Undergrad A version of the twin paradox without accelerations

[George Plousos]

TL;DR

The classic twin paradox is "explained" because there is acceleration, but in the following version of the twin paradox there is no acceleration and I do not see how the dilation of time could be explained.

Bob is standing on Earth and Alice is on a distant planet at a constant distance from Earth. Their watches are already synchronized in the following sense: Suppose Alice's planet is a light-year away from Earth. Bob emits a light signal to Alice at time t = 0 according to Bob's watch. When Alice catches the signal, she sets her watch to position t = 0. Bob sets his watch to t = 0 one year after the signal is sent. Now Bob and Alice's watches are synchronized.

Alex stands behind Bob, then accelerates and the moment he overtakes Bob his speed stabilizes. At that moment they are synchronizing their watches, and this marks the beginning of the experiment. It does not matter what Alex did before the clocks were synchronized, so his acceleration does not count in the experiment.

Alex continues his journey until he gets too close to Alice. At that moment, Alex photographs his watch and Alice's watch at the same time, and immediately after that, he slows down and stops. If at that moment Alex compares his watch with Alice's watch, he should not find a significant difference from the difference between the photographed watches. Because Alice and Bob's watches are synchronized, it is not necessary for Alex to return to Earth to compare his watch with Bob's watch.

In this experiment we can assume that Alex is motionless while Bob and Alice are on the move. However, both alternative reporting systems are different aspects of the same events. This means that the photographed watches will have the same difference in both reference systems. But these reference systems are equivalent, as there is no acceleration. Without acceleration we cannot justify the dilation of time for any observer of the experiment. So I conclude the photographed watches should not be different, ie they do not record any dilation of time. But despite these objections,

the question is

Comparing the photographed watches, did Alex's watch gain or lose time compared to Alice's watch?

 

[Ibix]

Now Bob and Alice's watches are synchronized.

They are synchronised in the planets' mutual rest frame, yes. In other frames, no - the distance between the planets is length contracted in those frames and the planets are moving, so the one year correction applied by Bob does not synchronise watches in those frames. Thus Alex, once he is moving, does not agree that Bob and Alice's watches are synchronised. They tick at the same rate but are not zeroed the same.

If you work out the details of the synchronisation failure seen by Alex, you will find that Alex's expectation of Alice's watch when they pass is exactly the same as Bob's - Alice's watch is ahead. Bob and Alice say this is because Alex's watch was ticking slow. Alex says this is because his watch is zeroed to Bob's, but Alice's slow-ticking watch was not correctly zeroed and started the experiment already ahead of Bob's.

You didn't specify a speed for Alex. I recommend 0.6c or 0.8c if you want to work through the maths - the numbers are easier.

 

[Ibix]

The underlying point is that if two clocks are not in the same place there is no way to say whether they are correctly zeroed or not in an absolute sense. Your method of exchanging light pulses is the best that can be done, but it is not free of personal choice (admittedly there's a natural choice, but the natural choice depends on your state of motion).

This is why the regular twin paradox contains a round-trip, so the clocks can be directly compared without having to deal with interpreting light travel times.

 

[PeroK]

Without acceleration we cannot justify the dilation of time for any observer of the experiment. So I conclude the photographed watches should not be different, ie they do not record any dilation of time.

Time dilation is not the only aspect of SR that you must take into account. There is also the relativity of simultaneity. Both these aspects, plus length contract, are encapsulated in the Lorentz Transformation between inertial frames.

What you have found is that time dilation on its own does not lead to a consistent theory. You need the Lorentz Transformation.

 

[George Plousos]

The underlying point is that if two clocks are not in the same place there is no way to say whether they are correctly zeroed or not in an absolute sense. Your method of exchanging light pulses is the best that can be done, but it is not free of personal choice (admittedly there's a natural choice, but the natural choice depends on your state of motion).

This is why the regular twin paradox contains a round-trip, so the clocks can be directly compared without having to deal with interpreting light travel times.

From what you say I understand that it is difficult to further improve the synchronization of Bob and Alice's watches. I wonder if this can be corrected by placing Bob and Alice in the middle of the distance that separates the two planets. Then they would synchronize their clocks and then return to their planets at the same speeds.

Despite these shortcomings, I think the experiment described in the original question is feasible and there would be some difference in the photographed watches. If it is necessary for Alex to return to Earth, he will have to use the same method by which he traveled to Alice to rule out acceleration from the experiment. In this case, the final question will only include the photographed watches of Bob and Alex.

I do not know if there is any range of error in the accuracy of the results determined by the parameters of the experiment, so I chose to leave open the choice of speed of Alex.

 

[George Plousos]

Time dilation is not the only aspect of SR that you must take into account. There is also the relativity of simultaneity. Both these aspects, plus length contract, are encapsulated in the Lorentz Transformation between inertial frames.

What you have found is that time dilation on its own does not lead to a consistent theory. You need the Lorentz Transformation.

These concepts seem difficult to me. However, is the experiment not feasible with the existing data? Wouldn't it give some results even with some limited accuracy limits?

 

[PeroK]

These concepts seem difficult to me. However, is the experiment not feasible with the existing data? Wouldn't it give some results even with some limited accuracy limits?

The experiment is not yet feasible because we cannot build spacecraft that move at relativistic speeds. But, as a thought experiment it is clear that Alex's watch will show less time than Alice's. That is a simple consequence of spacetime geometry and the Lorentz Transformation.

 

[Ibix]

From what you say I understand that it is difficult to further improve the synchronization of Bob and Alice's watches.

It's impossible to improve it - absolute synchronisation of clocks just isn't possible in a relativistic universe.

I wonder if this can be corrected by placing Bob and Alice in the middle of the distance that separates the two planets. Then they would synchronize their clocks and then return to their planets at the same speeds.

No. This is called "slow clock transport" and is equivalent to the light-pulse synchronisation you were using before. There's only one frame in which they are both traveling at the same speed. In that frame their clocks remain synchronised, but in other frames they are moving at different speeds so tick at different rates and de-synchronise. There's no way around this, unless anyone ever detects a violation of Lorentz covariance, which no one ever has (and not for lack of trying).

I think the experiment described in the original question is feasible and there would be some difference in the photographed watches.

Of course you could do the experiment, although not as written. You'd need shorter distances, slower speeds, and very precise clocks. The twin paradox experiment was first carried out in the 1970s by Hafele and Keating using an airliner for transport. The result of your version would be as PeroK and I have said - Alex's watch would show a lower reading than Alice's when you took the photograph.

 

[vanhees71]

These concepts seem difficult to me. However, is the experiment not feasible with the existing data? Wouldn't it give some results even with some limited accuracy limits?

Time dilation (and also gravitational effects similar to it like Shapiro delay) is one of the best-tested predictions of relativity ever. For SR, here's a very nice website about the empirical foundations:

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

One experiment related to the additional issues with clock synchronization between accelerated observers is the Sagnac effect (also described on this page).

 

[Dale]

I wonder if this can be corrected by placing Bob and Alice in the middle of the distance that separates the two planets. Then they would synchronize their clocks and then return to their planets at the same speeds.

What you described first is basically what is called Einstein synchronization. What you describe here is called slow clock transport synchronization. They are equivalent.

I think the experiment described in the original question is feasible and there would be some difference in the photographed watches.

If you did this with particles then you could use the half life as Alex’s clock. Then your experiment would be looking at decay times in a linear particle accelerator. I know such experiments have been done for ring accelerators, but I am not sure about linear accelerators. I suspect yes, but don’t have a reference

 

[George Plousos]

From all the answers I realize that there are inherent problems in the nature of the experiment. There is one point I do not understand:

Alternatively, we can assume that Alex is motionless while Bob and Alice are on the move. In this view of the experiment, Bob overtakes the immobilized Alex. When Alice approaches Alex, then Alex photographs his own watch and Alice's watch and then accelerates to be able to accompany Alice on her journey. Shouldn't the photographed watches now show that Alice's time is slower?

1. If the answer is yes, then this contradicts the previous result. But this is impossible because we are not referring to two different experiments but to two different perspectives of the same and unique experiment.

2. If the answer is no, then why? are the two reference systems not symmetrical?

 

[PeroK]

2. If the answer is no, then why? are the two reference systems not symmetrical?

The answer is no. The scenario is not symmetrical because you have two people at rest in one frame and the relativity of simultaneity applies. Bob and Alice's watches are not synchronised in Alex's frame.

 

[Ibix]

Shouldn't the photographed watches now show that Alice's time is slower?

A photograph can't show if a watch is running slow or fast, just what it shows at an instant. What it will show is that Alice's watch is ahead of Alex's. As I noted previously:

Alice's watch is ahead. Bob and Alice say this is because Alex's watch was ticking slow. Alex says this is because his watch is zeroed to Bob's, but Alice's slow-ticking watch was not correctly zeroed and started the experiment already ahead of Bob's.

 

[Janus]

From what you say I understand that it is difficult to further improve the synchronization of Bob and Alice's watches. I wonder if this can be corrected by placing Bob and Alice in the middle of the distance that separates the two planets. Then they would synchronize their clocks and then return to their planets at the same speeds.

Despite these shortcomings, I think the experiment described in the original question is feasible and there would be some difference in the photographed watches. If it is necessary for Alex to return to Earth, he will have to use the same method by which he traveled to Alice to rule out acceleration from the experiment. In this case, the final question will only include the photographed watches of Bob and Alex.

I do not know if there is any range of error in the accuracy of the results determined by the parameters of the experiment, so I chose to leave open the choice of speed of Alex.

It's not a matter of lack of accuracy. It is a matter of not all inertial frames agreeing that t Bob's and Alice's clocks were ever synchronized to each other.
Imagine two other observers who, before Alice and Bob tried to sync their clocks, were already in motion with respect to Bob and Alice. Carl is moving in the Bob to Alice direction, and David in the Alice to Bob direction. Let's say that they both have a speed of 0.5c relative to Bob and Alice.
Bob sends his signal.
From Carl's frame, Both Bob and Alice are moving at 0.5c relative to him, with Alice "chasing after" Bob.
One of the key ideas of SR is that light (in a vacuum) travels at c in all inertial reference frame, regardless of the velocity of the source. What this means is that Carl must measure the signal spent by Bob as moving at c relative to himself. So, according to him Alice is rushing to meet Bob's signal. and the they are closing in on each other at 1.5c.
For David, Bob and Alice are also moving at 0.5c, but with Bob chasing after Alice. He must also measure the signal as traveling at c relative to himself and thus Alice is running away from the signal, so that the closing speed between them is only 0.5c.
The upshot is that for Carl, less time passes between the signal leaving Bob and arriving at Alice than does according to David. Both will agree on the distance between Bob and Alice, and how much time passes between Bob emitting the signal and his setting his local clock to 0.

From what you say I understand that it is difficult to further improve the synchronization of Bob and Alice's watches. I wonder if this can be corrected by placing Bob and Alice in the middle of the distance that separates the two planets. Then they would synchronize their clocks and then return to their planets at the same speeds.

Despite these shortcomings, I think the experiment described in the original question is feasible and there would be some difference in the photographed watches. If it is necessary for Alex to return to Earth, he will have to use the same method by which he traveled to Alice to rule out acceleration from the experiment. In this case, the final question will only include the photographed watches of Bob and Alex.

I do not know if there is any range of error in the accuracy of the results determined by the parameters of the experiment, so I chose to leave open the choice of speed of Alex.

It's not a matter of lack of accuracy. It is a matter of not all inertial frames agreeing that t Bob's and Alice's clocks were ever synchronized to each other.
Imagine two other observers who, before Alice and Bob tried to sync their clocks, were already in motion with respect to Bob and Alice. Carl is moving in the Bob to Alice direction, and David in the Alice to Bob direction. Let's say that they both have a speed of 0.5c relative to Bob and Alice.
Bob sends his signal.
From Carl's frame, Both Bob and Alice are moving at 0.5c relative to him, with Alice "chasing after" Bob.
One of the key ideas of SR is that light (in a vacuum) travels at c in all inertial reference frame, regardless of the velocity of the source. What this means is that Carl must measure the signal spent by Bob as moving at c relative to himself. So, according to him Alice is rushing to meet Bob's signal. and the they are closing in on each other at 1.5c.
For David, Bob and Alice are also moving at 0.5c, but with Bob chasing after Alice. He must also measure the signal as traveling at c relative to himself and thus Alice is running away from the signal, so that the closing speed between them is only 0.5c.
The upshot is that for Carl, much less time passes between the signal leaving Bob and arriving at Alice than does according to David*. Both however will agree on the distance between Bob and Alice, and how much time passes between Bob emitting the signal and his setting his local clock to 0.
So according to both, Bob will not reset his clock to zero at the same moment that the signal reaches Alice and she set her clock to zero, despite the fact that according to both Bob and Alice these events are simultaneous.
This is the gist of the Relativity of Simultaneity; frames of reference in relative motion with respect to each other will not agree on the simultaneity of spatially separated events.

So, let's consider Alex. We'll assume that he, after accelerating will pass Bob moving at 0.5c (so now he is at rest in the same inertial frame as Carl.)
According to both Bob and Alice, Alex passes Bob when both of their clocks read some time T, and he will arrive at Alice when both clocks read T+2 yrs. Alex's clock, having undergone time dilation will have advanced 1.732 years (If we assume that Alex matched his clock to Bob as he passed, then his clock will read T+ 1,732 yrs when he meets up with Alice). Thus according to them( Bob and Alice), less time passed Alex during the trip than did for either of them.

Now let's look at the same events according to Carl.
He and Alex are at rest with respect to each other, thus their clocks tick at the same rate. Bob's and Alice's clocks are moving at 0.5c, and their clocks are time dilated at a rate of 0.866. The distance between Alice and Bob is also length contracted to 0.866 ly. In addition, as covered above, Bob's and Alice's clocks are not in sync. But will be offset from each other by 0.5 yr with Alice's clock reading later. (If Alex passed Bob when Bob's clock reads T, Alice's clock will already be reading T+0.5y).

At a relative speed of 0.5c it will take 1.732 yrs from the moment Bob and Alex pass and Alex and Alice meet. So this is how much time Carl will measuring passing for Alex.
Both Bob's and Alice's will be seen as time dilated, so Carl will measure 1.732 x 0.866 = 1.5 years passing for them. Since Alice's clock already read T+0.5 years when Alex was next to Bob, Alice's clock will read T+2 yrs when she and Alex meet up. ( Thus according to Carl, Bob's clock reads T+1.5 yrs When Alex meets up with Alice.)

So Bob, Alice and Carl will all agree that Bob's and Alex's clocks both read T when they passed each other, and that Alex's clock read T when passing Bob, and T+1.732 yrs while when meeting Alice' s clock, and that Alice's clock reads T+2 years when Alex and she meet up. However they will not agree as which clocks ticked faster over the interval between Alex and Bob passing and Alex and Alice meeting, with Bob and Alice saying Alex's clock advanced by 1.732 yr for their 2, and Carl saying that Bob's and Alice's clock advanced only 1.5 years for Alex's 1.732 yr.

This is why a "round trip" is generally used. If Alex goes from Bob to Alice and back to Bob, then we are comparing Just Bob's and Alex's Clock while they are side by side, and everyone, (including Carl*) will agree that Alex will have aged less when they meet up again.

* In this case, while Carl says that Bob ages less during the "Bob to Alice" leg, He we also say that Alex's clock runs much slower than Bob's clock during the "Alice to Bob" leg( during this leg, while Bob's clock runs 0.866 the rate of Carl according to Carl, Alex's clock would run 1/7 as fast as Carl's).

 

[Dale]

From all the answers I realize that there are inherent problems in the nature of the experiment. There is one point I do not understand:

I actually don’t think there are real problems with the experiment. These observations have already been done on particles using the half life as a clock. This only differs from your setup “cosmetically”.

Shouldn't the photographed watches now show that Alice's time is slower?
...
2. If the answer is no, then why? are the two reference systems not symmetrical?

The answer is no.

The symmetry of the frames isn’t even relevant. You have two synchronized clocks at rest in one frame vs one clock at rest in the other frame. Two clocks is not symmetric with one. Everyone agrees which are the two and which is the one.

 

[George Plousos]

I have identified the various points that create conceptual problems in this hypothetical experiment from the clarifications given and thank you all. Beyond that, despite the fact that Relativity is based on solid foundations, I have to admit that I can not convince myself that nature works this way, although I do not find logical counter-arguments. I can only follow the current experimental research and wait for the results. I also follow related threads like this and any information or explanation added here could help more in understanding or clarifying individual issues related to the "twin paradox".

 

[PeterDonis]

I can not convince myself that nature works this way

Why not? Do you not believe the experimental data that says it does?

 

[George Plousos]

Why not? Do you not believe the experimental data that says it does?

In the classic twin paradox the astronaut travels to the nearest star and returns to Earth, gaining 1 year. If the same astronaut, with the same accelerations and decelerations, travels to the edge of the galaxy and returns to Earth, he will win 100 years.

In my mind the Earth and the astronaut are symmetrical bodies for most of the journey and I would expect these two bodies to share a common rate of time during this time, while the time the astronaut gains should be proportional to the accelerations and its decelerations. This intuitive logic contradicts what I say in the previous paragraph.

As far as I know, the experiments that have been done so far are results of GR, since gravity and acceleration are involved in them, and it seems that conducting experiments in the context of SR is very difficult.

Relativity may be something like a distorting lens. The phenomena that take place behind a distorting lens have consistency and predictability, but we must remove the lens to see the real image. So I'm wary, especially of SR.

 

[PeterDonis]

As far as I know, the experiments that have been done so far are results of GR

Not as far as time dilation due to relative motion is concerned. Gravity plays no role in the experiments that show that.

since gravity and acceleration are involved in them

You do not need GR to deal with acceleration. You only need GR if you cannot ignore the fact that spacetime is not flat due to the presence of gravitating masses. There are plenty of experiments where that is the case (for example the Pound-Rebka experiment showing gravitational time dilation due to height change near the Earth's surface, and the GPS system where both gravitational time dilation and time dilation due to motion play a role), but there are also plenty where it is not (for example, the detection of muons produced by cosmic rays, and the experiments with muons in circular traps).

 

[PeterDonis]

Relativity may be something like a distorting lens.

If you insist on using your intuitive logic instead of understanding how the theory actually works, then yes, it's going to seem that way to you. The only fix for that is to give up your intuitive logic--at least until you have learned enough to retrain your intuitions based on a correct understanding of the actual physics.

 

[Isaac0427]

Can someone help me out with the work to solve this problem relativistically (such as guidance on drawing the Minkowski diagrams or working out the relevant spacetime intervals)? I’m trying to relearn relativity (it’s been over a year since I last did a problem like this and I can’t seem to get a grasp on this situation). I don’t know if I should make this a new thread or not.

 

[PeterDonis]

I don’t know if I should make this a new thread or not.

I don't think it needs to be a new thread, since a more detailed working of the problem might help the OP in this thread.

I would recommend picking a speed for Alex relative to Alice and Bob (@Ibix recommended two good choices in post #2), working in Alice's and Bob's common rest frame, and choosing the spacetime origin $(x, t) = (0, 0)$ to be the event where Alex passes Bob. Then work out the coordinates of the event where Alex passes Alice, based on Alex's speed and the distance from Bob to Alice in the chosen frame, and calculate the spacetime interval between those events, which will be the time elapsed on Alex's clock between them, using the standard interval formula.

Once you have coordinates of the relevant events in the Bob-Alice rest frame, drawing a spacetime diagram should be easy. If you want to then transform into Alex's rest frame to see how things look there, the standard Lorentz transformation formulas will give you the coordinates of the relevant events in that frame so you can draw another diagram.

 

[PeroK]

In the classic twin paradox the astronaut travels to the nearest star and returns to Earth, gaining 1 year. If the same astronaut, with the same accelerations and decelerations, travels to the edge of the galaxy and returns to Earth, he will win 100 years.

In my mind the Earth and the astronaut are symmetrical bodies for most of the journey and I would expect these two bodies to share a common rate of time during this time, while the time the astronaut gains should be proportional to the accelerations and its decelerations. This intuitive logic contradicts what I say in the previous paragraph.

As far as I know, the experiments that have been done so far are results of GR, since gravity and acceleration are involved in them, and it seems that conducting experiments in the context of SR is very difficult.

Relativity may be something like a distorting lens. The phenomena that take place behind a distorting lens have consistency and predictability, but we must remove the lens to see the real image. So I'm wary, especially of SR.

This is a science forum and what you write is totally unscientific. SR is at the heart of particle physics and is tested every day in thousands of experiments round the world. Not in the "intersteller astronaut" setting, but in the particle physics setting. Same theory, but small particles that can be accelerated to near the speed of light.

SR is difficult to learn. The fact that you don't understand it doesn't mean it's wrong.

 

[Ibix]

This intuitive logic contradicts what I say in the previous paragraph.

To be blunt, "intuitive logic" is a euphemism for "guess based on poorly specified assumptions". Common sense is a poor guide to uncommon situations.

The first step in fixing your intuition is to learn the maths. As long as you are happy to limit yourself to instantaneous accelerations, you don't need anything more complex than a square root and possibly some basic algebra. I'd recommend looking up the Lorentz transforms and the interval (even Wikipedia will do for that) and trying what @PeterDinis suggested in #22. Additionally, I'd calculate the time Alex's rest frame assigns to Alice's time zero. Then I'd draw a Minkowski diagram in both Bob/Alice's frame and Alex's frame - that's the thing that really made relativity "click" for me.

I wouldn't worry about understanding where the maths comes from until you've done a couple of problems, although your mileage may vary.

Post if you get stuck.

 

[Ibix]

The fact that you don't understand it does mean it's wrong.

Um - I think you made a small but important typo...

 

[PeroK]

Um - I think you made a small but important typo...

Corrected, thanks.

 

[Isaac0427]

I don't think it needs to be a new thread, since a more detailed working of the problem might help the OP in this thread.

I would recommend picking a speed for Alex relative to Alice and Bob (@Ibix recommended two good choices in post #2), working in Alice's and Bob's common rest frame, and choosing the spacetime origin $(x, t) = (0, 0)$ to be the event where Alex passes Bob. Then work out the coordinates of the event where Alex passes Alice, based on Alex's speed and the distance from Bob to Alice in the chosen frame, and calculate the spacetime interval between those events, which will be the time elapsed on Alex's clock between them, using the standard interval formula.

Once you have coordinates of the relevant events in the Bob-Alice rest frame, drawing a spacetime diagram should be easy. If you want to then transform into Alex's rest frame to see how things look there, the standard Lorentz transformation formulas will give you the coordinates of the relevant events in that frame so you can draw another diagram.

Based on how I solved the problems...would it be correct to say that EVERYONE agrees that Alex passes Bob at t=0, but Alex will think his trip is shorter (in time) than Alice and Bob believe it is? It seems to me that Alex would still agree that Alice and Bob's clocks are synchronized, but based on this thread that seems to not be the case.

 

[Janus]

Based on how I solved the problems...would it be correct to say that EVERYONE agrees that Alex passes Bob at t=0, but Alex will think his trip is shorter (in time) than Alice and Bob believe it is? It seems to me that Alex would still agree that Alice and Bob's clocks are synchronized, but based on this thread that seems to not be the case.

Alex will also say that the distance between Bob and Alice is shorter in distance than Bob and Alice measure it as being.
As far as Bob's and Alice's clocks being in sync. let's assume that Alex has been traveling for some time at 0.5c when he passes Bob. At the moment they pass each other, they both are pointing very powerful telescopes at Alice and reading what they see on her clock. Bob and Alice are 1 ly apart in their frame and the image of Alice's clock will read as being 1 year behind Bob's clock reading. Because Bob knows that Alice's distance from him is a constant, he (and anyone at rest with respect to Bob and Alice) will conclude that Bob's and Alice's clocks are in sync.
Alex will see the same time reading for Alice; 1 yr behind Bob's clock reading. However, for him, Alice has not maintained a constant distance from him. That means that the distance between Alice and himself has not been a constant.
As I mentioned above, the distance between Bob and Alice is less than 1 ly according to Alex (0.866 ly). But that is the distance to Alice as he passed Bob. Since it took time for the image to reach him, He and Alice had to have been further apart when the light carrying that image left Alice. That light also was traveling at c relative to Alex. This means that the time between the light leaving Alice and Bob seeing it is longer than 1 yr ( 1.732 yrs) by Alex's clock. Alice's clock will be time dilated according to Alex and and 1.5 yrs will pass on it.
Thus, when Alex sees a reading of 1 yr behind Bob's clock in Alice;s clock as he passes Bob, he knows that 1.5 yrs has passed for Alice since that light left. This is 0.5 yr more than what Bob says passed on Alice's clock.
From this, alex has to conclude that. at the moment he passes Bob, Alice's clock reads 0.5 yr later than Bob's clock.

 

[Isaac0427]

Alex will also say that the distance between Bob and Alice is shorter in distance than Bob and Alice measure it as being.
As far as Bob's and Alice's clocks being in sync. let's assume that Alex has been traveling for some time at 0.5c when he passes Bob. At the moment they pass each other, they both are pointing very powerful telescopes at Alice and reading what they see on her clock. Bob and Alice are 1 ly apart in their frame and the image of Alice's clock will read as being 1 year behind Bob's clock reading. Because Bob knows that Alice's distance from him is a constant, he (and anyone at rest with respect to Bob and Alice) will conclude that Bob's and Alice's clocks are in sync.
Alex will see the same time reading for Alice; 1 yr behind Bob's clock reading. However, for him, Alice has not maintained a constant distance from him. That means that the distance between Alice and himself has not been a constant.
As I mentioned above, the distance between Bob and Alice is less than 1 ly according to Alex (0.866 ly). But that is the distance to Alice as he passed Bob. Since it took time for the image to reach him, He and Alice had to have been further apart when the light carrying that image left Alice. That light also was traveling at c relative to Alex. This means that the time between the light leaving Alice and Bob seeing it is longer than 1 yr ( 1.732 yrs) by Alex's clock. Alice's clock will be time dilated according to Alex and and 1.5 yrs will pass on it.
Thus, when Alex sees a reading of 1 yr behind Bob's clock in Alice;s clock as he passes Bob, he knows that 1.5 yrs has passed for Alice since that light left. This is 0.5 yr more than what Bob says passed on Alice's clock.
From this, alex has to conclude that. at the moment he passes Bob, Alice's clock reads 0.5 yr later than Bob's clock.

This seems to be an argument based on how the clocks were synchronized. What if Alex starts going at .5c right as the light signal from Alice reaches Bob? Wouldn't Alex see that Alice's clock reads .866 years?

 

[Janus]

This seems to be an argument based on how the clocks were synchronized. What if Alex starts going at .5c right as the light signal from Alice reaches Bob? Wouldn't Alex see that Alice's clock reads .866 years?

It has nothing to do with how the clocks were synced to each other. We just start with the assumption that according to Bob and Alice their clocks are synchronized. Alex however, has to conclude that Alice's and Bob's clocks can't be synchronized.
When Alex and Bob pass each other, they have to see the same image of Alice's clock. You can't have Alex and Bob being right next to each other and seeing different readings for Alice's clock.
If we assume that Alex's acceleration is instantaneous*, he will see Alice's clock reading 1 yr behind Bob's clock both before and after the acceleration. What will change is what time he would conclude is actually on Alice's clock.

*you have to be cautious when assuming "instantaneous" changes in velocity, This equates to a non zero change in velocity in 0 time, or a = dV/0. Since division by zero is undefined, you risk producing contradictory results if you are not careful. You can use a "near-instantaneous" velocity change instead, But then you have to accept that Alex will be in an non-inertial frame at some point, and account for how that effects what he says is happening to Alice's clock.