshifted, and since reflection does not change the frequency in Alice's frame, she will observe the reflected signal from Bob to be redshifted.

2) Bob, on the other hand, from within his reference frame, will detect both his and Alice's signal at the same time, but the time elapsed will be well after 20 seconds according to his clock.

Yes. Specifically, the time elapsed from Bob emitting his signal to his receiving both reflected signals, by Bob's clock, is

$$
\tau = 20 \sqrt{\frac{1 + v}{1 - v}}
$$

where $v$ is Bob's velocity relative to Alice/Z. For $v = 0.5$, this works out to $\tau = 20 \sqrt{3}$.

Furthermore, both his and Alice's signal will read less than 1000 kH on his detector

I think so, yes. But see below.

although his emitted signal's frequency will read even less than Alice's, the reason being that Bob's reading of his own signal has a doppler shift going in both directions, while Bob's reading of Alice's signal only has a doppler shift after being reflected from planet Z.

I think your answer is correct (Bob will observe Alice's signal to have less redshift than his own), but your reasoning is not.

First, remember Bob's measurements are relative to Bob's frame. Relative to Bob's frame, his own signal has no redshift when it is emitted; it gets redshifted when it is reflected, because the reflector is moving away from the source of the signal (Bob).

Second, remember that, unlike the case of Bob as observed by Alice above, Alice's emitted signal is headed *away* from Bob, not towards him. That makes a difference in the doppler shift of Alice's emitted signal, relative to Bob; it turns out that Alice's signal actually gets *blueshifted*, relative to Bob's emitted signal, relative to Bob's frame. (The best way to see this is to carefully draw a spacetime diagram of the light signals, relative to Bob's frame. I don't have time to do that right now, but I'll try to later if it's still not clear.)

Both signals then get redshifted, relative to Bob's frame, when they are reflected at Z, and by the same factor. I'm pretty sure this redshift factor is greater than the blueshift factor of Alice's emitted signal, relative to Bob; which means that both reflected signals will be observed by Bob to be redshifted, but Alice's will be redshifted by less. However, I haven't done a detailed calculation to confirm this.

 

[DiracPool]

DiracPool, a general comment: when specifying scenarios, it really helps to be painfully explicit about exactly what frame everything is relative to. That helps to avoid a lot of potential misunderstandings. You'll see this theme repeated in a number of my comments below.

Thank you for the response Peter, and yes, I will pay closer attention the frame details in the future. I thought I was, but that shows you how subtle these things can get and how easy it is to miss things. But I guess that's where the learning comes in.

I assume you intend that the distance it takes Bob to accelerate (in the rest frame of Alice and Z) is much less than 10 light-seconds, correct? In other words, the acceleration itself plays no role in the problem; we can idealize it away by saying that Bob accelerates basically instantaneously, without affecting anything of interest to you, correct?

Correct on both accounts.

I assume you mean, at the end of 20 seconds by Alice's clock; and that the signals being detected are the *reflected* signals. (Strictly speaking, the emitted signals are the ones going out towards the reflector, not coming back from it.)

Correct. Re-reading that, I see how being explicit is important there.

We also have to assume that the reflection of each signal does not change its frequency. (Actually, this assumption is frame-dependent, as we'll see in a moment; but it holds in the Alice/Z frame, so it applies at this point.)

Correct.

First, remember Bob's measurements are relative to Bob's frame. Relative to Bob's frame, his own signal has no redshift when it is emitted; it gets redshifted when it is reflected, because the reflector is moving away from the source of the signal (Bob).

That's a good point, rather than my comment that his signal is "redshifted in both directions." I made that comment in relation to your next comment about Alice's signal being blueshifted relative to Bob. I was imagining it from the Alice-Z frame where Bob's signal is redshifted coming in towards planet Z's detector, and Alice's signal isn't doppler shifted at all. So I see what you mean, you'd need to switch between Bob's and Alice's frame to get a redshift in both directions, seeing as Bob's redshifted signal coming into Alice-Z's frame won't appear to be doppler shifted at all from Alice-Z's frame once it leaves the reflector on Planet Z, although that same signal will be observed to be redshifted by Bob. Did I get that right?

Both signals then get redshifted, relative to Bob's frame, when they are reflected at Z, and by the same factor. I'm pretty sure this redshift factor is greater than the blueshift factor of Alice's emitted signal, relative to Bob; which means that both reflected signals will be observed by Bob to be redshifted, but Alice's will be redshifted by less. However, I haven't done a detailed calculation to confirm this.

My reasoning in thinking that Bob's own emitted signal would be a lower frequency than Alice's signal after it was reflected from Z and detected on his (Bob's) detector is the following: Since Alice is traveling in the same frame as Z, as you mentioned, her signal is blueshifted relative to Bob's when they hit the reflector on Z. Therefore, Alice's signal is being reflected from Z toward Bob at a higher frequency than Bob's signal emitted from his frame. Therefore, he will read his reflected frequency as being lower than Alice's. Is that reasoning sound?

 

[PeterDonis]

I was imagining it from the Alice-Z frame where Bob's signal is redshifted coming in towards planet Z's detector, and Alice's signal isn't doppler shifted at all.

I see--so in this frame, we have Bob's signal being redshifted on emission (since Bob is moving away from the emitted signal), and again on detection by Bob (since Bob is moving away from the reflector). Whereas Alice's signal is only redshifted once, on detection by Bob.

Bob's redshifted signal coming into Alice-Z's frame won't appear to be doppler shifted at all from Alice-Z's frame once it leaves the reflector on Planet Z, although that same signal will be observed to be redshifted by Bob. Did I get that right?

I think this is basically saying the same thing I said above; if so, yes, it's right.

Since Alice is traveling in the same frame as Z, as you mentioned, her signal is blueshifted relative to Bob's when they hit the reflector on Z. Therefore, Alice's signal is being reflected from Z toward Bob at a higher frequency than Bob's signal emitted from his frame. Therefore, he will read his reflected frequency as being lower than Alice's. Is that reasoning sound?

Yes, this is more or less a different way of saying what I said in my previous post.

 

[Orodruin]

Both signals then get redshifted, relative to Bob's frame, when they are reflected at Z, and by the same factor. I'm pretty sure this redshift factor is greater than the blueshift factor of Alice's emitted signal, relative to Bob; which means that both reflected signals will be observed by Bob to be redshifted, but Alice's will be redshifted by less. However, I haven't done a detailed calculation to confirm this.

There really is no calculation necessary, the frequency of the reflection Alice's light in her frame is the same as the light she emitted. Since the reflected light is traveling in the same direction as Bob, it will get redshifted in his frame.

 

[PeterDonis]

the frequency of the reflection Alice's light in her frame is the same as the light she emitted. Since the reflected light is traveling in the same direction as Bob, it will get redshifted in his frame.

Ah, good point.

 

[ghwellsjr]

Thought experiment: Alice, Bob, and the planet Zolan.

Alice is floating in space 10 light seconds from the planet Zolan (Z) with a radio emitter and detector. She and planet Z are in the same rest frame. Bob has the same devices as Alice but is on planet Z. Bob then accelerates from planet Z toward Alice and then stops accelerating once he reaches a velocity of 0.5c relative to planet Z (and Alice, of course).

At the instant Bob reaches Alice's position, they both simultaneously shine their radio emitters at a reflector on planet Z and leave the emitters running. Say the frequency of the radiation is 1000 kiloHertz.

Here are my assumptions. 1) At the end of 20 seconds, Alice's detector will detect both her's and Bob's emitted signal, and read that the signal reflected by her emitter is still 1000 kH, but Bob's is less than 1000 kH.

2) Bob, on the other hand, from within his reference frame, will detect both his and Alice's signal at the same time, but the time elapsed will be well after 20 seconds according to his clock. Furthermore, both his and Alice's signal will read less than 1000 kH on his detector, although his emitted signal's frequency will read even less than Alice's, the reason being that Bob's reading of his own signal has a doppler shift going in both directions, while Bob's reading of Alice's signal only has a doppler shift after being reflected from planet Z.

Are these assumptions correct?

Yes, you show a very good understanding of what is happening in your scenario.

Here are some spacetime diagrams depicting your scenario. Alice is shown as the thick blue line and her radio emissions are shown as the thin blue lines. Rather than show a million such lines per second, I show only one which is enough to illustrate the Doppler relationships. Planet Z is shown as the thick red line and the return signals off its reflector are shown as the thin red lines. Bob is shown as the thick black line starting at the bottom on planet Z and accelerating to the left toward Alice (blue). Before he gets to her, he has achieved a speed of 0.5 c in the mutual rest frame of Alice and planet Z. When he gets to her, they both start emitting their signals toward the reflector on planet Z. Bob's signals are shown as the thin black lines. The dots mark off one-second intervals of Proper Time for each object/observer.

First is the rest frame of Alice showing her radio emissions and its reflections:

She receives the reflected signal at the same rate that she sent it but Bob receives ten seconds worth of signal in about 17.4 seconds which is a Doppler ratio of 0.57. This conforms to the theoretical value for a speed of 0.5c.

Second is the rest frame of Alice showing Bob's radio emissions and its reflections:

I show 7 seconds of Bob's signals being received on planet Z in 13 seconds which is approximately the same Doppler ratio of 0.54 that Bob received Alice's signals. But then they get Doppler shifted by the same amount again going from planet Z to Bob so that he receives the 7 seconds of signals in about 20.7 seconds for a Doppler ratio of 0.34 which is close to the theoretical value of 0.33.

Third is the same as the first diagram but Lorentz Transformed to the rest frame of Bob after his acceleration:

​

Note that the same Doppler relationships are indicated in this diagram, showing that Doppler ratios are frame invariant.

Finally, we have the second diagram transformed to the Bob's rest frame after he accelerates:

Again, we see that the Doppler ratios are the same as in the second diagram.