Acceleration and the twin paradox

[PAllen]

Thank You. Excuse me, I am old enough to remember, that the Physics is the same in all inertial reference systems. So I find it hard to believe, that the Physical laws are looking the same way in all systems: sir Newton has not used name "non-inertial system" in his three laws. Be well.

Who said they look the same in inertial versus non-inertial frames? Nobody in this discussion said this.

Newtonian physics certainly covers non-inertial frames (even if that word wasn't used). Centrifugal force, coriolis force, are new 'fictitious' forces that have to be added to the laws true in inertial frames to describe motion in non-inertial frames.

 

[stevendaryl]

And the correct answer to this question is "mu"; as Nugatory pointed out, the question is ill-formed. IMO it's better to just face that up front, rather than trying to salvage people's pre-relativistic intuitions in some form. Understanding why the question is ill-formed is a key part of understanding relativity.

Well, there are two different aspect to the claim that the question has no answer. The first aspect is that it's relative to the observer. For any inertial observers, there is a more-or-less unique, best answer to the question: "How old is twin A when twin B is age X?" But in the case of noninertial observers, "relative to the observer" doesn't even give a unique answer.

 

[stevendaryl]

Radar coordinates work just fine with non-constant accelerations. They also work equally fine with constant accelerations. They also work equally fine with no acceleration (inertial observers). They work equally fine with multiple observers/objects accelerating in any arbitrary manner. They work fine in all circumstances.

I suppose. I don't actually think of radar coordinates as being very meaningful though, because they make the answer to the question: "Relative to twin A, how old is twin B when twin A is age X?" dependent on the future behavior of the twins, right?

You say that event $e$ takes place at time $\tau$, according to twin A, if there is a time $\delta \tau$ such that a light signal sent from Twin A at proper time $\tau - \delta \tau$ will reach event $e$ and a return light signal from $e$ will reach Twin A at proper time $\tau + \delta \tau$. So in a sense, $e$ is halfway between times $\tau - \delta \tau$ and $\tau + \delta \tau$.

But the time at which the return signal from $e$ reaches Twin A depends on how A accelerates after time $\tau$. How old twin B is at time $\tau$ depends on what A does after time $\tau$.

 

[m4r35n357]

It does not, if the other twin is outside the tube. Nor will any other coordinate system, as there's an assumption about simultaneity embedded in the word "when" in any question that starts "When the traveling twin is X years old..."

Quite, and that's precisely my motivation for looking at "moving" clocks and visualization, IMO these should always be considered as part of the solution to the twin "paradox", at the very least via a spacetime diagram showing light paths and the doppler effect. At least then you can then show the novice what X and his clocks look like from Y's POV the whole time, and vice versa. You can talk sensibly and point out the rates of the clocks as well as the readings in a very clear visual manner.
Without tying all this together, beginners tend to (justifiably) come away with a vague unease that you have used some kind of mathematical trickery to make time disappear for one or both of the twins. This is particularly true of the "instantaneous turnaround" description. The acceleration argument attempts to mitigate this, but I've never seen a concise and simple description of the effect from this standpoint; I think acceleration is strictly for experts only!

 

[A.T.]

Don't assume that the non-inertial reference frame that Brian Greene promotes is the only way to do it.

Isn't that true for both twins? The standard simultaneity convention for inertial frames is just a convention as well.

 

[ghwellsjr]

Radar coordinates work just fine with non-constant accelerations. They also work equally fine with constant accelerations. They also work equally fine with no acceleration (inertial observers). They work equally fine with multiple observers/objects accelerating in any arbitrary manner. They work fine in all circumstances.

I suppose. I don't actually think of radar coordinates as being very meaningful though, because they make the answer to the question: "Relative to twin A, how old is twin B when twin A is age X?" dependent on the future behavior of the twins, right?

Right, but not just the future behavior, the past behavior too, just like for inertial observers.

Radar coordinates, just like any other coordinates, derive their meaning from the definitions that are assumed in setting them up. None are any more meaningful than any other. They are all dependent on their definitions, including Einstein's.

You say that event $e$ takes place at time $\tau$, according to twin A, if there is a time $\delta \tau$ such that a light signal sent from Twin A at proper time $\tau - \delta \tau$ will reach event $e$ and a return light signal from $e$ will reach Twin A at proper time $\tau + \delta \tau$. So in a sense, $e$ is halfway between times $\tau - \delta \tau$ and $\tau + \delta \tau$.

Exactly, just like Einstein's convention for inertial observers.

But I think there is a better way of expressing it because twin A doesn't know the value of τ or Δτ while he is going through the exercise. If we set τ₁ = τ-Δτ and τ₂ = τ+Δτ, he doesn't even know the value of τ₁ until he sees the return echo at τ₂ and he sees the time on Twin B's clock. Then he simply averages τ₁ and τ₂ to get τ, the time on his own clock that is simultaneous with the time on Twin B's clock. Keep in mind that this is exactly what inertial observers also have to do when establishing simultaneity according to what you called "a more-or-less unique, best answer" in post #72.

But the time at which the return signal from $e$ reaches Twin A depends on how A accelerates after time $\tau$. How old twin B is at time $\tau$ depends on what A does after time $\tau$.

Yes, before and after, just like for your "more-or-less unique, best answer".

 

[ghwellsjr]

Don't assume that the non-inertial reference frame that Brian Greene promotes is the only way to do it.

Isn't that true for both twins? The standard simultaneity convention for inertial frames is just a convention as well.

Yes.

The standard simultaneity convention is identical to the radar convention, just one of many ways of establishing simultaneity.

 

[dmitrrr]

Newtonian physics certainly covers non-inertial frames... 'fictitious' forces that have to be added to the laws true.

Opponent: sir Newton has included non-inertial systems in his three laws: there are real fictive forces out there! Me: fictive force is the same as fictive marriage: it is not force nor marriage. An observer U on a free body B does not feel overloads and is not such crazy researcher to assign a force to the body B; however opponent O in non-inertial frame in a galaxy far, far away from U subjects a fictive force to this body B.

 

[PAllen]

I suppose. I don't actually think of radar coordinates as being very meaningful though, because they make the answer to the question: "Relative to twin A, how old is twin B when twin A is age X?" dependent on the future behavior of the twins, right?

You say that event $e$ takes place at time $\tau$, according to twin A, if there is a time $\delta \tau$ such that a light signal sent from Twin A at proper time $\tau - \delta \tau$ will reach event $e$ and a return light signal from $e$ will reach Twin A at proper time $\tau + \delta \tau$. So in a sense, $e$ is halfway between times $\tau - \delta \tau$ and $\tau + \delta \tau$.

But the time at which the return signal from $e$ reaches Twin A depends on how A accelerates after time $\tau$. How old twin B is at time $\tau$ depends on what A does after time $\tau$.

The future dependence of radar always seemed to me a feature rather than a 'bug'. You only know about something when it is in your past light cone. Radar coordinates have the feature that anything outside your past light cone can be included only by extrapolation. Since this is true of reality, forcing you to accept this seems good.

 

[A.T.]

The standard simultaneity convention is identical to the radar convention, just one of many ways of establishing simultaneity.

So instead of talking about "relativity of simultaneity", shouldn't we be talking about "arbitrariness of simultaneity".

 

[PAllen]

Isn't that true for both twins? The standard simultaneity convention for inertial frames is just a convention as well.

True, but it's the only one that preserves isotropy and homogeneity of physical laws.

 

[PAllen]

Opponent: sir Newton has included non-inertial systems in his three laws: there are real fictive forces out there! Me: fictive force is the same as fictive marriage: it is not force nor marriage. An observer U on a free body B does not feel overloads and is not such crazy researcher to assign a force to the body B; however opponent O in non-inertial frame in a galaxy far, far away from U subjects a fictive force to this body B.

What exactly is your point? Books on classical mechanics routinely cover non-inertial coordinate systems. Laws of motion simply take a more complex form in such, but this form can be useful for describing the experience of non-inertial system. Special relativity is no different. The Lorentz transform only applies between inertial frames, and physical laws are simplest, with isotropy and homogeneity, in inertial frames. However, non-inertial frames are perfectly possible and useful for the same purposes as in classical mechanics.

 

[A.T.]

True, but it's the only one that preserves isotropy and homogeneity of physical laws.

And I guess there is no such criteria for choosing the most sensible simultaneity convention in non-inertial frames?

 

[PAllen]

And I guess there is no such criteria for choosing the most sensible simultaneity convention in non-inertial frames?

Correct. More completely, you can note the following:

1) For inertial frames, several reasonable approaches for setting up coordinates agree globally (radar; manifesting isotropy and homgeneity; geometric definition). Thus you can say that while there is no absolute simultaneity SR, there is a preferred notion for inertial observers.
2) For non-inertial frames, they all disagree globally (or are not possible at all) but converge locally.
3) Therefore you can say distant simultaneity is completely arbitrary for non-inertial observers (up to the limitation that causally connected events not be labeled simultaneous). HOWEVER, locally you can talk about preferred simultaneity because what different methods converge to locally is Fermi-Normal coordinates.

 

[ghwellsjr]

So instead of talking about "relativity of simultaneity", shouldn't we be talking about "arbitrariness of simultaneity".

"Relativity of simultaneity" is the same as "relativity of time". We already talk about time being relative.

I don't think we need another new term like "arbitrariness of simultaneity". It's the definitions that are arbitrary. Once we make that clear we can meaningfully talk about the fact that simultaneity is relative to the particular frame that we arbitrarily choose.

 

[stevendaryl]

The future dependence of radar always seemed to me a feature rather than a 'bug'. You only know about something when it is in your past light cone. Radar coordinates have the feature that anything outside your past light cone can be included only by extrapolation. Since this is true of reality, forcing you to accept this seems good.

Well, that's an interesting perspective. But really, if you're piecing together what happens when afterwards, then there is no particular reason to use a coordinate system centered on your own rocket. Just pick an inertial coordinate system, and use that.

 

[stevendaryl]

So instead of talking about "relativity of simultaneity", shouldn't we be talking about "arbitrariness of simultaneity".

That's right. In a sense, the way SR is taught is a little weird, because you introduce new concepts which are then discarded. The concept of time being "relative to the observer" is a new concept to students--it's not true in Newtonian physics. But the concept is really only used in introductory relativity courses. When you get to advanced topics such as relativistic physics or General Relativity, time being relative to the observer plays essentially no role. If anything, it's relative to a coordinate system, which doesn't need to have anything to do with an observer.

 

[PeterDonis]

The first aspect is that it's relative to the observer.

But this still tries to preserve the pre-relativistic intuition that simultaneity is something "real". Yes, it's relative to the observer, but for each observer, it's still "real" somehow. That's the intuition I'm saying should be broken and discarded up front. Simultaneity is not real. There is no such thing as "now", period. That's the big roadblock that I see to people really grasping, for example, the twin paradox, and anything one says that doesn't drive home that point (like saying "for an inertial observer, there is a more or less unique answer...") just delays understanding, IMO.

Once a person really groks that there is no such thing as "now", then yes, you can talk about choosing coordinates, and how "simultaneity" comes into the picture once you've chosen coordinates, and how doing that can help to make it easier to calculate answers. But in my experience in plenty of discussions here on PF, any statement along those lines before a person has really discarded the intuition that "now" has a real physical meaning hinders, not helps, understanding, because it holds out the false hope to that person that the intuition might not have to be discarded.

 

[stevendaryl]

But this still tries to preserve the pre-relativistic intuition that simultaneity is something "real". Yes, it's relative to the observer, but for each observer, it's still "real" somehow. That's the intuition I'm saying should be broken and discarded up front. Simultaneity is not real.

I'm agreeing with you. I'm just describing the way that SR is usually taught. You start off with the Newtonian idea that simultaneity is absolute. Then you learn that it's relative to the observer. Then you learn that in general, it's merely a convention, with no physical meaning. The middle concept, that simultaneity is relative to the observer, is something that is introduced only to be discarded later. Now, it may be that this middle concept is important to get students to make the transition from Newtonian spacetime to Einsteinian spacetime, but it's unfortunate to have to work so hard to get across an idea that isn't even used much.

 

[PeterDonis]

The middle concept, that simultaneity is relative to the observer, is something that is introduced only to be discarded later.

I agree, and I see that you made that point in previous posts. Yes, I agree it would be better to get rid of intermediate concepts like this, that aren't Newtonian and aren't fully relativistic either.

 

[dmitrrr]

What exactly is your point? Books on classical mechanics routinely cover non-inertial coordinate systems.

The considerations of non-inertial reference frames are not well studied. Or you pretend, that "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" (supposed to be said by Lord Kelvin)? As simplest example. Inside accelerating rocket holds $d^2x/dt^2=a={\rm const}$. What is the theoretical derivation of this formula? Answer: 1) take Newton's second law in inertial frame, latter is co-moving with the rocket for a given moment $d^2x'/dt^2=0$, 2) make coordinate transformation $x' = x-a\,t^2/2$. Please give theoretical derivation of this coordinate transformation. The linear transformation was derived in Special Relativity from only two postulates. Which postulates would lead to the above non-linear transformation?

 

[stevendaryl]

The considerations of non-inertial reference frames are not well studied. Or you pretend, that "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" (supposed to be said by Lord Kelvin)? As simplest example. Inside accelerating rocket holds $d^2x/dt^2=a={\rm const}$. What is the theoretical derivation of this formula? Answer: 1) take Newton's second law in inertial frame, latter is co-moving with the rocket for a given moment $d^2x'/dt^2=0$, 2) make coordinate transformation $x' = x-a\,t^2/2$. Please give theoretical derivation of this coordinate transformation. The linear transformation was derived in Special Relativity from only two postulates. Which postulates would lead to the above non-linear transformation?

There doesn't need to be any derivation of a coordinate transformation. You can use any coordinates you like. What is special about the Lorentz transformations is that they preserve the following properties:

1. For an object with no external forces acting on it, $\frac{d^2 x^i}{dt^2} = 0$ (an object in freefall has zero coordinate acceleration)
   
2. The path of a pulse of light obeys $\frac{d^2 x^i}{dt^2} = 0$ and $\sum_i (\frac{dx^i}{dt})^2 = c^2$
3. The proper time $\tau$ on a standard clock at rest ($\frac{d x^i}{dt} = 0$) satisfies $\frac{d\tau}{dt} = 1$.
4. Etc.

The inertial coordinate systems are ones for which the laws of physics take a particularly simple form, when written in terms of coordinates. But you can use whatever coordinate system you like to describe physics, provided that you are careful to work out what the laws of physics look like in this new coordinate system.

 

[PAllen]

The considerations of non-inertial reference frames are not well studied. Or you pretend, that "There is nothing new to be discovered in physics now.

I claim exactly that: "non-inertial reference frames are well studied in classical mechanics for over 200 years, and in SR for over 90 years". Coordinates don't change physics, and the study of non-inertial frames is question of appropriate mathematics not physics. If you disagree, well that is not just 'your choice' as far as physicsforums goes, because that means you reject established mainstream physics, which is not permissible in pysicsforums.

 

[PAllen]

1) I am sorry, physicsForums administration. Am I talk too much? But I am not using f-words or else insults, so please be tolerant enough.
2) By having two definite coordinate systems: one inside the rocket and one outside, we have no freedom of choosing the coordinates. Therefore, there is unique coordinate transformation. I argue, if you have system A and system B, then you can derive the transformation f, so A = f(B).

It is nonsense claim coordinate systems are uniquely chose. If you ignore gravity, you can, at most, say there is family of coordinates in which laws of physics take their simplest form - standard inertial coordinates (if you include gravity, there is no such thing as a global inertial coordinates at all). However, nothing requires use of these to get the correct physical prediction from SR. Further, there is no concept of a preferred coordinates for an accelerating rocket. There are many choices for coordinates in which the rocket is the origin, with no reason to prefer one. Even more, there is no good reason for the rocket to use coordinates for which the rocket remains the origin.

 

[PhoebeLasa]

Many physicists have criticized this statement. In fact, it describes an invalid coordinate system (the same distant event has multiple coordinates, while a well formed coordinate chart is one-one with events).

Brian Greene didn't seem to regard his example (of simultaneity at a distance, under acceleration) as being "one choice among many". And he didn't seem to regard the fact, that the accelerating traveler's perspective that he described isn't invertible, to be a problem. Apparently, he doesn't believe that the accelerating traveler's perspective is required to be a chart. Brian's credentials seem impeccable to me. I certainly wouldn't be so quick to dismiss his example.

 

[Dale]

PAllen is correct. This is one reason that pop sci books are NOT considered acceptable references on PF, even when written by someone with impeccable credentials.

Here is a good reference explaining the mathematical requirements of a valid coordinate chart (chapter 2). As PAllen says one of the requirements is that it must be 1-to-1.

http://arxiv.org/abs/gr-qc/9712019

 

[DiracPool]

Thanks to everyone that has responded to this thread so far, it has been quite the learning experience. I feel I've gained some insight on the matter and wanted to check my understanding against the group here.As far as I see it, there are really two issues here, one is the math involved relating to Minkowski space versus ordinary “folk” conceptions of Euclidean space, and the other is a more visuo-spatial conceptual understanding of space, time, light, and moving matter. The good news is that I think I’ve come to terms with the former, but I’m still struggling with the latter.In post #7 of this thread, I put up a video of Susskind discussing the twin paradox at 1:29:00. It was only a couple minute discussion, but right after that he goes on to talk about how the “hypotenuse” of the space-time (ST) graph in the example of the twin paradox is actually shorter than the sum of the sides because of how the Minkowski 4-vector is set up, and this is where the sort of counter-intuitive nature of spacetime bites people. For instance, if we take the ST diagram ghwellsjr posted in another thread,

Here is a spacetime diagram to illustrate what I have been saying. The Earth twin is shown as the thick blue line with dots marking off increments of one year. The thick red line is the rocket twin. The thin lines show yearly signals propagating at the speed of light from each twin to the other one.

we see that the traveling twin A (we’ll call her Alice)’s “path” through this ST diagram visually looks longer than Bob’s. However, the sides of the symmetrical stacked right triangles are subtracting not adding, as in the Euclidean case, and thus the magnitude of Alice’s resultant vector, or proper time/invariant interval, is actually shorter than Bob’s, not vice-versa. That is, for Alice’s journey out to her turnaround point 3 light years away, her time/age is (tau)^2=(5)^2-(3)^2= (tau)=4 years, whereas Bob’s is (tau)^2=(5)^2-(0)^2=(tau)=5 years.

Now I do have a little confusion here in the above example. That is, who’s proper time are we measuring or using here, Bob’s or Alice’s or both? (tau)=proper time is supposed to be invariant, right? But in the above example (tau) is 5 for Bob and 4 for Alice. Should I have re-arranged one of those equations?As m4r35n357 mentioned, everything is contained within the ST interval, particularly the Lorentz transformations, because if you plug in Alice’s speed into the time dilation equation (3/5c, or 0.6c), then you come up with the exact same answer as in the 4-vector example above. So, if you can wrap your head around the 4-vector and how space and time subtract from one another, the twin paradox makes sense to me from this perspective, mathematically at least.

 

[atyy]

we see that the traveling twin A (we’ll call her Alice)’s “path” through this ST diagram visually looks longer than Bob’s. However, the sides of the symmetrical stacked right triangles are subtracting not adding, as in the Euclidean case, and thus the magnitude of Alice’s resultant vector, or proper time/invariant interval, is actually shorter than Bob’s, not vice-versa. That is, for Alice’s journey out to her turnaround point 3 light years away, her time/age is (tau)^2=(5)^2-(3)^2= (tau)=4 years, whereas Bob’s is (tau)^2=(5)^2-(0)^2=(tau)=5 years.

Now I do have a little confusion here in the above example. That is, who’s proper time are we measuring or using here, Bob’s or Alice’s or both? (tau)=proper time is supposed to be invariant, right? But in the above example (tau) is 5 for Bob and 4 for Alice. Should I have re-arranged one of those equations?

We are just comparing the total elapsed proper time for Bob and Alice from when they are separated to when they meet again. Each person has his own elapsed proper time which depends on his spacetime trajectory. The elapsed proper time is what we are colloquially calling their "age".

Simultaneity is relative, so there isn't really an "absolute" proper time elapsed for Bob at Alice's turnaround point.

The turnaround point is absolute for Alice, since there is an acceleration there, which is an absolute event. The moment of separation and the moment of reunion are also absolute events, since they are defined by the intersection of worldlines. The total elapsed proper time for Alice from the moment of separation to the moment of reunion is 2 X 4 years = 8 years, but the total elapsed proper time for Bob is 10 years.

 

[DiracPool]

While mathematically the twin paradox appears to make sense to me, trying to understand the twin paradox from a more intuitive visuo-spatio-temporal perspective is more tricky. I will gratefully borrow another one of ghwellsjr fine ST diagrams in order to set up an example scenario:

Here is another spacetime diagram to illustrate this scenario:

In the scenario I want to set up around this ST diagram, we have Alice and Bob each residing on their own planets in a common rest frame 3 light years apart. For simplicities sake, let’s say that Alice and Bob “see” each other through a constant mutual radio signal of 100 hz emitted from each planet directed at the other. We can say the information sent is amplitude modulated. So now perhaps we can define the spacetime between Bob and Alice as these mutually transmitted signals at 100 hz 3 light years apart. Using the above ST diagram, then, we can almost look at the ST between Alice and Bob as sort of a “box” of ST enclosing their mutual rest frame. If Alice and Bob stay on their planets, then each will see and continue to see the other as they were 3 years in the past.

Now, say, all of a sudden Alice gets lonely for Bob, jumps on a spaceship, and starts heading for Bob’s planet. What happens here? Well, we can see clearly that a symmetry in this situation indeed has been broken. That is, Alice will immediately begin to see signals sent from Bob that weren’t supposed to reach her until much later. So, in effect, she will begin to see Bob aging at an accelerated rate. Furthermore, the signal she receives from Bob will be blue shifted from 100 hz to 100+ hz. Bob, on the other hand, experiences none of these differences. So, seeing this was interesting to me because it demonstrates how an asymmetry can exist between Alice and Bob that can identify which one has broken the symmetry. Again, as m4r35n357 mentioned in post #37, it doesn’t look as if acceleration per se makes much difference here, other than the initial breaking of the mutual rest frame, if she travels to Bob’s planet at 0.6c, she will arrive there in 4 years, while in the same interval, Bob will have aged 5 years. It’s all in the velocity and the spacetime interval/LT.

What I am having trouble seeing from this scenario, however, is how this happens from the perspective of the mutually sent radio signals. That is, it seems as though, granted, Alice will see Bob getting older at an accelerated rate while Bob will not see the reciprocal for a full 3 years, but it seems that, as Alice approaches Bob’s planet, the signals she has been sending will simply “stack up” on Bob during her final approach. Thus, as Alice is coming in the final stretch, Bob will see the entirety of Alice’s aging over her trip at an extremely accelerated rate. I guess I don’t see how, at the end of Alice’s trip, the balance of the sent radio signals (say there was one “marker” signal sent each year as in ghwellsjr example) don’t add up at the end. Again, in ghwellsjr example, at the point at coordinate time 0 whereby Alice begins her trip toward Bob, I see that Alice crosses 8 blue lines before she arrives on Bob’s planet, while in the same time frame, Bob only crosses 7 red lines coming from Alice. Am I reading that correctly and is that significant quantitatively? And if so, how?

In any case, again, I’m just trying to piece this all together as best I can and see where I may be on the right track and where I’m off. Thanks again for everyone’s comments.

 

[ghwellsjr]

...
Now, say, all of a sudden Alice gets lonely for Bob, jumps on a spaceship, and starts heading for Bob’s planet. What happens here? Well, we can see clearly that a symmetry in this situation indeed has been broken. That is, Alice will immediately begin to see signals sent from Bob that weren’t supposed to reach her until much later. So, in effect, she will begin to see Bob aging at an accelerated rate. Furthermore, the signal she receives from Bob will be blue shifted from 100 hz to 100+ hz.

It would be exactly 200 Hz. You can see this in the diagram as the red dots mark off 1-year increments for Alice and the thin blue mark off the Bob's 1-year increments as seen by Alice. There are twice as many thin blue lines as red dots during the time that Alice is traveling.

...
What I am having trouble seeing from this scenario, however, is how this happens from the perspective of the mutually sent radio signals.

Radio signals, like light signals, don't have a perspective, so I'm not sure what you mean by this.

That is, it seems as though, granted, Alice will see Bob getting older at an accelerated rate while Bob will not see the reciprocal for a full 3 years, but it seems that, as Alice approaches Bob’s planet, the signals she has been sending will simply “stack up” on Bob during her final approach. Thus, as Alice is coming in the final stretch, Bob will see the entirety of Alice’s aging over her trip at an extremely accelerated rate.

Both these accelerated rates are exactly double, that is, Alice sees Bob's clock going twice the rate of hers and Bob sees Alice's clock going twice the rate of his.

I guess I don’t see how, at the end of Alice’s trip, the balance of the sent radio signals (say there was one “marker” signal sent each year as in ghwellsjr example) don’t add up at the end. Again, in ghwellsjr example, at the point at coordinate time 0 whereby Alice begins her trip toward Bob, I see that Alice crosses 8 blue lines before she arrives on Bob’s planet, while in the same time frame, Bob only crosses 7 red lines coming from Alice. Am I reading that correctly and is that significant quantitatively? And if so, how?

Yes, you are reading that correctly and it is significant quantitatively. It shows how when Alice traverses the distance between the two planets, she "loses" one year because of her speed. If she had gone slower, she would lose less time and if she had gone faster she would lose more time. But the thin lines show how Bob observes her gaining time so that he sees her originally as being 3 hours behind him to being only 1 hour behind him. Alice, on the other hand, sees Bob originally as being 3 hours behind her to being 1 hour ahead of her.

In any case, again, I’m just trying to piece this all together as best I can and see where I may be on the right track and where I’m off. Thanks again for everyone’s comments.

It appears that you are pretty much on the right track and maybe just don't realize it.

 

[m4r35n357]

we see that the traveling twin A (we’ll call her Alice)’s “path” through this ST diagram visually looks longer than Bob’s. However, the sides of the symmetrical stacked right triangles are subtracting not adding, as in the Euclidean case, and thus the magnitude of Alice’s resultant vector, or proper time/invariant interval, is actually shorter than Bob’s, not vice-versa. That is, for Alice’s journey out to her turnaround point 3 light years away, her time/age is (tau)^2=(5)^2-(3)^2= (tau)=4 years, whereas Bob’s is (tau)^2=(5)^2-(0)^2=(tau)=5 years.

Now I do have a little confusion here in the above example. That is, who’s proper time are we measuring or using here, Bob’s or Alice’s or both? (tau)=proper time is supposed to be invariant, right? But in the above example (tau) is 5 for Bob and 4 for Alice. Should I have re-arranged one of those equations?

Alice's dots are spaced further apart, too. But you have the right idea about pythagoras and subtracting, that is the essence of the relationship between time and space (the spacetime interval, to labour the point), notice we have a "3-5-4" triangle, not by accident! . Everyone has their own proper time, and you have worked out both Alice's and Bob's in this instance. I think you have it, just don't panic ;)

 

[PeterDonis]

it seems that, as Alice approaches Bob’s planet, the signals she has been sending will simply “stack up” on Bob during her final approach. Thus, as Alice is coming in the final stretch, Bob will see the entirety of Alice’s aging over her trip at an extremely accelerated rate. I guess I don’t see how, at the end of Alice’s trip, the balance of the sent radio signals (say there was one “marker” signal sent each year as in ghwellsjr example) don’t add up at the end.

Alice sees Bob's signals blueshifted for her entire journey--that is, she sees Bob's clock running faster than hers for the entire 4 years of her trip. So Bob's clock starts out 3 years behind hers (that is, the signal she receives from Bob when she starts her journey shows his clock 3 years behind her clock's reading when she starts); Bob's clock gains 4 years on Alice's during the journey (she sees 8 years' worth of Bob's signals in 4 years of her own time); so Bob's clock ends up reading 1 year ahead of Alice's when she reaches him.

Bob, however, only sees Alice's signals blueshifted for 2 years (5 years total time elapsed on Bob's clock, minus the 3 years of light travel time that it takes before Bob receives the signal Alice emits when she starts her journey). So even though he sees all of Alice's signals during the journey "stacked up" during that 2 years, it still isn't enough for her clock to "catch up" to his. He sees her clock reading 3 years behind his when Alice's "start of journey" signal reaches him; Alice's clock gains 2 years on his during the journey (he sees 4 years' worth of Alice's signals in 2 years of his own time); so when Alice reaches him, her clock is 1 year behind his.

 

[PhoebeLasa]

[...]
Here is a good reference explaining the mathematical requirements of a valid coordinate chart (chapter 2). As PAllen says one of the requirements is that it must be 1-to-1.

http://arxiv.org/abs/gr-qc/9712019

I'm not saying that Brian Greene apparently thinks that charts aren't required to be invertible.

I'm saying that Brian Greene apparently thinks that the accelerating traveler's perspective isn't required to be a chart.

 

[Dale]

I'm saying that Brian Greene apparently thinks that the accelerating traveler's perspective isn't required to be a chart.

Simultaneity requires a chart. Simultaneity means that two events share the same time coordinate. A time coordinate is part of a chart. So simultaneity necessarily implies a chart.

A "perspective" is not well-defined. But insofar as a "perspective" says anything about simultaneity then it must involve a chart.

 

[CKH]

what people want most from a "coordinate system of the traveling twin" is to be able to say: When the traveling twin is X years old, how old is the stay-at-home twin?

And the correct answer to this question is "mu"; as Nugatory pointed out, the question is ill-formed. IMO it's better to just face that up front, rather than trying to salvage people's pre-relativistic intuitions in some form. Understanding why the question is ill-formed is a key part of understanding relativity.

I think this is in fact what confused folks (like myself) what to know in order to remove the paradox. On the outbound leg, in the travelers frame, the stay at home clock runs slower, after the turnaround point in the travelers new frame, the stay at home clock still runs slower. These are both true statements from the Lorentz transform. Just as true as the calculation made above by the stay at home twin using the Lorentz transforms to predict that the returning twin will be younger.

So the paradox is how can we explain the fact that the travelers clock is behind the stay at home clock at the end of the trip from the travelers viewpoint, while on both legs of the journey, to the traveler, the stay at home clock ran slower. We can say what happens in the traveling twin's frame (which is not inertial at the turning point, but is on both legs). I believe the following is an answer.

At the turn around point, an abrupt change in the traveler's time coordinates occurs because the inertial frame of the traveler changes. When the traveler changes inertial frames in turning back, what was the current time on the stay at home clock in the outgoing frame changes abruptly to a later time in the new frame. The size of this jump depends on how far the traveler is from the stay at home twin and the travel velocity. This jump in the stay at home clock more than compensates for the slower running of the stay at home clock on the outbound and inbound legs of the journey as observed in the traveler's frame.

This is just a closer look at how the simultaneity relationship for the traveler changes. The change in time frames accounts for the apparently "missing" aging of the stay at home twin according to the traveling twin.

So if you want to ask in the travelers frame, how old is the stay at home twin, the answer is that the stay at home twin ages slower on the outgoing leg. At the turn around point, the stay at home twin suddenly ages by a large amount, on the return trip, the stay at home twin ages more slowly than the traveler.

If you make the turn around less abrupt, then for the traveler the stay at home twin will age very quickly during turn around but not so abruptly.

The situation is asymmetrical because the stay at home twin remains in the same inertial frame during the whole trip, while the travelers frame changes.

I don't know if that helps anyone else, but it helps me account for the "missing" aging of the stay at home twin. What this scenario looks like to the traveler when watching the stay at home clock with a telescope is different because the time delay of the observation is changing with distance.