Another Twins Paradox question

[JesseM]

What makes my "attack" unfounded? And how is it an attack on mainstream science? I can see it as an attack on lazy defining of a problem and an attack on equally lazy explanations of a theoretical phenomena.
I've also seen many posts from some users who "wish to argue Relativity's validity, or advertise their own personal theories", that didn't earn a warning about forum guidelines. What's so much worse about mine? Does it make too much sense? Scared? Go ahead and delete it if you must... If you want to stifle intelligent debate...

My understanding of the rules is that you are free to say that certain aspects of relativity don't make sense to you, and ask questions about how relativity would explain things, as you did in that long post. What's not allowed is just making definite assertions that relativity is wrong without any room for further discussion or calls for explanation.

 

[Physical_Anarchist]

Thank you for your replies, JesseM. I also read the other thread you referenced. I understand how, using the Lorentz transformation, one can demonstrate that the Twin in the spaceship ages less.
The problem I have with the whole concept, however, is that, by showing how one twin ages less, the paradox is trying to demonstrate that time dilates when traveling at speeds close to c. Why then is it OK to use a formula that assumes time dilation within our demonstration? It becomes a circular argument.
It's exactly as if I were trying to demonstrate that 5=7 by assuming that 1=3 and demonstrating that 1+4=3+4.

 

[RandallB]

Never fear, I am here to ……… tell it like it is.

Seriously, though, let's define the problem
How do we then define ……without needlessly confusing accelerations?

They accelerate at the same rate, …..
Afterwards, one of engages his thrusters in reverse, to decelerate,
and then to accelerate
and finally decelerate again
Meanwhile, the second one has stopped accelerating,
later, he does the same decelarating and accelerating
later, he decelerates the way his twin did and also ends at the point of origin.
I challenge anyone to prove …………….

What makes my "attack" unfounded?

OH I fear – If that was to remove acceleration confusion – I fear what you might say when you get to something not so simple as SR like GR or QM – I fear a great deal!

Your "attack" is unfounded, because it is irrationality incomplete.

In the Twins Paradox there are no acceleration issues to deal with.
It is easy to eliminate acceleration calculations in relativity.
Just use infinite instantaneous accelerations that take zero time to make transfers.
All reference frames will agree that the time elapse for any object going though such acceleration will slow to zero.
But since it also takes zero time in all reference frames there is also no argument as to when and where it started or ended as measured in any frame from any frame.
They will all agree.

If you can not handle that simple assumption, just use high speed snap shots of clocks with a fresh stop watch attached to each photo image. Then you can track total time for a clock and its images without anything actually having to accelerate anything at all. Just keep track of exactly where and when in each reference frame each image was taken and recorded.

Both methods will give the same agreement with SR, which is the Twins issue is not a paradox at all.
Do the work and you will know it like it is.
Just be sure to be absolutely complete and detailed about the where and when of each event in all three reference frames.

 

[Physical_Anarchist]

While true that I didn't eliminate acceleration from the sequence of events, I did eliminate it from the equation by having both twins experience the same amount of it.
You say that "In the Twins Paradox there are no acceleration issues to deal with."
JesseM also said we can make the acceleration instantaneous in order to eliminate it from the equation.
Why then is it that in explanations of the Twin Paradox, acceleration always rears its ugly head, by claiming that it can't really be instantaneous after all and some (or all) of the discrepancy between the clocks happens there, or that we have to only consider the worldline of the one who remained in an uniform inertial frame?
Well, the way I arranged the problem, these strategies can no longer be used to avoid the issue.
Thinking of the problem as I stated it, where both experience the same accelerations, the difference between them is only the speed that they are experiencing. Since that is relative, how can you tell which one is moving and thus remaining younger?

 

[RandallB]

Why then is it that in explanations of the Twin Paradox, acceleration always rears its ugly head, by claiming that it can't really be instantaneous after all and some (or all) of the discrepancy between the clocks happens there, or that we have to only consider the worldline of the one who remained in an uniform inertial frame?

Acceleration is not always used to explain the differences in the twin’s ages. As you've already noted, JesseM certainly didn’t use that.
And as you said, when acceleration on the traveling twin is used as the reason for the difference in ages, it is ugly because it is wrong.

 

[pervect]

What makes my "attack" unfounded? And how is it an attack on mainstream science? I can see it as an attack on lazy defining of a problem and an attack on equally lazy explanations of a theoretical phenomena.

Your attack is unfounded because you set up a "straw man" which you then proceed to be demolish.

This is a rather shabby form of debating practice.

The general purpose of this forum is to answer questions that people have about relativity. You are not "asking questions", you are playing silly little debating games.

I don't really believe you for a second when you say that you are attacking the "lazy writers" of that article and not attacking relativity. But I'll pretend that I do, for the sake of politeness.

In that case, I will simply say that it is not the fault of the article that it does not address your particular questions. It was not intended to provide a "proof" of relativity.

So, now let us pretend that you politely asked us - if this article doesn't provide a "proof" of relativity, and that it is not necessarily a bad article for omitting such a "proof", for it never intended to provide such a "proof", where do I find an article that does?

We will then politely answer you that that science does not provide such proofs. Back in the days of Aristotle, it was thought that "man's mind could elucidate all the laws of the universe, by thought alone, without recourse to experimentation"

Nowadays, we know better. Or at least most of us do. If we take your post at face value, you apparently do not know better.

So now we will politely attempt to explain to you that the scientific method is based on doing experiments - not on "proof".

This is really basic stuff. I'll conclude to a link to the wikipeda with some basic introductory info on the scientific method:

http://en.wikipedia.org/wiki/Scientific_method

a link to Aristotle's view on science

http://en.wikipedia.org/wiki/Aristotle#Science

and a suggestion that if you want to debate the foundations of science that you try the philosphy forum and not the relativity forum.

Now, if you ever manage to progress to the question: "What sort of experimental evidence makes us believe in relativity" this would be a resonably good forum to ask such a question. Of course we'd have to believe that you were actually interested in the answer...

 

[JesseM]

Thank you for your replies, JesseM. I also read the other thread you referenced. I understand how, using the Lorentz transformation, one can demonstrate that the Twin in the spaceship ages less.
The problem I have with the whole concept, however, is that, by showing how one twin ages less, the paradox is trying to demonstrate that time dilates when traveling at speeds close to c. Why then is it OK to use a formula that assumes time dilation within our demonstration? It becomes a circular argument.

I think you're misunderstanding the point of the twin paradox. The original idea of the "paradox" was to show that there was an internal logical inconsistency in the theory of relativity, so that even if you started out assuming the laws of relativity were correct, you would get inconsistent predictions if you analyzed the same situation from the point of view of different reference frames. The basic idea of the paradox is something like "from the Earth twin's point of view, the twin in the rocket is the one moving so his clock will be running slower, therefore he'll have aged less when they reunite; but you could equally well look at things from the point of view of the twin in the rocket, who sees the Earth moving, therefore he should predict the Earth twin will have aged less." The flaw in this argument is that the standard rules of time dilation only work in inertial frames, and the rocket twin does not stick to a single inertial frame (this is true regardless of whether he changes velocities instantaneously or if the acceleration is spread out over a finite period of time). As long as you analyze the paths of both twins from the point of view of an inertial frame, you will always get the same answer to how much each twin will have aged along their entire path, even if you use a frame where the Earth is moving and the twin on the rocket is at rest during one leg of the journey (but in such a frame, the twin on the rocket will have to move even faster than the Earth on the other leg of the journey in order for them to reunite).

If you are looking for actual experimental evidence of time dilation, rather than just arguments for why the theory of relativity is internally consistent, that's a separate subject. There's certainly plenty of experimental evidence, like the longer decay time of particles moving at very high velocities, or the fact that the GPS satellite system is designed to factor time dilation into all its calculations and would not work correctly if time dilation did not exist.

 

[Physical_Anarchist]

First of all, perv, I did not set up a "straw man". And stop defending that one specific article. It was merely an example. I read multiple articles on the twins paradox, some of which I was directed to from numerous threads on the subject in this forum. There was one that had "faraday" and, I believe, the university of toronto in the url, and another one linked to from the end of that one, for instance. They all claim to be the resolution of the paradox, and none did so satisfactorily in my view. That is why I registered here. Having read a few threads around here, I thought this would be a place where I could possibly get some clarifications. I am not looking for proof of relativity, experimental or otherwise. I merely wanted to analyse the twins paradox.
JesseM: you said: "The flaw in this argument is that the standard rules of time dilation only work in inertial frames, and the rocket twin does not stick to a single inertial frame". In my version of the twin paradox, I had both twins accelerating the same way. That leaves us then only the parts of the trip where each twin's speed is uniform to consider for comparison. I have yet to see a resolution of that scenario that doesn't use the conclusion as an assumption in the process. (I still believe that if I ask "Why is the sky blue?", "Because blue is the color of the sky" is not a complete and satisfactory answer. Circular reasoning is just not my thing... Shoot me!)

 

[clj4]

I have yet to see a resolution of that scenario that doesn't use the conclusion as an assumption in the process. (I still believe that if I ask "Why is the sky blue?", "Because blue is the color of the sky" is not a complete and satisfactory answer. Circular reasoning is just not my thing... Shoot me!)

Read this. If you are still confused, ask me questions:

http://sheol.org/throopw/sr-ticks-n-bricks.html

 

[JesseM]

JesseM: you said: "The flaw in this argument is that the standard rules of time dilation only work in inertial frames, and the rocket twin does not stick to a single inertial frame". In my version of the twin paradox, I had both twins accelerating the same way.

They didn't accelerate in the same way though. One accelerated twice between the time they began to move apart and the time they reunited--first to turn around after having left the earth, then to match his speed to the Earth to wait there for the other twin to return. The second twin, who spent longer away from the earth, only accelerated once, to turn around (you don't have to have him accelerate again once he reaches earth, you can just have the two compare clocks at the moment they pass at constant velocity).

I like to think of the twin paradox in terms of the "paths through spacetime" explanation. If you draw two points on a piece of paper, and one straight-line path between them and another with a bend in it, then you will always find that the straight-line path is shorter. Similarly, if you draw a spacetime diagram for the twins (with just one spatial dimension and one time dimension for convenience), they are taking two different paths between two points in spacetime (the point where they leave each other and the point where they reunite), and the way the time elapsed on a path is calculated in relativity insures that a straight path through spacetime will always have a greater proper time than any non-straight path between the same two points. So while the duration of the acceleration is not really important, the fact that acceleration leads to a bend in a twin's path through spacetime insures that it will have a smaller proper time. In your example where both twins accelerate, it's as if I had drawn two non-straight paths between the same two points in space, one consisting of two straight line segments joined at an angle, and one consisting of three straight line segments joined at an angle. Here, which path is longer really depends on the shape of the paths. Similarly, in your example it also depends on the shape of the path--it would actually be possible for the twin who spends most of his time on Earth to nevertheless be younger when they reunite, if he had been traveling at a much greater velocity relative to the Earth during his trip away and back. But if you specify they were both traveling at the same speed relative to the Earth during their trip, he will always have aged more. Similary, if you specify that the three-line-segment path through space has one segment that is parallel to a straight line between the points (analogous to the section of the twin's path through spacetime spent at rest on earth), and the other two segments are at exactly the same angle relative to this straight path as the two segments of the second path (analogous to the fact that in your example both twins have the same velocity during the inbound and outbound legs of their trip), then the path with three segments will always be longer than the path with two segments. If my descriptions are unclear I could provide a diagram as well.

(I still believe that if I ask "Why is the sky blue?", "Because blue is the color of the sky" is not a complete and satisfactory answer. Circular reasoning is just not my thing... Shoot me!)

You're equivocating on what kind of question you're asking though. If your question is about the internal logic of why relativity predicts that one twin will be younger, then in answering it we will take for granted the rules of relativity, and explain why the rules lead to these predictions. But if you're asking for experimental evidence that the rules of relativity are actually the ones that are seen in the real world, that's a totally separate question, the answer would involve various pieces of evidence for these rules such as the increased decay time of fast-moving particles or the workings of the GPS satellite system. If you want experimental evidence, than don't ask theoretical questions about the twin paradox, and if you want theoretical explanations of why relativity predicts one twin will be younger, then don't complain about "circular reasoning" when we assume the laws of relativity in our answer. Either one is inconsistent and illogical on your part.

 

[Al68]

Since this thread is still going, I have another question.

I personally don't like the Twins Paradox explanation with instantaneous turnaround. Or the two ship explanation where the incoming ship passes the outgoing ship at the distant star system. Considering this as two independent frames illustrates the math of time dilation, but, observers in the two ship frames will disagree with each other about the moment of departure from Earth and the moment of arrival back to earth. For example, with the two different ships passing each other in opposite directions at the star system 10 light years from Earth (earth frame), v = 0.866c, from Earth's frame the incoming ship arrived 23.1 years after the outgoing ship left. If you add up the times for the one way trip of each ship, we get 11.55 years. But, if we assume that Earth sends out a signal when the outgoing ship leaves earth, and the incoming ship received this signal, the incoming ship will calculate that he reaches Earth 46.2 years after the outgoing ship left earth. And similarly, if Earth sends out a signal when the incoming ship arrives at earth, and the outgoing ship eventually receives it, an observer on the outgoing ship will calculate that the incoming ship arrived at Earth 46.2 years after he left Earth (I hope I got this math right).

So, from the point of view of either ship frame, although it only took 5.77 years for them to get from Earth to the star system or vice versa, the total proper time (for either ship frame) between the event of the outgoing ship leaving Earth and the event of the incoming ship arriving at Earth is 46.2 years. Is this correct?

And this explanation also sidesteps some of the questions that arise from the Twins Paradox. With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)? Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)? And then get 5 light years closer during the brief acceleration when the ship leaves the star system?

Is there a good explanation of the Twins Paradox available on the internet that addresses these kinds of questions?

I would like to find a good, comprehensive explanation to read before I ask a lot more questions.

And I hope nobody interprets this as a challenge to SR. Of course SR is mainstream science, and we have plenty of experimental evidence. But obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct. But it's explanations are different and vary even in accepted textbooks. Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox. But, as far as I can tell, there is not universal agreement by mainstream sources about the details of this issue. And some details are not addressed at all.

Thanks,
Alan

 

[clj4]

Since this thread is still going, I have another question.

I personally don't like the Twins Paradox explanation with instantaneous turnaround. Or the two ship explanation where the incoming ship passes the outgoing ship at the distant star system. Considering this as two independent frames illustrates the math of time dilation, but, observers in the two ship frames will disagree with each other about the moment of departure from Earth and the moment of arrival back to earth. For example, with the two different ships passing each other in opposite directions at the star system 10 light years from Earth (earth frame), v = 0.866c, from Earth's frame the incoming ship arrived 23.1 years after the outgoing ship left. If you add up the times for the one way trip of each ship, we get 11.55 years. But, if we assume that Earth sends out a signal when the outgoing ship leaves earth, and the incoming ship received this signal, the incoming ship will calculate that he reaches Earth 46.2 years after the outgoing ship left earth. And similarly, if Earth sends out a signal when the incoming ship arrives at earth, and the outgoing ship eventually receives it, an observer on the outgoing ship will calculate that the incoming ship arrived at Earth 46.2 years after he left Earth (I hope I got this math right).

So, from the point of view of either ship frame, although it only took 5.77 years for them to get from Earth to the star system or vice versa, the total proper time (for either ship frame) between the event of the outgoing ship leaving Earth and the event of the incoming ship arriving at Earth is 46.2 years. Is this correct?

And this explanation also sidesteps some of the questions that arise from the Twins Paradox. With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)? Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)? And then get 5 light years closer during the brief acceleration when the ship leaves the star system?

Is there a good explanation of the Twins Paradox available on the internet that addresses these kinds of questions?

I would like to find a good, comprehensive explanation to read before I ask a lot more questions.

And I hope nobody interprets this as a challenge to SR. Of course SR is mainstream science, and we have plenty of experimental evidence. But obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct. But it's explanations are different and vary even in accepted textbooks. Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox. But, as far as I can tell, there is not universal agreement by mainstream sources about the details of this issue. And some details are not addressed at all.

Thanks,
Alan

The twins paradox has not been tested per se but there are plenty of other practical situations that received theoretical and experimental attention. Since you want something that you can read off the net, the best that comes to mind is the SR AND GR corrections that need to be applied prior to the launch of the GPS satellites. There may be more but this one is one of the best. See here:

http://relativity.livingreviews.org/open?pubNo=lrr-2003-1&page=node5.html

The other one that comes to mind is the Haefele - Keating experiment .You'll need to look up their paper.

 

[jtbell]

With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)?

Yes. The key concept here is that of instantaneously co-moving inertial reference frames. During the time period (as measured by the ship's clock, say) that the ship is accelerating or decelerating, it is not stationary in any single inertial reference frame. Nevertheless, at any point in time according to the ship's clock, it is instantaneously stationary in an inertial reference frame which is moving along with the ship. At that point in (ship) time, the distance between the Earth and the star system is contracted according to the relative speed of that instantaneously co-moving inertial reference frame with respect to the inertial reference frame in which the Earth and star system are stationary.

Loosely speaking, we can say that the ship "passes through" a continuous series of instantaneously co-moving inertial reference frames, with infinitesimal relative velocities between each pair of successive frames.

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c. The Earth's apparent superluminal velocity comes about because the ship is not moving inertially. The v <= c restriction applies to velocities of objects observed in a single inertial reference frame. (There's probably a more precise way to state this, but I can't think of it off the top of my head.)

obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct. But it's explanations are different and vary even in accepted textbooks.

Ever hear the saying, "There's more than one way to skin a cat?"

 

[robphy]

But obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct.

Of course, one way to test the twin paradox is to travel to a distant planet [insert: and back] at speeds close to the speed of light... then compare wristwatches. However, with an accurate enough clock, you don't need to go that far or that fast. Consider the clocks described here http://www.newscientist.com/article.ns?id=dn7397 "The first atomic clocks could pin this down to an accuracy of 1 part in 10^10. Today's caesium clocks can measure time to an accuracy of 1 in 10^15, or 1 second in about 30 million years." You can figure out the order of magnitude of v that corresponds to a gamma of (say) 10^(-15). I would think that such an experiment is possible in the forseeable future.

But it's explanations are different and vary even in accepted textbooks. Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox. But, as far as I can tell, there is not universal agreement by mainstream sources about the details of this issue. And some details are not addressed at all.

In my opinion, a "standard, mainstream textbook", especially one written by a non-relativist, is generally not the best place to find a definitive statement about "resolving the twin paradox", together with the various issues that may be raised. Such a textbook's explanation is usually based (read as "limited") by what material has been presented thus far in that textbook.

The variety of explanations arise from the many symmetries of Minkowski spacetime (see #5 in https://www.physicsforums.com/showthread.php?t=118994 ). In my opinion, the best explanations are the ones that use the fewest number of those symmetries... because they focus on the key physical idea: the proper-time [arc-length in spacetime] between two events is longest for the inertial observer.

As I have often said on this topic,
here's one of my favorite papers on the clock paradox:
http://links.jstor.org/sici?sici=0002-9890(195901)66%3A1%3C1%3ATCPIRT%3E2.0.CO%3B2-L
"The Clock Paradox in Relativity Theory"
Alfred Schild
The American Mathematical Monthly, Vol. 66, No. 1. (Jan., 1959), pp.1-18.
This addresses many of the approaches that have been suggested.

 

[RandallB]

I personally don't like the Twins Paradox explanation with instantaneous turnaround. Or the two ship explanation where the incoming ship passes the outgoing ship at the distant star system. Considering this as two independent frames illustrates the math of time dilation, but...

But nothing,
I thought you were getting the Twins but from the above I can see your still in the weeds.
You cannot do the twins where one returns the start with just two reference frames,
the traveler cannot get back to the other twin without a third frame.
And again if you don’t like transferring people at infinite accelerations – use the third returning frame to hold a clock and camera to take a photo back to earth.
Just collect ALL the data from all three frames with each photo to analyze what has happened.
Be sure Earth collects photos that include the WHERE and WHEN of all three reference frames as can be seen locally at Earth in all three frames for every event. Including: Three photos taken at Earth when that location is simultaneous with the traveling twin reaching the star for each of the three frames – That means three different photos of three unique events that hold 18 different pieces of information about Where and When those three events took place at earth.
Then do the same for star, based on a) when the Twin leaves Earth and b) when the photo of the twin and the star is brought back to Earth by someone in that third frame. That will be 36 pieces of information.
All these photos can be collected in one place after the fact for your review by whatever accelerations or data transfer is OK by you.
The conclusions you draw from this mathematical exercise using SR rules will correlate to the same kind of results that are always seen in experiments that confirm SR.

Take your time don’t lose track of a frame or locations and distances in it.

 

[JesseM]

With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)?

Yes. The key concept here is that of instantaneously co-moving inertial reference frames. During the time period (as measured by the ship's clock, say) that the ship is accelerating or decelerating, it is not stationary in any single inertial reference frame. Nevertheless, at any point in time according to the ship's clock, it is instantaneously stationary in an inertial reference frame which is moving along with the ship. At that point in (ship) time, the distance between the Earth and the star system is contracted according to the relative speed of that instantaneously co-moving inertial reference frame with respect to the inertial reference frame in which the Earth and star system are stationary.

I agree with this description of what happens if you measure the distance in a series of instantaneously co-moving inertial frames, but I think it's misleading terminology to say this is what will be "observed by the ship", period. The series of co-moving inertial frames do not together define a single well-behave non-inertial coordinate system for an accelerating observer, because the same event could happen simultaneously with more than one point on the observer's worldline. And when dealing with non-inertial coordinate systems, there is no reason to see one choice as more physical than another, so you could equally well invent a very different non-inertial coordinate system in which the accelerating observer is at rest but the distance at any given moment does not correspond to the distance in the instaneous inertial frame at that moment. I think the word "observed" should only be used without qualification when talking about inertial observers, where there is a single well-known convention for how to define the coordinate system that constitutes their "rest frame", while it shouldn't be used for non-inertial observers, at least not unless you define in advance what coordinate system they are using to make "observations", with it being understood that this choice of coordinate system is a somewhat arbitrary one.

 

[pervect]

There is a shortish summary of many of the varioius methods of addressing the twin paradox at

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

including an addendum about "too many explanations", which has a lot of very useful diagrams.

There are two general subsets of the many approaches that are worth some attention.

The first approach considers only what the two space-ships actually see. By this I mean the signals that they actually receive from each other. This is the "doppler approach". See figure 2 in http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html for instance.

A second subset of explanations focuses on which events different obsevers think of as being "simultaneous".

The most basic thing one must understand with the later approach is that simultaneity is relative. One can actually draw "lines of simultaneity" on a space-time diagram that represent different observer's concepts of simultaneity.

As we have mentioned in another thread, the slope of a line of simultaneity for an inertial obsever is always c^2 / v, also written as c/$\beta$ where $\beta$=v/c.It is probably better NOT to get too mired in the working out of "what events are simultaneous to other events according to which observer" but it seems that some people just can't help it.

See figures 3 and 4 in http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html

for diagrams which show the lines of simultaneity.

 

[Physical_Anarchist]

They didn't accelerate in the same way though. One accelerated twice between the time they began to move apart and the time they reunited--first to turn around after having left the earth, then to match his speed to the Earth to wait there for the other twin to return. The second twin, who spent longer away from the earth, only accelerated once, to turn around (you don't have to have him accelerate again once he reaches earth, you can just have the two compare clocks at the moment they pass at constant velocity).

I don't get it. You change the way I formulated the question, just so you can say they didn't accelerate in the same way? To have them experience the exact same accelerations overall is precisely why I had the second one decelerate before re-joining the point of origin.

Let me illustrate:
1.@@@@@@DDDDDD-@-@-@-@-@-@DDDDDD////////////////////////
2.@@@@@@////////////DDDDDD-@-@-@-@-@-@////////////DDDDDD

Legend: @=accelerate, D=decelerate, /=one month.
Since deceleration is merely acceleration in the opposite direction and wee're disregarding accelerations, since they are equivalent overall, we only have to compare 24 months at "low speed" to the 2 segments of 12 months at "high speed". Relativity tells us that the one at low speed could actually be the one moving at high speed. None has a more legitimate claim than the other. To understand this, imagine that their point of origin is actually moving, without them realizing it. Their trip, that they imagined as in the above illustration, actually could look like this:

1.DDDDDD@@@@@@DDDDDD-@-@-@-@-@-@///////////////////////
2.DDDDDD////////////@@@@@@DDDDDD////////////-@-@-@-@-@-@
This illustration is just as legitimate as the first one for the purposes of determining speed in a relative context.

As for your other argument, I'll always complain about circular resoning, because circular reasoning is simply bad logic. It should always be possible to work out the theory using logic.

Al68 said: "Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox." SR creates the Twins Paradox. That's why it's a paradox.

clj4: I read that thing, but it was really late at night and I'm still confused. I'll give it another try.

 

[clj4]

clj4: I read that thing, but it was really late at night and I'm still confused. I'll give it another try.

Yes, read again.

 

[JesseM]

I don't get it. You change the way I formulated the question, just so you can say they didn't accelerate in the same way? To have them experience the exact same accelerations overall is precisely why I had the second one decelerate before re-joining the point of origin.

My point is that as long as you're assuming instantaneous acceleration, accelerations at the start or end of each one's path (with the start being where they depart at a common time and place, the end being where they reunite at a common time and place) don't affect the length of the path in between those points, so they're irrelevant. And if the acceleration is brief but not instantaneous, the difference between accelerating right near an endpoint or not accelerating will be very small, it won't substantially change the answer to which twin is older when they reunite.

Did you read everything I wrote about the "paths through spacetime" way of thinking about the problem, and did you understand why, in your example, their two paths will be quite different, regardless of whether the second one accelerates when he reunites with his twin who's already at rest on earth? I'll try to render a diagram here if it helps:../\
./..\___
*...*
.\.../
..\.../
...\../
...\/

Here position is the vertical axis, and time is the horizontal axis (ignore the rows of dots, they're just there to keep everything spaced right since the forum automatically deletes multiple spaces in a row...if the diagram is unclear I can redraw it as a nicer-looking image file on request). The *'s are the endpoints of the path, the top path involves first moving away from Earth (the part of the path slanted like /), then moving back towards (the part of the path slanted like \), then resting on Earth while waiting for the other twin to return (the flat part of the path which looks like ___ ), while the bottom just involves moving away from the Earth (\) and returning (/)between the two endpoints. If you understand the diagram, it should be obvious that it doesn't matter if the twin on the bottom path accelerates to come to rest on Earth right as he is about to reach the endpoint or not, it will have no significant effect on the overall length of the path between the two endpoints, and that's all that's important.

As for your other argument, I'll always complain about circular resoning, because circular reasoning is simply bad logic.

You should review the meaning of the term "circular reasoning", because proving that there is no logical inconsistency in a theory using the axioms of the theory itself is definitely not circular reasoning. Of course this can't prove whether or not the theory is true in the real world, only whether it contains any internal inconsistencies.

It should always be possible to work out the theory using logic.

Complete nonsense. There is not a single scientific theory that can be proven using only "logic" without any need for observation. Both Newtonian mechanics and relativity are internally consistent, for example, it is only experimental tests which can tell you which is actually true in the real world.

 

[Physical_Anarchist]

I don't know why a position diagram is relevant. A speed graph would be more appropriate and I think it would look like this:
.../\
../...\
/...\...__________
...\.../
....\../
.....\/

versus
..._____
.../...\
../....\
/....\
......\.../
.....\.../
......\_____/

 

[JesseM]

I don't know why a position diagram is relevant. A speed graph would be more appropriate and I think it would look like this:
.../\
../...\
/...\...__________
...\.../
....\../
.....\/

versus
..._____
.../...\
../....\
/....\
......\.../
.....\.../
......\_____/

Either a speed graph or a position graph can be used to determine the total spacetime "length" of each path (ie the proper time along each path). Similarly, if you have two paths drawn on an ordinary piece of graph paper with x and y coordinate axes drawn on, then if you can describe the paths in terms of the y-position as a function of x-position, like y(x), or if you can describe the slope of each path at every point along the x-axis with the function S(x), then you can use either one to find the total length of the path between the two points.

You can see in your diagram above that the speed graphs for the two paths look quite different, and that your addition of that third acceleration at the very tail end of the last graph didn't make their overall shapes the same (also, if that acceleration was very brief compared to the overall time spend in space, the last upward section should be much shorter along the t-axis, but maybe they weren't meant to be to scale). In terms of figuring out the proper time, if you know the functions for speed as a function of time v(t) in a particular inertial reference frame, and you know the times of the two endpoints $$t_0$$ and $$t_1$$ in that frame, then to find the total proper time you'd integrate $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$. The value of this integral for the second path will be only barely changed if you add a brief acceleration immediately before the time $$t_0$$ when they reunite.

So again, what's important is the precise nature of the position vs. time or speed vs. time functions for each one. The "whichever twin accelerates will have aged less" is not meant to be a general answer that covers all cases, it's only meant to cover the case where one twin moves inertially between the two endpoints while the other does not (as long as this is true, then no matter what specific position vs. time or speed vs. time function you pick for the second twin, you will find that his proper time is less). But in the case where both twins accelerate, you obviously can't apply this rule, you have to consider the two paths in a more specific way.

All this is directly analogous to the example of two paths drawn on an ordinary 2D piece of paper; if one is straight while the other has a bend in it, you can say "whichever path has the bend will be longer", but this is not a general rule that would cover all cases, in an example where neither path is straight you have to consider the specific shape of each path.

edit: I just noticed something about your graphs--is there a reason that the changes in speed in the first graph are sharp, while the changes in speed in the second graph have those flat intervals? If the flat intervals are meant to be extended periods of time at rest relative to the earth, so that both ships spend the exact same amount of total time at rest relative to the Earth from beginning to end, and also the same amount of time moving away from Earth at speed v and the same amount of time moving towards it at speed v, then in that case they will be the same age when they reunite. If this is what you meant all along and I misunderstood, then sorry for the confusion.

edit 2: OK, another thing I missed, but shouldn't constant speed always be a flat section of the graph? On a graph of speed vs. time, a non-flat slope would be a period of acceleration (speed changing at a constant rate), is that what you meant the sloped parts of the graph to represent? I thought in your example the idea was that each ship spent most of the time moving at constant velocity, with only brief periods of acceleration.

 

[Al68]

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c. The Earth's apparent superluminal velocity comes about because the ship is not moving inertially. The v <= c restriction applies to velocities of objects observed in a single inertial reference frame. (There's probably a more precise way to state this, but I can't think of it off the top of my head.)

Why would we not call this v > c genuine? If we are calling the v < c restriction for the ship's velocity (on the way to star system) relative to Earth's position genuine. If we do not call Earth's position of 5 light years away from the star system "genuine", then our velocity of v = 0.866c would not be "genuine". Are you just saying this v > c is not genuine because this is not a restriction for accelerating observers?

Also, I have read all of the referenced explanations on the internet (except the one on jstor, I don't have access), and none of them address the questions I have. That's probably because I've read them all before, and no longer have the questions that they do address.

Also, it's my understanding that Einstein initially thought he should be able to consider the ship at rest with the Earth and star system moving back and forth relative to the ship, and still resolve the Twins Paradox in SR. But he gave up on this and tried to resolve it with GR. And physicists generally consider his GR resolution erroneous. Is this correct? I think this is what wikipedia says, also.

Thanks,
Alan

 

[pervect]

Why would we not call this v > c genuine? If we are calling the v < c restriction for the ship's velocity (on the way to star system) relative to Earth's position genuine. If we do not call Earth's position of 5 light years away from the star system "genuine", then our velocity of v = 0.866c would not be "genuine". Are you just saying this v > c is not genuine because this is not a restriction for accelerating observers?

Here's a quote from Baez's et al paper on GR ("The Meaning of Einstein's equations" which helps explain this point.

http://math.ucr.edu/home/baez/einstein/node2.html

(also available at http://arxiv.org/abs/gr-qc/0103044 in pdf if you don't like the chopped-up version)

Before stating Einstein's equation, we need a little preparation. We assume the reader is somewhat familiar with special relativity -- otherwise general relativity will be too hard. But there are some big differences between special and general relativity, which can cause immense confusion if neglected.

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

This is the issue that you are running into with apparently FTL velocities. You are using a non-inertial coordinate system, and expecting it to act like an inertial coordinate system.

Note that because the underlying problem is in flat space-time, one actually CAN talk about the relative velocities of two particles. But in order to do so and get the right answer, one must restrict oneself to inertial coordinate systems.

Note that even in flat space-time, if the velocity between two objects is changing (because one of them is accelerating), the velocity of a distant object "at the same time" is ambiguous, because "at the same time" is an ambiguous concept in SR.

Thus the main problem is in your expectations. You are applying concepts which work in inertial coordinates and expecting them to apply in generalized coordinates.

You might also notice (or maybe you haven't) that the velocity of light is not constant in your accelerated coordinate system. Thus when you say that the distant object is moving "faster than light", it is actually not moving faster than light moves at that particular location. In your non-inertial coordinate system, light does not have a constant coordinate velocity, and in the region where the Earth appears to be moving faster than 'c', light appears to be moving even faster than the moving Earth.

Also, I have read all of the referenced explanations on the internet (except the one on jstor, I don't have access), and none of them address the questions I have. That's probably because I've read them all before, and no longer have the questions that they do address.

I also do not have access to the Jstor article.

As far as I can tell, you are trying to run before you can walk. It is possible to understand and work with non-inertial coordinates in relativity, but it requires some sophisticated mathematical techniques, such as the process of "parallel transport" that Baez alludes to.

It is both easier and more productive (IMO) to start to learn about relativity in a coordinate independent manner. This means learning about 4-vectors, and space-time diagrams. You need to have a firm grasp on SR, especially on the relativity of simultaneity (which you apparently still are struggling with from what I can infer from your remarks) before you can go on to handle GR and arbitrary coordinate systems.

You might also give some thought to the philosophical idea that coordinate systems are not the fundamental basis of reality.

Rather than treat coordinates as the basis of reality, think of the arrival of signals, and the readings of clocks, as being the fundamental basis - after all, that is actually what you can observe. You do not directly perceive the coordinates of some distant object, you percieve signals from that object.

The "coordinate" of a distant object are just something that you compute. Coordinates are supposed to be a convenience to make your life easier (and not a millstone around your neck that drags you into confusion). What you actually physically observe are signals emitted from and sent to said object (such as radar signals, or observations you make with a telescope).

The Doppler explanation of relativity, for instance, tells you all about how to compute the arrival time of such signals.

If you get into a muddle, think not about coordinates, but think instead about physical signals - when they were sent (and by whose clock that time deterimnation was made!), and when they arrived. Think about things that you actually could directly observe (i.e. NOT coordinates, which are things that you compute, not observe).

Also, it's my understanding that Einstein initially thought he should be able to consider the ship at rest with the Earth and star system moving back and forth relative to the ship, and still resolve the Twins Paradox in SR. But he gave up on this and tried to resolve it with GR. And physicists generally consider his GR resolution erroneous. Is this correct? I think this is what wikipedia says, also.

Thanks,
Alan

I can't make heads or tails of this remark. If you could provide a specific quote from the Wikipedia I might be able to say more.

 

[Al68]

pervect,

I was only objecting to the use of the phrase "not genuine". And I note that using observation as a basis for reality is objected to by some on this forum. I have my own reasons for asking these questions. I don't have a problem with apparent FTL velocities. I was just remarking that observations should not be referred to as "not genuine". And I would think that, after your comments here, that you would agree.

And here is a link to that wikipedia article: http://en.wikipedia.org/wiki/Twin_paradox#Resolution_of_the_Paradox_in_General_Relativity

It is part of their Twins Paradox article.

Thanks,
Alan

 

[pervect]

I might quibble with the exact way Jesse worded his remark, but I agree with the spirit. There may be some way to explain it more concisely than the rather long quote I gave, but I'm not sure how to do it :-(.

As far as the Wikipedia goes, I agree that there are some issues with the "gravitational time dilation" explanation of the twin paradox, but those problems don't actually come up in the twin paradox itself in my opinion.

My personal opinion is that the explanation IS good enough to explain the twin paradox, but has troubles further down the road.

I suspect you are traveling down that road right now, so I'll give you some idea of where I see the roadblock occurring. The roadblock is that "the" coordinate system of an acclerated observer does not cover all of space and time. It's only good locally.

This is covered in various textbooks, unfortunately I've never seen a textbook that covers this well that does not use tensors.

The problem can be illustrated with a simple diagram without the math, though. Basically, if you draw the lines of simultaneity for "the" coordiante system of an accelerated observer, they eventually cross.

Example: accelerated observers follow a "hyperbolic" motion, whose equations are just:

x^2 - t^2 = constant

If you draw the "lines of simultaneity" for such an accelrated motion, all of them cross at the origin of the coordinate system.

(I've got a picture of this somewhere, can't find the post though).

The fact that all of the coordinate lines of simultaneity cross leads to a coordinate system that is good only in the region before the lines cross. After the crossing occurs, one point has multiple coordinates - the origin, for instance, has an infinite number of "time" coordinates. This is very bad behavior, it does not meet the standards that every point must have one and only one set of coordinates.

The fact is closely related to the existence of the "Rindler horizon" for an accelerated observer. Another reason the acclerated observer does not have a coordinate system that covers all of space-time is that he cannot see all of space-time. An observer who accelerates away from Earth at 1 light year/year^2 will never actually see any event on Earth that occurs at a time later than 1 year, unless he stops accelerating, for example. All events more than 1 light year behind the accelerating observer are "behind" his Rindler horizon.

Detailed treatments of this do exist (my favorite is in MTW's "Gravitation")- unfortunately, as I 've said, most of them use tensor notation.

 

[JesseM]

I was only objecting to the use of the phrase "not genuine". And I note that using observation as a basis for reality is objected to by some on this forum.

If you're referring to me, my objection was not to "using observation as a basis for reality", but rather to the notion that the v>c thing constitutes an "observation" in the first place. In an inertial frame, what you "observe" is not what you actually see at the time, but what you reconstruct in retrospect based on the assumption that light always moves at c in your frame. For example, if I look through my telescope in 2006 and see the image of an explosion 10 light-years away as measured in my frame, and then in 2016 I look through my telescope again and see the image of an explosion 20 light-years away in my frame (which is the same one as before since I'm moving inertially), I can do a calculation of speed/distance for each and say that I "observed" these explosions to happen simultaneously in my frame, even though I certainly didn't see them happen simultaneously.

Now, have you thought about what type of calculation an accelerating observer would have to make to say in retrospect that he "observed" the Earth moving faster than c? For one thing, it would involve using a different set of rulers to measure distance at each point during his acceleration. And if you want things to work out so that he always "observes" an event's distance at a given moment to be identical with the distance in his instantaneous inertial rest frame, then he can't just take the time he actually sees the event and divide the distance in his inertial rest frame at the moment he saw it by c like in the inertial case, because his inertial rest frame at the moment he sees the event will be different from the inertial rest frame he was in at earlier moments when the light was on its way. I'd suggest that you take a shot at figuring out just how he could calculate in retrospect when he "observed" different events based on the moments he actually sees the light from them, and then perhaps you will change your mind about whether this highly abstract (and not very well-motivated physically, unlike the inertial case) calculation really deserves the commonsense word "observation".

 

[jtbell]

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c.

Why would we not call this v > c genuine?

I shouldn't have said "not genuine." It has loaded connotations. A better statement is simply that coordinate velocities (and other physical quantities based on coordinate measurements) behave differently in non-inertial coordinate systems than in inertial ones. Restrictions that apply in inertial coordinate systems don't necessarily apply in non-inertial ones.

Consider a different, but somewhat similar example. Suppose you are studying the behavior of a distant star in a coordinate system that has your head as its origin, and its x-axis sticking out straight ahead from your nose. Turn your head. The coordinate system is now rotating. In that coordinate system, the star now has a huge velocity, many times the speed of light, perpendicular to the x-axis.

 

[Al68]

Jesse,

I was not referring to you. And I wasn't referring to a direct observation of a measurement of Earth's velocity relative to the ship, while the ship decelerated. Just the observation by the ship's observer that the Earth's apparent position relative to the ship changed by a few light years in a matter of days, according to the ship's clock. I was only suggesting that the observation of Earth at 5 light years from the star system prior to deceleration, then the observation of Earth at 10 light years from the star system after deceleration should be considered genuine. After all, if we cannot call this length expansion genuine, how could we say the ship ended up 10 light years from earth, while its clock only showed 5.77 years since it left earth?

Thanks,
Alan

 

[JesseM]

Jesse,

I was not referring to you. And I wasn't referring to a direct observation of a measurement of Earth's velocity relative to the ship, while the ship decelerated. Just the observation by the ship's observer that the Earth's apparent position relative to the ship changed by a few light years in a matter of days, according to the ship's clock.

But again, this won't be a straightforward "observation". If he looks through his telescope, he won't see the position of the Earth change by a few light years in a matter of days (or if he does, it'll only be in the sense that he has switched which set of rulers he is using to define distance, the apparent size of the Earth as seen in his telescope won't change significantly). Like I said above, even in SR, "observation" involves a process of calculating dates of events in retrospect, long after they were observed. And the calculation for an accelerated observer needed to insure that what he "observes" matches his instantaneous inertial reference frame at each moment would be very complicated and not too physically well-motivated. Again, I invite you to figure out what he will actually see through his telescope during the acceleration, and then to figure out exactly what sorts of calculations he'd have to do on the dates he saw things to get the dates he "observed" them to work out the way you want them to.

I was only suggesting that the observation of Earth at 5 light years from the star system prior to deceleration, then the observation of Earth at 10 light years from the star system after deceleration should be considered genuine.

Why? Again, please try to figure out what measurements and calculations he'd have to do to get this "observation", and explain why this series of complicated calculations should be considered more "genuine" than some different set of calculations corresponding to a different non-inertial coordinate system.

After all, if we cannot call this length expansion genuine, how could we say the ship ended up 10 light years from earth, while its clock only showed 5.77 years since it left earth?

This comment is only true in the Earth's inertial reference frame, it's not an objective coordinate-independent statement about reality, any more than the statement "the velocity of the Earth is zero". Even in other inertial frames, it is not true that the ship "ended up" 10 light years from earth. And there's certainly no reason to think this would have to be true in a non-inertial coordinate system.

 

[Al68]

Jesse,

I was only referring to the fact that the ship's observer could look at his clock when he arrived at the star system, it would read 5.77 years. He could then use radar to measure his distance to earth, it would be 10 light years. He could calculate how far apart the Earth and star system were in his frame before he decelerated as 0.866c * 5.77 years = 5 light years. This would be approximate, neglecting the distance he would have needed to decelerate.

Please don't read more into this than I intended. I'm not under the impression that the ship could observe Earth's position in real time. And as far as calculating the magnitude of the ship's decceleration relative to Earth's apparent position, and Earth's apparent position relative to the ship as a funtion of ship's time, as observed by the ship, I don't think I'll try it right now. Especially since this was my point. These are the kinds of questions I have. I'm only asking them because they are not covered in most textbooks. And textbooks are not interactive (neither are most professors for that matter).

And I think it is normally considered reality that, theoretically, a ship's crew could end up 10 light years from earth, at rest with earth, after an elapsed proper time of 5.77 years by the ship's clock. Is this not considered objective reality by physicists?

Thanks,
Alan

 

[JesseM]

Jesse,

I was only referring to the fact that the ship's observer could look at his clock when he arrived at the star system, it would read 5.77 years. He could then use radar to measure his distance to earth, it would be 10 light years. He could calculate how far apart the Earth and star system were in his frame before he decelerated as 0.866c * 5.77 years = 5 light years. This would be approximate, neglecting the distance he would have needed to decelerate.

OK, but now you're comparing distances in two different inertial reference frames. So he did not "observe" that the Earth moved from 5 light years to 10 light years, he just figured out that the distance to the Earth is different in two different inertial frames. The fact that each frame happened to be his instantaneous rest frame at that moment is irrelevant, it doesn't imply something like "the Earth moved from 5 light years to 10 light years from his point of view" unless you define his "point of view" as a non-inertial coordinate system which defines distances at each moment to be identical to the distance in his instantaneous inertial rest frame at that moment.

And I think it is normally considered reality that, theoretically, a ship's crew could end up 10 light years from earth, at rest with earth, after an elapsed proper time of 5.77 years by the ship's clock. Is this not considered objective reality by physicists?

My impression is that the only thing a physicist would call a truly "objective reality" is something that is coordinate-independent, like the proper time along a worldline between two events on that worldline, or the answer to whether one event lies in the past light cone of another, or what the readings on two clocks will be at the moment they meet at a single point in spacetime. Any statement that's true in one coordinate system but not true in another can't be considered a simple physical truth, such statements can only be true relative to a particular choice of coordinate system (which means you have to specify which coordinate system you're talking about when making the statement).

 

[Al68]

OK, how about from the (non inertial) frame of the ship. Again, I know the ship cannot observe Earth in real time. When the ship accelerates away from earth, then cuts its engines, the ship's crew and Earth will disagree about what time the engines were cut. And they will disagree about the magnitude of the ship's acceleration. If the ship's crew calculates that their velocity relative to Earth is 0.866c after acceleration, will Earth also agree that the ship's velocity is 0.866c relative to earth? Is velocity invariant between inertial reference frames?

Thanks,
Alan

 

[pervect]

OK, how about from the (non inertial) frame of the ship. Again, I know the ship cannot observe Earth in real time. When the ship accelerates away from earth, then cuts its engines, the ship's crew and Earth will disagree about what time the engines were cut. And they will disagree about the magnitude of the ship's acceleration. If the ship's crew calculates that their velocity relative to Earth is 0.866c after acceleration, will Earth also agree that the ship's velocity is 0.866c relative to earth? Is velocity invariant between inertial reference frames?

Thanks,
Alan

The short answer is yes, everyone in an inertial coordinate system will agree that the velocity is .866 c after the acceleration stops, as long as inertial coordinates are used.

The procedure I would recommend to get a meaningful velocity from the coordinate system of an accelerting spaceship is rather involved, but it will give the same answer as the velocity measured by inertial observers, which is why I recommend it.

Let us say that we have an object, O, and that we want to measure it's velocity relative to the accelerating spaceship, using the coordinate system (viewpoint) of the spaceship.

Basically, one constructs a second space-ship, that maintains a "constant distance" away from the first.

This second space-ship is placed at the location of the object O.

Spaceship #2, at the location of object O, then measures the velocity of object O, using its own local set of clocks and rulers. (These will be different from the clocks and rulers in spaceship #1).

It can be shown that object O will measure the same velocity for spaceship #2 as spaceship #2 measures for object O, as long as measurements are made with standard clocks and rulers.

Technically one constructs a tetrad of "orthonormal basis vectors" at the second space-ship (even more exactly, an orthonormal tetrad of one-forms).

To make this work, one has to have a meaningful concept of what it means for spaceship #2 to be "stationary" (maintaining a constant distance) with respect to spaceship #1. If space-time is flat, or even if it's non-flat and static, this can be defined by the fact that light signals exchanged between spaceship #2 and spaceship #1 will always have the same travel time.

If the space-time is not static, there isn't any way that I'm aware of to define a meaningful concept of a "stationary" distant spaceship, and this procedure for defining relative velocity breaks down.

 

[JesseM]

OK, how about from the (non inertial) frame of the ship.

There is no single standard way to construct a coordinate system for a non-inertial object. If you specify how the coordinate system is constructed, then you can figure out how things will behave in that coordinate system.

Again, I know the ship cannot observe Earth in real time. When the ship accelerates away from earth, then cuts its engines, the ship's crew and Earth will disagree about what time the engines were cut. And they will disagree about the magnitude of the ship's acceleration.

It depends whether you're defining "acceleration" in terms of change in coordinate velocity, or in terms of the acceleration felt by the ship itself (which means acceleration at a given moment is the rate of change of velocity as seen in the ship's instantaneous co-moving frame at that moment). Often in relativity "acceleration" is used to refer to the latter, so "constant acceleration" would mean a constant G-force experienced by passengers, not a constant rate of change of velocity as seen in a single inertial frame. But I assume you were talking about the former type of acceleration, in which case different inertial frames will indeed disagree about the ship's acceleration.

If the ship's crew calculates that their velocity relative to Earth is 0.866c after acceleration, will Earth also agree that the ship's velocity is 0.866c relative to earth? Is velocity invariant between inertial reference frames?

Between the two rest frames of the objects who are in relative motion, yes. Other frames besides the rest frame of the ship and the rest frame of the Earth won't see the distance between the ship and the Earth increasing at 0.866 light-years per year, though.