Acceleration and the twin paradox

[ghwellsjr]

In the special case of constant acceleration, there is a nice coordinate system, Rindler coordinates, but if the acceleration is nonconstant, then it's pretty hopeless to come up with a coordinate system in which the rocket is always at rest. For instance, if the rocket accelerates for a while, drifts for a while, decelerates, then drifts back to where it started, there is no good way to describe that using a noninertial coordinate system. The way you have to describe situations like that is either to use inertial coordinates, where the rocket is not at rest, or else use charts, which are coordinate systems that only apply to small regions of spacetime. (Then you have to worry about relating coordinates of one chart with coordinates of the other).

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.

 

[PAllen]

Okay, but I think 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? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.

No, it won't. You would need some other coordinate chart. One based on radar simultaneity will provide an answer for any situation where the the origin world line is inertial before some event, and inertial again after some event, no matter what happens in between. However, without this restrictions, radar simultaneity also fails for arbitrary non-inertial world lines.

 

[PAllen]

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.

Unfortunately, this is not true. For an eternally accelerating observer, radar coordinates have exactly the same limited coverage as Rindler coordinates.

 

[ghwellsjr]

Okay, but I think 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? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.

Most people may want an answer to the question of how old the home twin is for any age of the traveling twin but they should be told it's a multiple-choice problem where the last choice is "all of the above".

 

[pervect]

Okay, but I think 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? I'm not sure if the narrow world tube coordinate system that you describe would answer that question.

I'm not sure there is an answer to this question.

Let's talk about a specific case where the issue arises. Suppose we have a classic twin paradox set up, with traveling twin, accelerating at a constant 1g. Said twin asks "what time is it on Earth now" when when he's one light hear away from Earth, at the point where the Earth is at or past the accelerating twin's Rindler horizion, such that light signals emitted from the Earth will no longer be able to catch up with the accelerating twin, assuming he keeps accelerating at the same rate.

The most careful answer is to point out the constraints on the size of an accelerated frame, and discuss how simultaneity is relativeI think, though I suspect in many cases people want and expect an answer to the issue of simultaneity as stevendaryl points out. However, it i s reasonably likely that the people expecting/demanding an answer don't fully accept the fact that simultaneity is relative, part of their expectation of an answer is a carry over from the concepts of absolute time where simultaneity was universal rather than relative.

I don't have a better answer at this point, I think a lot of popularized answers in relativity have come about because it's easier than trying to explain to people that there is no more absolute simultaneity in SR. After a while of trying to explain that simultaneity is relative, I can see why people don't want to go through the hassle, especially when the people asking their questions are asking something that seems unrelated without realizing that the relativity of simultaneity even enters into a complete answer to their seemingly unrelated question.

 

[PeterDonis]

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.

 

[PeterDonis]

I can see why people don't want to go through the hassle, especially when the people asking their questions are asking something that seems unrelated without realizing that the relativity of simultaneity even enters into a complete answer to their seemingly unrelated question.

But part of understanding relativity is understanding how these things that don't seem related, to one's pre-relativistic intutions, actually are related. Again, IMO it's better to get that out on the table up front, to just bluntly say "your pre-relativistic intuitions are wrong and you need to unlearn them to really understand relativity", instead of going in circles trying to explain without really explaining (because the real explanation requires giving up the intuitions that you're going in circles trying to preserve in some form).

 

[pervect]

But part of understanding relativity is understanding how these things that don't seem related, to one's pre-relativistic intutions, actually are related. Again, IMO it's better to get that out on the table up front, to just bluntly say "your pre-relativistic intuitions are wrong and you need to unlearn them to really understand relativity", instead of going in circles trying to explain without really explaining (because the real explanation requires giving up the intuitions that you're going in circles trying to preserve in some form).

On the whole, I agree. Certainly I'm not going to oppose anyone who tries to explain things more fully, though I may not always feel motivated enough personally, especially if the target audience seems unreceptive or appears to lack needed background.

My current thinking is that it is good to point out that textbooks (specifically, MTW) do say that there are constraints on the size of an accelerated frame, and the reason for this restriction is to ensure that every event in space-time has one and only one set of coordinates in the chart - this is basically your position as well, if I'm reading your posts correctly.

Also I think it's worth pointing out that accelerating observers may not be able to exchange light signals (or any other sort of signals) with non-accelerating observers in certain cases, and to give some examples of such cases, such as the rocket accelerating at 1g who is 1 light year away (in the Earth frame). It's also needed for relevance to point out that this lack of ability to exchange any sort of signals does makes "clock synchronization" rather problematic. On the whole I don't think I need to say any more than that, really.

 

[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.

Unfortunately, this is not true. For an eternally accelerating observer, radar coordinates have exactly the same limited coverage as Rindler coordinates.

You're right, I keep forgetting about those eternally accelerating observers. I should have limited my comments to the twin scenario that the OP specified, which is the one that stevendaryl was referring to.

 

[dmitrrr]

Nikolic has introduced no new postulates, he's further developing the implications of the two basic postulates upon which SR is based.

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.

 

[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.