Struggling with a special relativity "paradox"

[KipIngram]

TL;DR Summary

I describe a scenario that seems to present a paradox. I understand what the "correct" answer is supposed to be, but one side of the analysis seems to be paradoxical, and I can't see where my error is. Please help!

Ok, I hope someone can help me see how to sort this out.

Alice has a full-frame (no rolling shutter) video camera that records exactly 30 frames per second. It's mounted to a telescope looking far out into space.

Bob is out there in space with a digital clock that reads out to the millisecond level. He is moving at half the speed of light such that when he makes his closest approach to Alice he will be in her telescope's field of view, and it will take him one second (in Alice time) to cross that field of view. His closest approach is far enough away that the line separating him from Alice is very nearly perpendicular to his direction of motion the entire time he's in her field of view.

Ok, so Alice is going to grab 30 images of Bob's clock as he moves through her field of view. The question is simple. What will be the time delta shown on Bob's clock in consecutive images?

If we regard Alice as stationary, then Bob is in motion and time dilation means his "time rate" is running slower than Alice's. Specifically his clock is running at about 86.7% as fast as Alices. Alices frames are 33.3 ms apart by her clock - during that interval Bob's clock should advance 33.3*0.867 = 28.9 ms, so she should get 30 frames that each show values 28.9 ms apart, frame to frame.

Ok, good. I believe this to be the correct answer, and in fact if Bob were also equipped with a camera and scope and Alice also with a clock, Bob would get the same result from imaging Alice's clock.

Here is the problem. Now I want to be Bob, and I want to predict what Alice is going to see. I see Alice in motion at half the speed of light. I know that means her clock runs 86.7% as fast as mine, so it takes longer than 30ms of my time for her camera to advance from frame to frame. In fact, it seems entirely obvious to me that she will capture images from my clock that are 33.3/0.867 = 38.5 ms apart.

So that is my problem. I believe Alice will get images that show 28.9 ms delta, frame to frame. What is wrong with my Bob-side analysis that predicts 38.5?

I think I could have framed this problem with Bob and Alice moving directly toward or away from one another, but I was trying to avoid having to deal with standard Doppler effect issues. My understanding is that the piece of the puzzle I'm focusing on is "relativistic transverse Doppler effect."

I've been pulling my hair over this for about two days now - if someone can straighten it out for me I'll be much obliged.

 

[PeroK]

You have quite a complicated scenario here. The whole idea of "coming into a field of view" looks a bit suspect to me. Let's try to sort that out first.

Alice has two markers out in space, a certain distance apart, at rest in her frame. She will start the experiment when Bob passes the first marker and end the experiment when he passes the second. Let's say the markers are $1$ light second apart. For her the experiment lasts $2s$.

In Bob's frame, the markers are less than $1$ light second apart, owing to length contraction, and moving towards him at half the speed of light. From his point of view the experiment lasts less time.

That's an asymmetry in your set-up to begin with.

Now, we put two clocks at the markers, synchronised in Alice's frame. These allow Alice to measure the time dilation of Bob's clock as he moves past the markers.

From Bob's reference frame, the two clocks are not only showing time dilation, but are out of sync. The first clock (in his reference frame) lags behind the other. This is due to the relativity of simultaneity. So, although the second clock runs slow, it shows a time that is consistent with what Alice has recorded. Because, according to Bob, when the experiment started it was in advance of the first clock.

If you can, it is worth going through the maths of this three-clock scenario to check the self-consistency of SR.

PS I realize I didn't quite finish the answer.

As far as Alice is concerned the experiment lasted $2s$ and Bob traveled between the markers during it.

If we take the first clock as syncronised with Alice's camera, then in Bob's frame the experiment started when he passed that first marker/clock, but it lasted longer than $2s$ and the second marker/clock was behind him when it ended: the last tick of Alice's clock or click of her camera.

There is an asymmetry, therefore, in where and when the experiment ended. You hid this in your nonchalant "field of view", which was the conceptual root of the paradox, as it assumed the experiment had a universally well-defined start and end.

 

[KipIngram]

You have quite a complicated scenario here. The whole idea of "coming into a field of view" looks a bit suspect to me. Let's try to sort that out first.

Alice has two markers out in space, a certain distance apart, at rest in her frame. She will start the experiment when Bob passes the first marker and end the experiment when he passes the second. Let's say the markers are $1$ light second apart. For her the experiment lasts $2s$.

In Bob's frame, the markers are less than $1$ light second apart, owing to length contraction, and moving towards him at half the speed of light. From his point of view the experiment lasts less time.

That's an asymmetry in your set-up to begin with.

Now, we put two clocks at the markers, synchronised in Alice's frame. These allow Alice to measure the time dilation of Bob's clock as he moves past the markers.

From Bob's reference frame, the two clocks are not only showing time dilation, but are out of sync. The first clock (in his reference frame) lags behind the other. This is due to the relativity of simultaneity. So, although the second clock runs slow, it shows a time that is consistent with what Alice has recorded. Because, according to Bob, when the experiment started it was in advance of the first clock.

If you can, it is worth going through the maths of this three-clock scenario to check the self-consistency of SR.

PS I realize I didn't quite finish the answer.

As far as Alice is concerned the experiment lasted $2s$ and Bob traveled between the markers during it.

If we take the first clock as syncronised with Alice's camera, then in Bob's frame the experiment started when he passed that first marker/clock, but it lasted longer than $2s$ and the second marker/clock was behind him when it ended: the last tick of Alice's clock or click of her camera.

There is an asymmetry, therefore, in where and when the experiment ended. You hid this in your nonchalant "field of view", which was the conceptual root of the paradox, as it assumed the experiment had a universally well-defined start and end.

Hmmm. Ok, I'll try to mentally fold those aspects into it. One of the things I wanted to get was an inarguable "result" - the recorded camera images. Each one has to show some specific reading off of Bob's clock, and there's no "interpretation puzzle" around that - the pixels are what they are. I've nosed around online about this for a couple of days, and it seems generally agreed that the "Alice side analysis" (Bob's clock running slower, camera images show numbers 28.9 ms apart) seems to be the accepted one. So where I'm getting hung up is having Bob predict what Alice's camera images are going to be.

Regarding the beginning and end of the experiment, all that I think matters is that it's long enough for Alice to record multiple images. It's not really important that it's exactly one second. As long as Alice catches at least two pictures of Bob's clock, she can determine the difference of the numbers presented.

You're right about it being a bit complicated - the main thing I was going for there was that I wanted to be able to completely neglect the change in radial distance between Bob and Alice frame-to-frame. I figured with the setup as described, I could imagine the closest approach being as large as I wanted it to be - even unreasonably large if necessary, like a million light years or something. Since it is a "thought experiment" and all.

Thanks for the feedback - I'll ponder on it.

 

[PeroK]

What Alice records is a physical fact. No one can dispute that.

 

[PAllen]

Look, I think the main thing here is just a form of transverse Doppler. The gist is that what is transverse Doppler in one frame (thus seeing only time dilation, no other frequency shift) is not transverse doppler in the other frame. Bob’s frame sees Alice receive the signals at an angle, with significant redshift expected per Bob. This compensates for Alice’s time dilation per Bob, so they both make the same predictions. This is all covered in any discussion of transverse Doppler, and you should be able to apply it to your situation.

 

[KipIngram]

What Alice records is a physical fact. No one can dispute that.

Right - exactly. So how would Bob go about predicting what Alice will record? My "simplest attempt" at that predicts exactly the opposite of what you expect looking at it from her side.

I guess Bob could say, "Well, Alice sees me moving, so she will see my clock running slower... etc." But that seems like a "shortcut" - it seems like he should be able to make a prediction based on his own perspective, and it should support relativity, instead of using relativity to make the prediction.

 

[PeroK]

Right - exactly. So how would Bob go about predicting what Alice will record?

Bob can predict what Alice will record using the laws of physics. He could analyse the problem using her reference frame; or, he could analyse the problem from his reference frame. Either way, he must get the same result.

If we look at my analysis in post #2, Bob will establish that Alice starts the experiment when he passes the first marker and ends the experiment when he passes the second marker. Those are two events and represent points in spacetime. The number of times that Bob's clock ticks between those two events is invariant (*). It doesn't matter who measures them. You could imagine that Bob records his clock reading at the first marker, and again at the second marker. He could make a copy of the readings and drop them off at the markers.

(*) this is because both those events take place at Bob's location.

Then anyone can come along and see that Bob's clock recorded about 1.7 seconds between those two events. That would be the result of the experiment. Bob's clock reading at the second marker minus his clock reading at the first marker is 1.7 seconds.

Everyone, regardless of their state of motion relative to Alice or Bob will agree on that.

Notice, for example, that I can do that analysis without being there or being part of the experiment. If someone describes the experiment in unambiguous terms, then I can calculate the result without any complications. As can Bob, as can Alice, as can anyone else.

PS If I choose Alice's frame, then I calculate: Bob traveled 1 light-second between the markers at half the speed of light. This was 2 seconds in Alice's frame, so Bob's clock recorded 1.7s.

If I do this using Bob's frame, then the markers are 0.85 light seconds apart and traveling at half the speed of light, so again Bob's clock records 1.7 s between the first and second markers passing him.

 

[KipIngram]

Bob can predict what Alice will record using the laws of physics. He could analyse the problem using her reference frame; or, he could analyse the problem from his reference frame. Either way, he must get the same result.

If we look at my analysis in post #2, Bob will establish that Alice starts the experiment when he passes the first marker and ends the experiment when he passes the second marker. Those are two events and represent points in spacetime. The number of times that Bob's clock ticks between those two events is invariant (*). It doesn't matter who measures them. You could imagine that Bob records his clock reading at the first marker, and again at the second marker. He could make a copy of the readings and drop them off at the markers.

(*) this is because both those events take place at Bob's location.

Then anyone can come along and see that Bob's clock recorded about 1.7 seconds between those two events. That would be the result of the experiment. Bob's clock reading at the second marker minus his clock reading at the first marker is 1.7 seconds.

Everyone, regardless of their state of motion relative to Alice or Bob will agree on that.

Notice, for example, that I can do that analysis without being there or being part of the experiment. If someone describes the experiment in unambiguous terms, then I can calculate the result without any complications. As can Bob, as can Alice, as can anyone else.

PS If I choose Alice's frame, then I calculate: Bob traveled 1 light-second between the markers at half the speed of light. This was 2 seconds in Alice's frame, so Bob's clock recorded 1.7s.

If I do this using Bob's frame, then the markers are 0.85 light seconds apart and traveling at half the speed of light, so again Bob's clock records 1.7 s between the first and second markers passing him.

I agree with that. But what Bob is wanting to predict is what the pixels recorded by Alice's camera look like. He knows Alice's camera is designed to catch a frame 30 times per second per Alice's clock. From his perspective, Alice's clock is running slower than his own, so her camera should fire less than 30 times per second, by Bob's clock. That should imply that the images of his clock that she acquires show differences larger than 1/30, shouldn't it?

 

[KipIngram]

I do want to re-iterate that I feel certain something is wrong with that simple Bob analysis. I just can't see what it is. :-|

 

[PeterDonis]

From his perspective, Alice's clock is running slower than his own

But Alice is also moving relative to him, so successive light signals from him take different times to reach her telescope and are frequency shifted relative to her. The correct equations for transverse Doppler take both of these effects into account. As @PAllen has already noted, they act in opposite ways. You certainly can't get the right answer by just considering time dilation alone, as you have already found.

 

[PeroK]

I agree with that. But what Bob is wanting to predict is what the pixels recorded by Alice's camera look like. He knows Alice's camera is designed to catch a frame 30 times per second per Alice's clock. From his perspective, Alice's clock is running slower than his own, so her camera should fire less than 30 times per second, by Bob's clock. That should imply that the images of his clock that she acquires show differences larger than 1/30, shouldn't it?

Okay, so here's the problem. You have put Alice a long way away so that you can disguise the relativity of simultaneity. What you want to do is use Alice's local clock back on Earth as a master clock not just for her, but for Bob's frame as well. Essentially you are saying: at time $t$ in Bob's frame, at any point in space, the time in Alice's frame (at any point is space) is given by Alice's master clock back on Earth.

That ignores the relativity of simultaneity and the "leading clocks lag".

Let's look at an example, using the analysis of the two markers.

1) In Alice's frame, Bob passes the first marker at $t = 0$ and the second at $t = 2s$.

2) What you want to say is that if Bob passes the first marker at $t'=0$ (his frame) and the second at $t' = 1.7s$ (his frame - that's what everyone agrees on), then using time dilation, Bob calculates that the second event occurs at $t = 1.5s$ in Alice's frame. Double time-dilation if you like.

And, we can clear out all the extraneous complications of cameras and everything else. This is the essense of the problem. You want to apply time dilation again using Bob's frame to get this contradiction:

1) Alice says Bob passes the second marker at $t = 2s$ (in her frame).

2) Bob says Alice should have recorded $t = 1.5s$ (in her frame): because, he recorded $1.7s$ and her clock is dilated relative to his.

The solution, as ever, is relativity of simultaneity and "leading clocks lag".

What Bob does calculate is that a clock at the first marker records only $t = 1.7s$ when he passes the second marker (in his frame). But, he is no longer at the first marker. He is at the second marker. Alice's clock at the second marker reads $t = 2s$. And, Bob can conclude that at the point Alice will record a photograph of him there. Sure, the clock at that marker is running slow, but it is ahead of the first of Alice's clocks. And, indeed, the clock at the next marker (where Bob will be when the third photo is taken) is ahead of the second clock. And, when Bob passes that clock it will indeed read $t = 4s$. The previous two clocks will read $3.7s$ and $3.4s$ (in Bob's frame).

In short, it is a combination of time dilation and relativity of simulaneity that ensure that the Alice's clocks read $2s, 4s, 6s, 8s$ in Bob's frame as he passes them. And, in that way, they agree on what physical photographs are taken.

 

[PeroK]

@KipIngram PS it's worth working through this basic scenario of time dilation plus relativity of simultaneity before you do anything else. This will save you from burying this basic apparent paradox in a more complicated scenario, which is essentially what you've done with these remote cameras.

 

[KipIngram]

But Alice is also moving relative to him, so successive light signals from him take different times to reach her telescope and are frequency shifted relative to her. The correct equations for transverse Doppler take both of these effects into account. As @PAllen has already noted, they act in opposite ways. You certainly can't get the right answer by just considering time dilation alone, as you have already found.

But we can make that part of the effect as small as we wish to make it, by making the "closest approach" distance further. We can make the angle Bob traverses during that one frame-to-frame interval that contains the closest approach as tiny as we want it to be. I don't think that's responsible for the problem.

Alternatively, we could have Bob going in a (very, very) large circle around Alice. Then the distance won't change at all, and what we'll be reducing to negligibility will be the change in Bob's direction of motion during that small bit of time. We can make his path approximate a straight line as closely as we wish.

 

[PeterDonis]

we can make that part of the effect as small as we wish to make it, by making the "closest approach" distance further. We can make the angle Bob traverses during that one frame-to-frame interval that contains the closest approach as tiny as we want it to be.

You have this backwards. The angle Bob has to traverse is constant; it's determined by the field of view of Alice's camera, which is an angle, not a distance. If Bob is farther away, he has to move faster to traverse the same angle with respect to Alice in the same time with respect to Alice (30 frames of Alice's camera), and covers a larger distance.

 

[PAllen]

But we can make that part of the effect as small as we wish to make it, by making the "closest approach" distance further. We can make the angle Bob traverses during that one frame-to-frame interval that contains the closest approach as tiny as we want it to be. I don't think that's responsible for the problem.

Alternatively, we could have Bob going in a (very, very) large circle around Alice. Then the distance won't change at all, and what we'll be reducing to negligibility will be the change in Bob's direction of motion during that small bit of time. We can make his path approximate a straight line as closely as we wish.

Look, if in Alice's frame, a signal is emitted along the line of sight when both are at closest approach, then in Bob's frame, Alice will receive this signal when Alice is way beyond closest approach. The longer you make the closest approach distance, the larger this delay will be, so all the angles will be the same. Thus, in Bob's frame, there is a substantial Doppler effect plus time dilation, and nothing you propose can reduce this. The problem just scales up. As I have mentioned, when you combine these affects, you find that the facts more easily derived in Alice frame also work exactly in Bob's frame (with different explanation). I am deliberately avoiding (so far) giving a complete, detailed computation for you, because I believe you should be able to work this out. Though I am sometimes reluctant to recommend it, you can consult wikipedia for transverse Doppler (their article is pretty good, but could be clarified in spots). Note also, that sequences of signals or image snapshots are frequency shifted just like a wave.

 

[PeterDonis]

we can make that part of the effect as small as we wish to make it, by making the "closest approach" distance further.

In addition to my previous comment, you need to bear in mind that the light recorded by Alice's camera for each frame is not light that was emitted at the same time (in Alice's frame) by Bob; it is light that is received at the same time (in Alice's frame) by the camera. Light at different pixels on the camera will have taken different times to arrive there. (Looking up Penrose-Terrell rotation might also be helpful.)

 

[KipIngram]

You have this backwards. The angle Bob has to traverse is constant; it's determined by the field of view of Alice's camera, which is an angle, not a distance. If Bob is farther away, he has to move faster to traverse the same angle with respect to Alice in the same time with respect to Alice (30 frames of Alice's camera), and covers a larger distance.

I assumed that if we wanted to move Bob further away we'd give Alice a higher magnification. I really shouldn't have even mentioned the field of view, or that it took one second for Bob to cross it, etc. All that really matters is that Alice sees Bob long enough to capture two frames. Those other details are irrelevant.

 

[KipIngram]

Look, if in Alice's frame, a signal is emitted along the line of sight when both are at closest approach, then in Bob's frame, Alice will receive this signal when Alice is way beyond closest approach. The longer you make the closest approach distance, the larger this delay will be, so all the angles will be the same. Thus, in Bob's frame, there is a substantial Doppler effect plus time dilation, and nothing you propose can reduce this. The problem just scales up. As I have mentioned, when you combine these affects, you find that the facts more easily derived in Alice frame also work exactly in Bob's frame (with different explanation). I am deliberately avoiding (so far) giving a complete, detailed computation for you, because I believe you should be able to work this out. Though I am sometimes reluctant to recommend it, you can consult wikipedia for transverse Doppler (their article is pretty good, but could be clarified in spots). Note also, that sequences of signals or image snapshots are frequency shifted just like a wave.

Yes, the light that Alice receives was emitted by Bob much earlier, But as long as Alice's scope is pointed in the direction of the point of closest approach, she will see the light he emitted while he was there. The fact that he has moved on in the mean time doesn't matter, right?

 

[KipIngram]

Okay, so here's the problem. You have put Alice a long way away so that you can disguise the relativity of simultaneity. What you want to do is use Alice's local clock back on Earth as a master clock not just for her, but for Bob's frame as well. Essentially you are saying: at time $t$ in Bob's frame, at any point in space, the time in Alice's frame (at any point is space) is given by Alice's master clock back on Earth.

That ignores the relativity of simultaneity and the "leading clocks lag".

Let's look at an example, using the analysis of the two markers.

1) In Alice's frame, Bob passes the first marker at $t = 0$ and the second at $t = 2s$.

2) What you want to say is that if Bob passes the first marker at $t'=0$ (his frame) and the second at $t' = 1.7s$ (his frame - that's what everyone agrees on), then using time dilation, Bob calculates that the second event occurs at $t = 1.5s$ in Alice's frame. Double time-dilation if you like.

And, we can clear out all the extraneous complications of cameras and everything else. This is the essense of the problem. You want to apply time dilation again using Bob's frame to get this contradiction:

1) Alice says Bob passes the second marker at $t = 2s$ (in her frame).

2) Bob says Alice should have recorded $t = 1.5s$ (in her frame): because, he recorded $1.7s$ and her clock is dilated relative to his.

The solution, as ever, is relativity of simultaneity and "leading clocks lag".

What Bob does calculate is that a clock at the first marker records only $t = 1.7s$ when he passes the second marker (in his frame). But, he is no longer at the first marker. He is at the second marker. Alice's clock at the second marker reads $t = 2s$. And, Bob can conclude that at the point Alice will record a photograph of him there. Sure, the clock at that marker is running slow, but it is ahead of the first of Alice's clocks. And, indeed, the clock at the next marker (where Bob will be when the third photo is taken) is ahead of the second clock. And, when Bob passes that clock it will indeed read $t = 4s$. The previous two clocks will read $3.7s$ and $3.4s$ (in Bob's frame).

In short, it is a combination of time dilation and relativity of simulaneity that ensure that the Alice's clocks read $2s, 4s, 6s, 8s$ in Bob's frame as he passes them. And, in that way, they agree on what physical photographs are taken.

I don't want to get rid of the camera - one of the things I like best about this "situation" is that we get a result that goes beyond just making calculations around things - Alice's camera will record pixels, and Alice's "analysis" and a correct Bob analysis should predict the same pixels. Think of it this way. Bob has a timepiece - it's actually a clock and is counting off milliseconds. Alice has a timepiece too - it's a camera that ticks of intervals of 1/30 of a second. In Alice's frame, Bob's clock is running slow, so he will tick off fewer than 33.3 ms between two "ticks" of Alice's timepiece. In Bob's frame, Alice's clock is running slow - the interval between two of her ticks will be more than 33.3 of Bob's ticks. But the images recorded by Alice's camera can only show one number each, and we can calculate the difference between any adjacent pair of those numbers.

As I said earlier, I believe that difference will be less than 33.3 ms; I think working in Alice's frame gives us the right answer. I'm just trying to figure out the right way to analyze the problem from Bob's frame.

Note that if we drop the whole one second wide field of view and just say that Alice is going to receive two images from Bob, we can stipulate that those are centered on the point of closest approach. Then both points are exactly the same distance from Alice. There will be a tiny radial component of motion (decreasing the distance on the first frame and increasing it on the second frame). But if Bob is far enough away that becomes extremely small.

I'm absolutely not trying to be obtuse about this, but so far none of the issues that have been brought up really feel like they address the issue. And I'd really like to get this sorted out.

 

[KipIngram]

I guess what I'm fretting over here is this: Alice predicts that she will see a difference of less than 33.3 ms in her images, because Bob's clock is running slower than her's. How can Bob predict that as well, while still claiming that Alice's clock is running slower than his?

 

[PAllen]

Yes, the light that Alice receives was emitted by Bob much earlier, But as long as Alice's scope is pointed in the direction of the point of closest approach, she will see the light he emitted while he was there. The fact that he has moved on in the mean time doesn't matter, right?

You keep going back to Alice frame. Per Bob, it is Alice who has moved way beyond the closest approach before receiving signals. Because Alice is moving per Bob, her telescope can receive oblique signals despite being perpendicular to the relative motion per Alice. The result is that Bob has to account for a combination of Doppler and time dilation to predict what Alice sees. The result is then consistent with the facts derived in Alice frame.

 

[PAllen]

I guess what I'm fretting over here is this: Alice predicts that she will see a difference of less than 33.3 ms in her images, because Bob's clock is running slower than her's. How can Bob predict that as well, while still claiming that Alice's clock is running slower than his?

Because Bob thinks Alice will be subject to redshift as well as time dilation. Thus, take snapshot frequency Alice computes for Bob. Bob expects this frequency will be red shifted per Alice (longer time between images), but also Alice clock is running slow so this frequency decrease will be decreased (not increased). The combination means Bob
predicts his 28.9 ms intervals get lengthened to 33.3, exactly consistent with Alice

 

[PeterDonis]

I assumed that if we wanted to move Bob further away we'd give Alice a higher magnification.

Okay, then let's take this to the extreme: Bob is so far away that the light rays coming from him into Alice's telescope are essentially parallel. Then Alice can place two cameras, one at the left edge of her telescope and one at the right edge, to record the light rays coming from Bob along those two parallel lines. (Or you can view these as two sets of pixels on her camera, recording the two images coming from Bob.)

Now we have, as you say, completely eliminated any change in the light travel time of the two successive light signals from Bob to Alice. So in Alice's frame, the two emission events are separated in time by exactly the same amount as the two detection events--1/30 of a second. And since Bob's clock is ticking slow relative to Alice, the times recorded on Bob's clock will differ by less than 1/30 of a second in the two images. So far so good.

Now, in Bob's frame, he doesn't have to worry about light propagation at all. He can calculate as though there were two "markers", at rest relative to Alice, spaced apart along his line of travel by exactly the right amount so that the light he emits at the first marker hits the pixels on the left edge of Alice's telescope, and the light he emits at the second marker hits the pixels on the right edge of Alice's telescope. (Or, if you like, there are two very long rods, at rest relative to Alice, extending out from the two edges of Alice's telescope along parallel lines; Bob emits two successive light signals along these rods.) The two markers (or rods) are spaced just right so that it takes, by Alice's clock, 1/30 of a second for Bob to travel between them. But as has already been pointed out, the distance between the markers (or rods) is length contracted in Bob's frame, and they are moving at the same speed relative to Bob that Bob is moving relative to Alice, so Bob will take less than 1/30 of a second, by his clock, to see the two markers in succession.

However, Bob will also compute that the Alice clock on the second marker is set ahead of the Alice clock on the first marker. (I am basically restating here what @PeroK said earlier.) In other words, if Bob computes what the reading is on the second marker's clock at the same time, in his frame, that he passes the first marker's clock, he will compute that the two readings are not the same: the second marker clock's reading is later. So when Bob passes the second marker clock, he expects to see a reading 33.3 ms later than the reading he saw on the first marker clock, even though it only took 28.9 ms by his clock to travel between them, because the second clock is set ahead in his frame. This is a manifestation of relativity of simultaneity: two Alice clocks show the same readings at the same times in Alice's frame, but not in Bob's frame.

 

[Nugatory]

I don't want to get rid of the camera - one of the things I like best about this "situation" is that we get a result that goes beyond just making calculations around things - Alice's camera will record pixels, and Alice's "analysis" and a correct Bob analysis should predict the same pixels

That's a reasonable approach - you are concentrating on the invariants, the things that have to come out the same no matter which frame you use to analyze the problem. However, there are some subtle pitfalls in the way that you're trying to use the the angle subtended by Bob's passage across Alice's field of view, so you might want to try a version that is easier to analyze first:

Instead of Alice sitting at one point with one camera we will position a bunch of cameras, all at rest relative to Alice, on a line that passes through Alice's position and parallel to Bob's direction of travel. All of these cameras are doing their 30 frames/second thing, and because they are all at rest relative to one another and Alice we can arrange so that they're synchronized, all capturing at the same moment. Now have Bob send the images of his clock using a tightly collimated beam of light exactly perpendicular to he's direction of travel; now only one of Alice's ensemble of cameras will capture an image, and it will be of Bob's clock at the moment when Bob is directly above that camera (using the frame in which Alice is at rest - "directly above" is frame dependent). This ensures that the light travel time from Bob to detecting camera is the same for every captured image and avoids the complications that result from the Bob-Alice distance not being constant in your original thought experiment.

When you analyze this problem using the frame in which Alice is at rest, you'll get the same result for Alice's observation as in your original thought experiment; it's basically the same problem except that instead of having thirty images on one camera Alice has one image on each of thirty cameras. However, you will finf it much much easier to use Bob's frame to calculate the same results. Two big hints:
1) If we use Alice's frame, the flash of light captured by one of Alice's cameras will have been emitted by Bob when he was directly above that camera. However, this will not be the case if we use Bob's frame; from his point of view the cameras are moving so that camera that catches the flash was not underneath him at emission time but moved into that position while the light was in flight.
2) The whole problem becomes easy if you write down the coordinates of each emission and image capture event using Alice's frame, and then Lorentz transform these into Bob's coordinates.

 

[PeterDonis]

Instead of Alice sitting at one point with one camera we will position a bunch of cameras, all at rest relative to Alice, on a line that passes through Alice's position and parallel to Bob's direction of travel.

For @KipIngram, this is basically equivalent to what I proposed in #23; @Nugatory's bunch of cameras is equivalent to my separate sets of pixels at the left and right edges of Alice's telescope.

 

[Nugatory]

For @KipIngram, this is basically equivalent to what I proposed in #23; @Nugatory's bunch of cameras is equivalent to my separate sets of pixels at the left and right edges of Alice's telescope.

Indeed it is

 

[PAllen]

Just a quick comment now, no time right now.

It is important not to mix up two complementary situation:

1) When emission occurs at mutual closest approach of the two observers. Then there is redshift (decrease of frequency) of reception compared to emission. This is an obvious consequence of pure time dilation when analyzed in receiver frame, but is due to dominance or Doppler redshift in emitter frame from receiver having moved well past closest approach at event of reception.

2) When reception occurs at mutual closest approach of the two observers. There is blue shift (increased frequency) of reception compared to emission. This is an obvious consequence of pure time dilation when analyzed in the emitter frame, but is due to dominance of Doppler blueshift in receiver frame from emitter having been approaching when signal was emitted to arrive at closest approach on reception.

I have interpreted the OP scenario is basically a case of (1). However, I think there might be some mix up in this thread with complementary scenario (2).

 

[KipIngram]

Ok, so I'm still thinking about this stuff, and I think I may see something interesting. Someone of you may have said something earlier that triggered this - if so, thanks!

So I'm presuming the closest approach distance between Alice and Bob is quite large - large enough to make the angle traversed during the "interesting part" of the experiment quite small. Ok, so that means that the signals Alice sees

Just a quick comment now, no time right now.

It is important not to mix up two complementary situation:

1) When emission occurs at mutual closest approach of the two observers. Then there is redshift (decrease of frequency) of reception compared to emission. This is an obvious consequence of pure time dilation when analyzed in receiver frame, but is due to dominance or Doppler redshift in emitter frame from receiver having moved well past closest approach at event of reception.

2) When reception occurs at mutual closest approach of the two observers. There is blue shift (increased frequency) of reception compared to emission. This is an obvious consequence of pure time dilation when analyzed in the emitter frame, but is due to dominance of Doppler blueshift in receiver frame from emitter having been approaching when signal was emitted to arrive at closest approach on reception.

I have interpreted the OP scenario is basically a case of (1). However, I think there might be some mix up in this thread with complementary scenario (2).

Ah - I need to think about this some, but this feels like it's on the right track to me; thanks so much.

I was about to post anyway with a question. I think I'm getting this right, but I'm not sure. Say we have this whole situation unfolding, and at some arbitrary point in time Bob's clock switches to a new millisecond reading. That first instant of it displaying the new value - that "signal front" will move away from Bob in a way that Bob perceives as a spherical wave front, and also in a way that Alice perceives as a spherical wavefront, right? Bob will see it as a spherical wavefront moving away from him (since he's stationary in his frame), and Alice will see it as a spherical wavefront moving away from where Bob was (in her frame) at the moment of emission - is that correct? And both will perceive those spherical wavefronts as moving at the speed of light?

I realize I'm sort of giving both of them a "God's eye view" here - I know that all they can really perceive is what's right with them right "now." But they can do calculations to reason about what's going on elsewhere, and shouldn't the above description fit the proper calculations?

If that's correct, I think I may be onto something. Bob emits some signal front when he's "at closest approach." In Alice's frame, that just move straight toward her on the line of closest approach. But in Bob's frame, Alice is moving, and by the time that wave catches up to her she's moved well away from the point of closest approach. From Bob's perspective, his signal front has to "catch up with her," and that means it will travel substantially further than the closest approach distance (by Bob's reckoning). And the signal front emitted one tick later will have to travel even further, and so on. It's late and I'm going to bed shortly, but I wonder if when I work the math for this out tomorrow I'll find that it makes things just right.

For Alice, the signals emitted near closest approach just travel straight to her, along the separation vector of closest approach. My earlier reasoning that the radial distance variation is negligible at this point holds, in Alice's frame. But in Bob's frame those signals follow an angled path to Alice, and the the change in direct distance frame to frame is certainly significant. And this is something that I can't "immunize against" by separating them further - it's an effect that's "congruent," so to speak - I can make the triangle bigger, but I can't change its shape. If I put them 10x further apart, it takes 10x longer for the signal to reach Alice, and Bob has seen her move 10x further during that period.

So, I think the flaw in my earlier reasoning was that I was allowing Bob to also assume the signals went to Alice in a straight shot over the closest approach separation vector, and that's clearly invalid.

Wow - I'm going to sleep well; I feel like I may finally be on the verge of sorting this out.

One (or more) of you may have already noted this, but it just didn't sink in for me until just now. You get full credit; I just had to let it roll around in my gray matter for a while.

Night, guys! Thanks for pitching in - I really appreciate it.

 

[jk22]

I think the paradox comes from the usage of formulas :

Time dilation factor : .86

Distance=speed*time hence space dilation.

This contradicts the known space contraction. I don't understand that.

 

[Ibix]

I think the paradox comes from the usage of formulas :

No, or at least not for the reason you suggest.

This contradicts the known space contraction. I don't understand that.

Because your clocks only tick slowly for me, and I don't use your clocks to measure time. Can I suggest you start a separate thread for a different confusion about frame changes? It's going to get very confusing in here if we have two intertwined conversations about different confusions on related topics.

 

[PeroK]

Alternatively, we could have Bob going in a (very, very) large circle around Alice. Then the distance won't change at all, and what we'll be reducing to negligibility will be the change in Bob's direction of motion during that small bit of time. We can make his path approximate a straight line as closely as we wish.

Let me try a different approach. First we have Einstein's theory of Special Relativity. This is based on:

1) Velocity-based time dilation
2) Length Contraction
3) Relativity of simultaneity
4) (1) - (3) can all be bundled into the Lorentz transformation.

These can all be deduced from the invariance of the speed of light. This theory can be shown to be self-constistent. I.e. (1)-(4) form a consistent set of rules. Note that (1) - (4) apply for inertial reference frames.

Then we have an alternative theory, which I'll call Ingram's theory of Special Relativity. This is based on:

1) Velocity-based time dilation

This theory is, fairly obviously, not self consistent. You can get some simple physical paradoxes based on a pair of moving clocks; and, you can also get some complicated physical paradoxes, based on cameras and observers, and light rays traveling across a large distance. As you have shown.

If you are asking whether your theory of special relativity is inconsistent, then I agree it is.

I say this because you only seem to accept time dilation, on its own. And not the other consequences of Einstein's theory.

On a second point, if Bob is moving in a circle, then he is accelerating (or repeatedly changing direction - at each marker, say). Then his rest frame is not a single inertial reference frame, and he cannot apply the rules of SR directly, as Alice or any inertial observer can.

For cicular motion the time dilation is asymmetric and Bob's clock will definitely lose time compared to any of Alice's that he passes. If his clock and the first of Alice's are synchronised, then when Bob returns to that clock after one full revolution, his clock will read 0.85 units and Alices's clock will read 1 unit.

In conclusion, there are literally dozens of previous threads on PF, based on the same misconceptions:

1) Time dilation is all there is.

and/or

2) Motion in a very large circle is the same as linear inertial motion.

 

[KipIngram]

Let me try a different approach. First we have Einstein's theory of Special Relativity. This is based on:

1) Velocity-based time dilation
2) Length Contraction
3) Relativity of simultaneity
4) (1) - (3) can all be bundled into the Lorentz transformation.

These can all be deduced from the invariance of the speed of light. This theory can be shown to be self-constistent. I.e. (1)-(4) form a consistent set of rules. Note that (1) - (4) apply for inertial reference frames.

Then we have an alternative theory, which I'll call Ingram's theory of Special Relativity. This is based on:

1) Velocity-based time dilation

This theory is, fairly obviously, not self consistent. You can get some simple physical paradoxes based on a pair of moving clocks; and, you can also get some complicated physical paradoxes, based on cameras and observers, and light rays traveling across a large distance. As you have shown.

If you are asking whether your theory of special relativity is inconsistent, then I agree it is.

I say this because you only seem to accept time dilation, on its own. And not the other consequences of Einstein's theory.

On a second point, if Bob is moving in a circle, then he is accelerating (or repeatedly changing direction - at each marker, say). Then his rest frame is not a single inertial reference frame, and he cannot apply the rules of SR directly, as Alice or any inertial observer can.

For cicular motion the time dilation is asymmetric and Bob's clock will definitely lose time compared to any of Alice's that he passes. If his clock and the first of Alice's are synchronised, then when Bob returns to that clock after one full revolution, his clock will read 0.85 units and Alices's clock will read 1 unit.

In conclusion, there are literally dozens of previous threads on PF, based on the same misconceptions:

1) Time dilation is all there is.

and/or

2) Motion in a very large circle is the same as linear inertial motion.

Thanks, PeroK. I was trying to create a situation where time dilation was the only applicable effect, by looking at motion that was (to as close an approximation as desired) perpendicular to the line connecting the parties. I am aware the the stationary party would perceive the other parties lengths as contracted in the direction of motion.

Similarly, the same thing that made that approximation valid (moving the parties very far apart) would also make the curving acceleration of a circular path that much closer to a straight line. I understand those effects, but feel they could be minimized.

But see my latest comment above - I think I saw through to the problem with my reasoning late last night. Alice can see the signals coming to her as approaching straight on along a path perpendicular to Bob's motion, but Bob cannot say the same - he will see the signals follow an angled path to Alice, because he sees Alice moving between emission and reception. Alice sees Bob moving between emission and reception as well, but it doesn't matter - once Bob "emits" it's done - his motion after emission is irrelevant. My originally presented "Bob analysis" of what Alice would see also assumed "straight shot" signals, and that was the invalid assumption. Even though Alice is at closest approach when Bob emits the signals, she's not when she receives them.

I'm going to work the algebra on this later today, but my gut tells me that all is well now.

 

[PeroK]

2) Motion in a very large circle is the same as linear inertial motion.

The argument goes like this:

We imagine Bob going in a very large circle, so large that its curvature can be neglected. There is a clock on the circumference of the circle, synchronised to a clock at the centre.

As Bob passes this clock, both that clock and his get synchonised to 0. Sometime later he passes that clock again. According to the dodgy version of SR (adopted by some) where time dilation applies equally to Bob and the clock on the circle, there is a paradox. According to each the other should have recorded less time during Bob's circular orbit. Which is physically impossible.

The resolution is that circular motion, no matter how large the circle, is not linear inertial motion. Bob's clock in non-inertial and records less time during a circular orbit than any clock at rest relative to this orbit.

 

[PeroK]

Thanks, PeroK. 1) I was trying to create a situation where time dilation was the only applicable effect,

2) Similarly, the same thing that made that approximation valid (moving the parties very far apart) would also make the curving acceleration of a circular path that much closer to a straight line. I understand those effects, but feel they could be minimized.

I'm going to work the algebra on this later today, but my gut tells me that all is well now.

Sorry, that's wrong on all counts.

1) The Lorentz Transformation always applies. There is no situation where you have only time dilation.

2) See my post above. A circle is not a straight line, no matter how large it is!

 

[PeroK]

PS Here's an analysis from a previous thread of the symmetric time dilation for a circular orbit. The time dilation depends only on the speed $v$ and not on the radius of the orbit:

https://www.physicsforums.com/threa...ther-in-a-circular-orbit.896607/#post-5660815