You misunderstand. Bob sees his own clock tick off four years ( from year 0 to year 4 ) on the way out and four years ( from year 4 to year 8 ) on the way back.

Bob sees Ann clock tick off 3.2 years ( from year 0 to year 3.2 ) on the way out and 3.2 years ( from year 6.8 to year 10 ) on the way back.

3.2 - 0 = 3.2
10 - 6.8 = 3.2
3.2 + 3.2 = 6.4

3.2 / 4 = .8
6.4 / 8 = .8

Bob sees his own clock ticking off 8 years on the whole trip, and he sees Ann's clock ticking off 6.4 years on the whole trip, but Ann's clock jumps at turn-around.

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"Actually"? There are more than one instance of the thought experiment, and in some of them, there is no setting to "00", they just have to be synchronous. In the case of the discussion about the twins, very few twins are truly synchronous--but it's close enough for the purpose, when the differences are measured in years.
Not necessarily, in SR.
In this case, that website is very close to Einstein's original treatment of the "twin paradox" (which he called a "peculiar consequence") in his 1905 paper.
But that webpage only introduces the twins.

Negative. I was simply responding to someone's comment that he feels like he understands less than he did before. Whether thats a reflection on you or not I won't say, though I'm sure you know my opinion.

Your error here is to (intentionally?) misinterpret what was written.

Nope.

Nope. Not what was written. Is this a test of comprehension?

Again, not what was written.

All false. Is this "Intentional Misinterpretation Week"?

How does a simple thought experiment generate such nonsense? Have I once said that each can see each other's clocks?

Your error here is that you neither understand special relativity nor general relativity but think you do. All of the statements you produced were fallacious.

Anne does not move. So her spacetime interval (tau) is 10 years because the space interval is zero.

Bob however does move. His total distance moved is s/2 moving out measured by a wristwatch he's carrying onboard the rocket which measures 4 years, then instantaneously gets on another rocket going back at 3/5c for another 4 years. At the end of this time, he flashes by Anne again. She has been waiting for ten years and he for 8 years.

Simple task: Complete the puzzle using the following formula: (tau)^2= t^2 - s^2

From this you can work out how far Bob has travelled (as measured by Anne)

For people who are joining this thread late, let me try to summarize the situation from my point of view.

Special Relativity (1905) “time dilation due to reltive motion” = not correct

Ann and Bob “twins paradox ‘solution’ based on Special Relativity) = not correct

General Relativity (1915) “atomic clock slowdown due to acceleration” = CORRECT

As I understand your thought experiment about Ann and Bob, you are trying to use Special Relativity (the theory with the error) to solve the paradox, and my contention is that you can’t do it because the Special Relativity theory contains an error.

Since there is no “acceleration” in the Special Relativity theory, you can’t use it in the Ann and Bob example, and since there is no “clock jumping” in the Special Relativity theory, you can’t use that in your Ann and Bob example either.

Your own first error is that you have Bob turning around at 4 hours, which is what Ann “sees” on his clock at turn-around time (while she sees 5 on her clock), and you have Bob “seeing” 4 on his clock at turn-around time too. Based on the terms of the original SR theory, that is incorrect. In SR theory, Bob would see 5 on his clock at turn-around time and 4 on Ann’s clock. At the end time, Bob would see 10 on his clock and 8 on Ann’s clock, while Ann would see 10 on her clock and 8 on Bob’s clock. And that reveals the paradox and error of SR theory.

In your attempted “resolution”, when you force Bob to turn around at 4 on his clock, he would see only 3.2 on Ann’s clock, and at end time Bob would see 8 on his clock and 6.4 on Ann’s clock, while you say Ann would see 10 on her clock and 8 on Bob’s clock. So you’ve still got a disagreement between Bob and Ann and a paradox.

You try to “resolve” this paradox by adding an extra 3.6 seconds to what Bob “sees” on Ann’s clock at end time, and you do that by claiming that Ann’s clock somehow “jumps” from 5 to 6.8 when Bob turns around. Then you add Bob’s second “seen” 3.2 to the “jumped” 6.8, and you get “10”, which is what you claim Bob “sees” on Ann’s clock at end time.

But, there are no “jumping clocks” in SR theory (or in GR theory). You’ve just added that “jumping clock” story, even though it is not in the SR theory.

There is no way that either Bob or Ann could see the “jump”, since at turn-around, Ann’s clock is only at 5 and hasn’t yet reached the time of 6.8. Bob can’t “see” the 6.8, since you said all he “sees” is the 3.2 on the way out and the 3.2 on the way back. And of course, in SR theory, none of the observers see a “clock jumping”, either backward or forward. All the observers ever see in SR theory is the the other frame’s clocks “running slow”, but never "jumping" or “running fast”.

Another error you’ve got in this thought experiment is that if Bob turns around at 4 by his clock, and if he “sees” 3.2 on Ann’s clock, then that (based on the Lorentz Transformation equation) means her clock has ticked off only 4 minutes, not 5, since the Lorentz Transformation clock slowdown factor of 3.2 / .8 = 4

So you wind up with a double paradox and too many different numbers “seen” on Ann’s and Bob’s clocks by Ann and Bob at end time. Your “resolution” contains several errors and the paradox still exists.

So the experimental results confirming time dilation due to relative motion mean?

The error that is being made here is that the correct redshift/blueshift formulae are not being used. Read my two posts in the other thread starting with this one:

The proper redshift during the first phase is sqrt((1-.6)/(1+.6)) = 0.5 and the blueshift during the second phase is 2.0. During the first 4 years Bob sees only 4*0.5 = 2 years of Anne's transmissions, but during the last 4 years he sees the remaining 4*2.0 = 8 years, for a total of 10 years of Anne's transmissions. Anne sees redshifted signals from Bob for (1+.6)/2 = 0.8 of the total time, that is, 8 years. In 8 years she sees 8*0.5 = 4 years of transmissions from Bob, and in the remaining 2 years she sees 2*2.0 = 4 years of transmissions from Bob, for a total of 8 years of Bob's transmissions. Resolved! QED!

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You have been provided with popular accounts of experiments that show that time dilation does occur--particles moving close to the speed of light take longer to decay than slower moving particles.
So far you have not convinced many people why this is not correct. SeanF has shown admirable patience in trying to demonstrate to you why there is no fallacy involved. More about this later.
GR is a superset of SR, since SR can explain the apparent paradox, it's not surprising that GR should manage as well.
You can phrase the twin problem in SR such that there is no acceleration--you have the outbound twin tell the inbound twin how much time the outbound twin thinks has elapsed since passing Earth.

SR says that "simultaneity" is not well defined--after four hours travelling away from earth, the outbound twin might think that only 3.2 hours had passed on Earth, whereas the inbound twin would think that 6.8 hours Earth time had elapsed. Switching from the outbound reference frame to the inbound reference frame involves a change in the interpretation of "Now".
Not an error
Again, "simultaneous" is not well defined--how much time the outbound twin thinks has elasped on earth is not the same as how much time the inbound twin thinks has elasped.
Right. However, when you switch reference frames you may see a jump. Of course, if you're watching the earth clock via telescope you don't see the jump even when switching reference frames, you only get the jump when you try to compensate for distance (when you try to figure out "What time is it NOW on Earth").

First, just consider, for a moment, that there are no “time dilations” due only to “relative motion”, since both of two relatively moving clocks don’t know the other clock exists or is moving.

However, there can be clock slowdowns due to the motion of clocks through electric, magnetic, and gravitational fields at the surface of the earth (or anywhere else), even if the clocks never experience “acceleration”.

Earlier I gave an example of a small electric-motor clock either slowing down or speeding up, when a large industrial magnet is moved near it. This clock rate change is NOT due to “relative motion”, it is due to the coil of wire in the armature of the clock experiencing a strong changing magnetic field, a changing “flux” or changing magnetic field “potential” at the clock’s coil.

Sure, the magnet has to “move” near the clock for the magnetic field potential to change at the coil, but the clock rate change is NOT due to the “relative motion”. If it was due only to “relative motion”, then the clock rate would change if the clock were in New York while the magnet moved in Los Angeles. But of course this does not happen, since the clock in New York can’t feel the magnetic field potential change of the moving industrial magnet while the magnet moves in Los Angeles, since the distance between the clock and the magnet is so great. So, the “relative motion” does not cause the speed up or slowdown effect, since at large distances the clock coil doesn’t “know” that the magnet is moving and at large distances it doesn’t “feel” any change in the local magnetic field potential at the coil inside the clock.

So, the “motion” of the magnet near the clock causes the magnetic field potential to change at the coil of wire inside the motor, and this causes the clock to either speed up or slow down.

I’ve seen “relativity” websites that claim the clock rate change IS due to “relative motion”, but they don’t provide any examples of the magnet and the coil being separated by great distances.

The truth comes out at last! So I was right all along. Bob "sees" no jump in Ann's clock. Nor does he "infer" a jump when he tries to figure out what time it is NOW on Earth. Why? Because Bob is not an idiot. He understands relativity and is familiar with thing like time dilation, Fitzgerald-Lorentz contraction and the relative nature of simultaneity. He does the math and is not surprised to discover that 5 years have passed on Ann's clock when he turns around, even though only four have passed on his clock.

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Well, I’ve pointed out several times that particles moving inside particle accelerators are accelerated, and I’ve also pointed out that acceleration causes slowdowns in atomic clocks. In fact, muons are accelerated as they move rapidly toward the earth’s surface. Anything falling from great heights toward the surface of the earth is accelerated.

You have probably heard that atomic clocks located inside an accelerator lab but outside the actual accelerator, do not tick more slowly as the result of the “relative motion” of the particle inside the accelerator. And the “relative motion” of the lab, relative to the so-called “stationary particle” moving inside the accelerator is NOT what causes the “time dilation" in the accelerating particle.

No, not exactly. There is still a mystery going on in the SR theory that I’m trying to work out right now. Based on many of the responses I’ve read here, I think we’ve both been “missing” something. You and I understand some of the phenomena of the SR theory differently, and I think I’ve recently begun to understand the reason why. I have been figuring out what I’ve been missing, and what you’ve been missing, and I think I’m just about ready to lock in on it. It is absolutely fascinating.

It has to do with a fundamental difference between plain ordinary “relative motion” at a distance, and the “relative motion” of atoms and electrons through fields at close range to where the two objects are “relativity moving”.

If the two objects are at a great distance from one another, such as our galaxy and a very distant galaxy, the “c” limit does not apply, since neither of the objects is moving through the fields of the other object. But if the objects are moving at close range to one another, such as a particle being accelerated here on the surface of the earth, then the “c” limit does apply, since their fields put up a “resistance” to the motion. At great universal distances, our local fields can’t put up any “resistance” to the motion of the most distance galaxies.

This is something like the Lorentz Force, that allowed NASA to generate electricity through a long tether wire that was pulled along behind the space shuttle. I’ve read several reports that say the motion of the metal tether through the magnetic field of the earth not only generates an electron flow in the wire, but it causes a slight slowdown in the space shuttle, because the earth’s magnetic field is putting up a resistance to the motion of the long tether through it. This resistance apparently isn’t noticed with short wires or vehicles moving fast relative to the earth’s magnetic field, or when short wires and vehicles move at slow speeds relative to the earth’s magnetic field, but it is noticed when the tether is several miles long and its speed is fast relative to the earth’s magnetic field.

So, apparently, the tether experiment would not work in deep space, since the wire would not be moving directly through any astronomical body’s local magnetic field.

And so, in the electrodynamical part of the SR theory, (the second half of the theory), Einstein might have correctly calculated a true speed limit of “c” for masses moving through strong fields, such as the fields that are located here at the surface of the earth and that travel through space with the earth. If I am correct about this, then that means the “speed limit” would apply here on earth (and near other astronomical bodies), but it would not apply to distant galaxies relative to our galaxy, since they aren’t moving through our own local fields or any other fields.

So, while the local “speed limit” and some kinds of “clock rate changes” are often thought of as “an SR effect” due to “relative motion” and the “relative motion speed limit”, they might actually be “an SR effect” due to the motion of masses through the strong local fields of other masses.

So my disagreement with Sean about what I call the fundamental “difference” between the small masses in the SR theory thought experiments moving relative to one another at great distances from one another, and the separate train example where a fairly small mass moves across the surface of a great mass, the earth, does represent a fundamental difference in the types of and the causes of the “observed” effects due to the “relative motions” in both cases. Meaning, the relative motion of the small masses at great distances in the SR theory’s first half, can NOT have any influence on either mass, while the motion of the train across the surface of the massive earth CAN cause an effect at the train. Sean and I disagree as to what that “effect” is, but we both agree that there IS an effect.

I was a bit careless in my earlier post in my haste to gallop off to the conclusion. I should have said that the redshift during the first phase is sqrt((1+.6)/(1-.6)) = 2.0 (consistent with my terminology in my "When Harry Meets Sally" posts) which results in a frequency shift of 0.5. Likewise, the blueshift is 0.5 which results in a frequency shift of 2.0. I still stand by my earlier calculations, all clock pulses are accounted for!

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If Bob were watching the earth through a telescope, he would see no jump in Ann's clock when he made the turnaround. His calculation of what time it is "Now" on earth changes when his reference frame changes. Outgoing Bob thinks that ingoing Bob was travelling very fast and his clock was running very slow, ingoing Bob thinks the same about outgoing Bob.

Glom,


Just so I can make this clear again about my point of view, when I say that a “relative motion” all by itself can not possibly slow down any clock: When the moving magnet is in Los Angeles and the electric clock is in New York, we’ve still got the original “relative motion” between the two, but the New York clock doesn’t “feel” it this time, since the magnet is so far away.

The magnet has to be right up close to the clock, so the electrons inside the wire coil of the motor of the clock can “feel” something, which is the changing magnetic field at those electrons.

So it isn’t the basic “relative motion” that slows down the electric clock, it is motion of the coil through the magnetic flux lines of the magnet, and this can be accomplished by either moving the magnet close to the coil or by moving the coil close to the magnet, but the magnet and the coil must be at close range to one another, or else the coil won’t “feel” the changing flux lines.

So, this is what I’ve been ranting about when I’ve said, “relative motion alone can’t possibly cause any clock to change its rate”, and I’ve said “something physical has to happen to the clock, and the clock has to “feel” some kind of ‘force’ upon it, or a changing force, before it can change its rate.”

I think maybe some of you thought I meant that unaccelerated motion can’t cause any kind of clock to slow down, but that’s not what I meant. I meant that the clock has to “feel” something as a result of the relative motion before it can change rates.

And I pointed out that the decay particles I'm talking about are generated outside of the accelerator--after the accelerated particle has rammed into the target. Was I unclear?

It sounds as if you're blaming the extended lifetime of muons produced by cosmic ray interactions with the atmosphere on GR. Does it bother you that the data fit SR, and that nobody seems to be invoking GR to explain it? Do you have equations that would predict the expected time dilation imposed by your version of GR? Is the time dilation dependent on the velocity of the particle, the distance it falls, or the time it takes it to fall?
Do you think that SR implies that a clock C1 near the high-speed particle runs at a different rate than a clock C2 away from the high-speed particle but in the same reference frame as C1?

Yes, and I've asked you for a scientific paper source of that information but you never provided one. We might be talking about two different things here.

No.

In SR theory, the “rate of clock slowdown” is determined by the relative “v” only.

And by this equation



the direction of the relative motion doesn’t matter, so a so-called “frame change” does not affect the rate of a clock, and it doesn’t cause any clock in the SR theory to “jump”. There are no “jumping clocks” in SR theory.

Here is the rate of the change of the relatively moving clock in SR theory:



You will notice that it doesn’t include any “jumps”. Since the motion is “inertial” with no acceleration, even a direction change has no affect on any of the clocks. Only the relative “v” determines the rate of the change, and of course the “v” doesn’t “jump” since there are no accelerations. Any real life “turn-around” acceleration affects are simply ignored in SR theory, because:

Don't stop there, Sam5. Read the very end of Section 3, right after that equality:
That first transformation equation, which converts time in one frame to time in a relatively moving frame, is clearly dependent on the sign of v (as well as x, the distance). In fact, I think you'll find that if you enter in 4 years for t (the time Bob travelled), 0.6c for v (Bob's velocity), and 2.4 light years for x (the distance Bob would be from Ann after 4 years at 0.6c), you'll get =3.2 years.

If you put in those same numbers for t and x but put in negative 0.6c for v, you'll get =6.8 years.

And just in case you forgot:

See? No paradox.

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There is no year 6.8 until Bob is most of the way back to the earth. The year 6.8 certainly doesn’t turn up at the end of 50% of the travel time or 50% of the waiting time.

8 – 6.8 = 1.2 years, when Bob is almost home, not when he’s just halfway home.

10 – 6.8 = 3.2 years when Ann’s clock is just 3.2 years away from 10, 32% away from 10, but NOT when her clock is 50% away from 10.

You are now trying to claim that Bob turns around when about 2/3 of the time has elapsed on Ann’s clock but you said earlier he turns around when 1/2 of the time had elapsed on her clock. [-X

When Bob is 50% away from 8, his clock is at 4, not 6.8, and when Ann is 50% away from 10, her clock is at 5, not 6.8.

You are trying to claim that the distance and time on the way out is longer than on the way back, and that’s not in SR theory.

So you've still got a paradox.

I pointed you to the Stanford page, which alludes to the tau particle decay times. I haven't tried too hard to check for other references.
There are no jumping clocks in SR theory until you start changing reference frames.
This equation is describing the relative rates of the clocks, it doesn't say anything about the actual values on the clocks. SeanF's previous post discusses this.
SeanF's post discusses this. Again, the formula concerns the rate at which the clocks are ticking, not the time the clock shows. Observers in different reference frames have different concepts of "Now". Two observers at the same point in space but in different reference frames would see the same time on a distant clock if peering at it through their telescopes, but would have different opinions about what time it showed if they compensated for the distance to the clock.

You’ve got things all mixed up.

You have Ann “seeing” Bob travel an un-contracted distance of 3 light years, 558000 miles for 5 years on her clock, while she “sees” 4 years transpire on Bob’s clock.

You have Bob “seeing” himself travel a contracted distance of 2.4 light years, 446400 miles while he “sees” 4 years on his clock and 3.2 years on Ann’s clock.

So you’ve got two different distances and three different times for just two clocks.

So you are having Bob travel one distance for Ann and a different distance for Bob.

Paradox.

If you had Bob travel 558000 miles at .6 c it would take him 5 years.

And remember:

“The shape of a body in the sense indicated we will call its ‘geometrical shape.’ The latter obviously does not depend on the state of motion of a reference body.” A. Einstein, 1907

So there is no real “length contraction”.

Anyway, if this is all “relative motion”, then Bob would “see” himself as stationary and Ann as “moving”. He would see himself “stationary” for 5 years until Ann “turns around”, and he would “see” 4 years transpire on her clock.

Paradox.

Remember, this is SR, not GR, so the motion is “relative” not “absolute”. Remember what you were saying about the train being “stationary” while the “earth moved”? Why don’t you apply that here? Bob is “stationary” while Ann “moves”.

Why is that a paradox? They're trillions of miles apart (by the way, you better check your conversion factor on light-years to miles). How can it possibly matter if they disagree on the time on Ann's clock?
Bob turns around after only 4 years of travel by his time, not 5. That's why he says Ann's clock has only ticked 3.2 years.
You really don't understand SR, do you? t and x are the measurements in the "stationary" reference frame in that equation, and I put Bob's numbers in there. That was with Bob being "stationary"!

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Is there any point in working through the math?

Call outgong Bob BobO, incoming Bob BobI. BobO thinks that when he passes BobI that 3.2 years have elapsed for Alice since BobO left. BobI thinks that when he passes BobO that 6.8 years have passed for Alice since BobO left. When BobO switches reference frames to become BobI, his concept of what time it is "Now" for Alice has to change. However, if BobO was keeping an eye on Alice through a telescope, he wouldn't see any sudden jump in her wristwatch.

The math isn't too complicated if you want to go through it--it doesn't involve anything worse than a square root (and these square roots are all rational). Take it from the viewpoint of both Bobs--start BobI 6 ly out (from Alice's point of view), BobO at earth, run through for each of them what time they SEE (1) BobO leave earth, (2) BobO pass BobI, and (3) BobI pass earth. Figure from there what time they THINK the various events occurred for the other people, compensating for distance. You'll find that (1) everyone agrees how much time has elapsed for each observer between the light from one event arriving and the light for another event arriving, and (2) they disagree about when the non-local events occur (BobI and BobO disagree about what time it is for Alice when BobI passes BobO, BobI and Alice disagree about what time it is for BobO when BobI passes Alice, BobO and Alice disagree about what time it is for BobI when BobO passes Alice).

I think the confusion between what time BobI and BobO think it is for Alice when BobI passes BobO is causing you problems.

Not quite sure what you mean by the Pinocchio nose Sam5. Could you be a bit more explicit?

Yes, thanks, you are right. I was discussing this subject on another board with a guy who was talking about light “seconds” rather than light years. Just multiply all my miles by 31557600 seconds in a year.


According to SR they are supposed to "agree" at the end. That's why you have to add the 3.6 seconds to what Bob "sees" on Ann's clock at the end. So you add the 3.6 to the first 3.2 to get the 6.8 at turn-around, and then you add the second 3.2 to that and you wind up with Bob "seeing" 10 on Ann's clock.

3.2 + 3.6 = 6.8

6.8 + 3.2 = 10

That's why you have to have Bob "see" Ann's clock "jump from 3.2 years to 6.8 years at turn-around, even though it's still only 5 years on her own clock and only 4 years on Bob's clock.

And you've still got them traveling different distances.

Paradox.

Well yes, because Sean is applying a double Lorentz Transformation to Ann’s clock as “seen” by Bob, while he’s applying only a single one to Bob’s clock as “seen” by Ann.

You can’t do that in SR theory.

I know the theory and I know what he is doing. I’m not a kid just reading the twins paradox thought experiment on the internet and just accepting his numbers on blind faith. I don't accept things I read on the internet on blind faith.

5 x .8 = 4

4 x .8 = 3.2

That’s a double application of the Lorentz Transformation, whereas Einstein in his actual theory says we are to apply it only once.

And also, Sean has Bob and Ann traveling different distances, which is not correct according to the SR theory.

When young kids go on a “twins paradox resolution” page on the internet, and if the page is done by an adult, the kids tend to believe the numbers without question. And then if they don’t understand what’s going on, then they feel bad about themselves and think they are “stupid”, when in reality the adult who put the “resolution” page on the internet is either mistaken or is just not telling the truth or is not telling the whole story about the SR theory, and the whole story is very complex. You can’t “solve it” all and “explain” the theory on just one computer-screen page.

Someone needs to tell the public that these guys who make up the internet pages are making mistakes themselves. This is not Einstein’s fault, it’s the fault of the guys who make up the internet pages, and a lot of people need to tell the kids what’s going on and how these adults are making their mistakes, and they need to assure the kids who don’t understand it that they (the kids) are NOT stupid.

It’s not the kids’ fault that they don’t understand this. It’s the fault of the guys who put the wrong “resolutions” on the internet.

Please look at my earlier posting in this thread for the resolution of the twin "paradox". The proper formulae for the redshift and blueshift are still not being used which is why you are arguing about a non-existent "jump". Please read my two "When Harry Meets Sally" posts in the "Einstein's Theories ..." thread where the K-calculus is used to derive the proper formulae.

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OK. Back to two Bob's. BobI is the inbound Bob, BobO the outbound Bob.

From Alice's point of view, the experiment starts at time 0 with BobI six light years out, coming towards her at .6 c. BobO leaves, heading towards BobI, also at .6 c. Alice flashes BobI a message at this time.

Alice thinks BobI receives the message 3.75 light years out. BobI thinks it is 3 light years to Alice when he receives the message, he thinks Alice must have sent it 7.5 years ago, although to Alice it would seem only 6.

Perhaps I should do the math. 6 l.y / (1 + .6) = 3.75 years for the signal to meet BobI. 3.75 * gamma = 3 light years for the distance from BobI to Alice. 3 ligiht years / (1 - .6) = 7.5 years BobI thinks the message was in transit. 7.5 * gamma = 6 years elapsed for Alice.

OK. Alice thinks BobI and BobO cross 3 light years out. BobI and BobO think they cross 2.4 light years out. BobI thinks that an additional year has passed since he received the message from Alice, although he thinks Alice would think only 0.8 years have passed, so BobI thinks 6.8 years have passed for Alice when he and BobO cross paths. BobO has been journeying for 4 years when he and BobI cross paths, and think only 3.2 years have passed for Alice.

BobI thinks that 10 years will have passed for Alice since BobO passed her. BobI thinks that four years will have passed for BobO since BobO passed Alice, and four years will have passed for him since he passed BobO. If BobO had passed BobI a stopwatch (probably a Timex), the watch will read eight years since BobO started.

Nobody is surprised. Everybody, at every well defined moment in the example agrees about how much time has elapsed for the participants. I haven't worked through all the numbers, but feel free. Alice, BobI, and BobO agree how much time has passed for BobO from when he passed Alice to when he passed BobI. Alice, BobI, and BobO agree how much time has passed from BobI from when he received Alice's signal to when he passed BobO. Alice, BobI, and BobO agree how much time passed for BobI from when he passed BobO to when he passed Alice. Alice, BobI, and BobO agree on how much time passed for Alice from when she was passed by BobO to when she was passed by BobI.

What they don't agree on are the non-local events--they will get different numbers for how much time had passed for BobO or Alice when BobI received the message from Alice, how much time passed for Alice when BobO passed BobI, and how much time passed for BobO when BobI passed Alice.