Acceleration and the twin paradox

[ghwellsjr]

Instead of answering the same questions about twin paradox again and again, I will just put a link to my already written (and published) answer:
http://lanl.arxiv.org/abs/physics/0004024 [Found.Phys.Lett. 13 (2000) 595-601]

Can you please go through the steps to show us how you would get the answer to the OP's first question in his section 2b?

 

[m4r35n357]

If this sounds bitter, it's probably because it is. I really hope that time dilation and other aspects of relativity are understood some day much better than now.

It is pretty well understood by now I think ;) You just need to understand the concept of spacetime interval, which is to spacetime what pythagoras is to space: $$d \tau ^2 = dt^2 - dx^2$$ and computes the elapsed proper time between two events. That one equation is all there is to it, you just need to imagine two different paths through spacetime between the same two events. Just as different paths through space between two points will usually have different lengths, different paths through spacetime between two events will usually have different durations. You don't even need to calculate or measure anything, just keep saying the previous sentence as many times as it takes.

Forget talk of acceleration, seriously; that is just overcomplicating things. You don't need to agonize over the Lorentz transform either, as it's built into the definition of the spacetime interval.

 

[Demystifier]

Can you please go through the steps to show us how you would get the answer to the OP's first question in his section 2b?

I can only sketch the procedure and leave the details as an exercise.

1) Take first the traveler A. Evaluate (6) for t'=t'_A and insert the result in (5) to obtain t=f_A(t'_A) with some explicit function f_A.
2) Now take the traveler B and repeat the same procedure to get t=f_B(t'_B) with another explicit function f_B.
3) To compare 1) and 2) use f_A(t'_A)=f_B(t'_B), which gives you an implicit relation between t'_A and t'_B. Try to rewrite this relation in an explicit form as t'_A=F(t'_B).

 

[WannabeNewton]

Just as different paths through space between two points will usually have different lengths, different paths through spacetime between two events will usually have different durations.

It's seriously as simple and as elegant as that. If one can understand this then I don't see any reason to find the twin paradox conceptually non-trivial.

 

[PhoebeLasa]

Then can you do it for any of the other scenarios listed in this thread where only one twin accelerates?

I can't remember how the calculations are done. I do remember seeing some old posts on this forum where it was explained how to do it, though, and it wasn't very hard. And I remember that the result was that the while the traveling twin is accelerating toward the home twin, the traveler will say that the home twin is rapidly getting older, and that that explains how the traveler can find the home twin older at the end of the trip, even though the traveler says that during most of the trip (when he isn't accelerating), the home twin is aging more slowly than the traveler.

 

[Demystifier]

It is pretty well understood by now I think ;) You just need to understand the concept of spacetime interval, which is to spacetime what pythagoras is to space: $$d \tau ^2 = dt^2 - dx^2$$ and computes the elapsed proper time between two events. That one equation is all there is to it, you just need to imagine two different paths through spacetime between the same two events. Just as different paths through space between two points will usually have different lengths, different paths through spacetime between two events will usually have different durations. You don't even need to calculate or measure anything, just keep saying the previous sentence as many times as it takes.

Forget talk of acceleration, seriously; that is just overcomplicating things. You don't need to agonize over the Lorentz transform either, as it's built into the definition of the spacetime interval.

This, indeed, is the simplest way to understand the twin paradox, provided that you accept the geometrical Minkowski view of special relativity. But the original Einstein formulation of special relativity did not have such a geometrical form, moreover Einstein at first didn't like the Minkowski geometrical formulation (he later changed his mind when he applied such a view to construct general relativity), and most importantly, special relativity in introductory textbooks is usually not taught in such a geometrical form. That's why many people still seek an explanation in terms of Lorentz transformations or something alike. My post #31 above offers such a coordinate-transformation explanation, for those who want it.

 

[jethomas3182]

The Twin Paradox is an application of the simple math of lorentz transforms, and unfortunately it was designed to mystify.

The lorentz transform says that when you assign velocity to another observer, his distances contract and his clock slows. At the same time, when he assigns the velocity to you, he calculates that your distances contract and your clock slows. The situations are strictly symmetric, but that does not mean that you have the same distances and clock speeds. To him, yours are shorter and to you, his are shorter. They are shorter than each other. In reality, of course, we can't tell who has the velocity, we can only assign it arbitrarily. What we can measure is relative velocity.

We don't need to consider acceleration, we can do the trick with magic. He starts out beside you, with no velocity. By magic, create a relative velocity that's close to lightspeed. You calculate that his clock runs slower.

Then after a suitable long time, by magic reverse his velocity so he approaches you again. You still calculate that his clock runs slower. the

t' = gamma (t-vx)

formula gives for him t+vx in place of t-vx on the return trip, which might appear to reverse his clock slowing because of your random choice of x axis, but the gamma part stays the same and makes the sum of leaving and coming back smaller.

So when he arrives at your location, still speeding by at close to lightspeed, you will calculate his time has passed slower while he calculates that your time has passed slower. Then by magic you cancel his velocity, leaving him with a slower clock than you. It follows straight from the math, and the magic.

We can do some of this without the magic. Like in the classic homework problem of the mesons arriving from space at nearly lightspeed. They weren't accelerated to nearly lightspeed, they were born that way. We calculate that their clocks run slow so they decay slow. Somebody traveling with them would decide that our clock runs slow, and the world has become paper-thin so they just punch right through before they decay. Which version is right? They both fit the facts, of course. From our point of view they "really" have their clocks run slow so they decay slow. We leave out the step where we magically slow their velocity to zero while resetting their own clocks to fit our calculation of them. ;-)

If you complicate it with acceleration, you can get reasonable results without magic. But that's a complication on the original idea.

 

[georgir]

That's why many people still seek an explanation in terms of Lorentz transformations or something alike. My post #31 above offers such a coordinate-transformation explanation, for those who want it.

More specifically, I think the key to it is understanding what coordinate transformations do to a "now" (simultaneity plane), and from that how you have two completely different definitions of "now" right before and right after switching reference frames (acceleration).

 

[m4r35n357]

As a learner, I find myself disagreeing more and more strongly about this Lorentz Transform approach to the twin "paradox". IMO SR needs to be taught in an operationally practical way, not as a "like wow, weird" sort of way. We are some way towards ditching the concept of relativistic mass, but we're still stuck with the Mr Tompkins view of moving clocks going slower and length contraction which we surely by now all know is NOT what you would really see. Relying on this more complicated Lorentz Transform analysis puts me in mind of Lorentz Ether theory. There's nothing to stop you doing it, but it's neither necessary or helpful to a novice, it comes across to me as just a historical throwback to the older, harder way of doing things (which of course is interesting in its own right to a more advanced student).

Relativity is confusing enough, let's keep it as simple as we can when talking to beginners. Let's not put artificial difficulties in their path. The key to my understanding of this was plotting events on a spacetime diagram and putting in light lines to observe the doppler effect and see the delay and rate changes that an observer would really see on various clocks in the system. These are real things, seen with the eyes or a telescope, rather than unobservable coordinate artifacts. I am unrepentant ;)

 

[georgir]

Relying on this more complicated Lorentz Transform analysis puts me in mind of Lorentz Ether theory.

But that's part of the beauty of it - you can start to think of it as LET centered on some preferred frame, and then realize that any other inertial frame will do the job equally well because of the symmetry of the transforms, getting to the "relativity" part of it in a natural way.

 

[m4r35n357]

I have no issue whatsoever with that approach for those who are interested, just don't expect poor newbies (like the ones in this thread) to follow that kind of discussion!

 

[dmitrrr]

My post #31 above offers such a coordinate-transformation explanation, for those who want it.

About arXiv: physics/0004024 The paper was published in reputable journal [Found.Phys.Lett. 13 (2000) 595-601]. It has no evident flaws. It has following characteristics: 1) the effect of memory in form of integration. 2) The Special Relativity is based on two postulates, isn't it? How many postulates has this paper? It can be important theoretical construction for wide range of the natural processes.

 

[Nugatory]

About arXiv: physics/0004024 The paper was published in reputable journal [Found.Phys.Lett. 13 (2000) 595-601]. It has no evident flaws. It has following characteristics: 1) the effect of memory in form of integration. 2) The Special Relativity is based on two postulates, isn't it? How many postulates has this paper? It can be important theoretical construction for wide range of the natural processes.

I don't understand what you mean by "the effect of memory in the form of integration". as for you rsecond question, Nikolic has introduced no new postulates, he's further developing the implications of the two basic postulates upon which SR is based.

 

[dmitrrr]

I don't understand what you mean by "the effect of memory in the form of integration".

Thank You and I am sorry for lack of clarity. Under the effect of memory is meant: the coming behavior of a body depends on the previous history of this body. In common worldview it is enough to know position and velocity of a body to know its future evolution. Here it is not the case, am I right? I understand, that it is not easy question for the rapid answering. Author of this "memory" remark is Dmitri Martila.

 

[Demystifier]

I don't understand what you mean by "the effect of memory in the form of integration". as for you rsecond question, Nikolic has introduced no new postulates, he's further developing the implications of the two basic postulates upon which SR is based.

I agree.

 

[Demystifier]

Thank You and I am sorry for lack of clarity. Under the effect of memory is meant: the coming behavior of a body depends on the previous history of this body. In common worldview it is enough to know position and velocity of a body to know its future evolution. Here it is not the case, am I right? I understand, that it is not easy question for the rapid answering. Author of this "memory" remark is Dmitri Martila.

The time integral appearing in the paper is not really a memory effect. It is just a summation of infinitesimal increments, not much different from a non-relativistic formula for a traveled distance as function of time $$x(t)=\int dt\,v(t),$$ where $v(t)$ is a time-dependent velocity.

 

[PhoebeLasa]

More specifically, I think the key to it is understanding what coordinate transformations do to a "now" (simultaneity plane), and from that how you have two completely different definitions of "now" right before and right after switching reference frames (acceleration).

I think you're right. But I'm not sure that the terminology "switching reference frames" is the best way to phrase it (even though that's commonly-used terminology). You can regard the accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip. It's not an inertial reference frame, but it is a reference frame. And that reference frame is such that, when he is accelerating toward the home twin, he will say that the home twin is rapidly getting older. That is the key to understanding the traveler's perspective in the twin paradox.

 

[stevendaryl]

I think you're right. But I'm not sure that the terminology "switching reference frames" is the best way to phrase it (even though that's commonly-used terminology). You can regard the accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip. It's not an inertial reference frame, but it is a reference frame. And that reference frame is such that, when he is accelerating toward the home twin, he will say that the home twin is rapidly getting older. That is the key to understanding the traveler's perspective in the twin paradox.

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).

 

[Dale]

I'm not sure that the terminology "switching reference frames" is the best way to phrase it (even though that's commonly-used terminology)

I truly hate that terminology. It conveys the idea that a reference frame is something which has a physical existence and some limited spatial extent, both of which I think lead to conceptual errors in novice students.

 

[PhoebeLasa]

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.
[...]

Brian Greene, in his book and NOVA series called "The Fabric of the Cosmos", gave an example where the acceleration wasn't constant (and wasn't even one-dimensional, it was small, slow circular motion at an extremely great distance). And the result was that the person who was riding (a bicycle) around in a small circle would say that the local time at a place an extremely great distance away was varying back and forth over several centuries, once for each completed circle.

Somewhere in some old posts on this forum, I've seen a description of how those kinds of calculations are done, but I haven't been able to find them lately.

 

[PAllen]

Brian Greene, in his book and NOVA series called "The Fabric of the Cosmos", gave an example where the acceleration wasn't constant (and wasn't even one-dimensional, it was small, slow circular motion at an extremely great distance). And the result was that the person who was riding (a bicycle) around in a small circle would say that the local time at a place an extremely great distance away was varying back and forth over several centuries, once for each completed circle.

Somewhere in some old posts on this forum, I've seen a description of how those kinds of calculations are done, but I haven't been able to find them lately.

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). This issue is discussed in the Nikolic paper referenced by Demystifier, which argues that non-inertial frames are only locally physically meaningful. This description by Greene comes from trying to give an inconsistent global interpretation to non-inertial frames.

 

[PAllen]

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).

You can generally use one physically motivated chart over any arbitrary non-inertial world line by restricting its scope to a world tube around the world line. The more extreme the mix of accelerations, the narrower the world tube must become to remain a valid chart. Basically, I'm speaking of Fermi-Normal coordinates. If you try to extend them far from the origin world line, you are forced to use multiple, overlapping charts. However, if you restrict them to a narrow world tube, the a single chart is possible. This is equivalent to the approach in the Nikolic paper, I believe.

 

[stevendaryl]

You can generally use one physically motivated chart over any arbitrary non-inertial world line by restricting its scope to a world tube around the world line. The more extreme the mix of accelerations, the narrower the world tube must become to remain a valid chart. Basically, I'm speaking of Fermi-Normal coordinates. If you try to extend them far from the origin world line, you are forced to use multiple, overlapping charts. However, if you restrict them to a narrow world tube, the a single chart is possible. This is equivalent to the approach in the Nikolic paper, I believe.

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.

 

[Nugatory]

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.

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..."

One of the keys to getting people through the twin paradox is getting them to understand that the question is ill-formed except when both twins are at the same place at the same time.

 

[ghwellsjr]

...You can regard the accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip. It's not an inertial reference frame, but it is a reference frame. And that reference frame is such that, when he is accelerating toward the home twin, he will say that the home twin is rapidly getting older. That is the key to understanding the traveler's perspective in the twin paradox.

There are other ways for an "accelerating traveler to have his own single reference frame (in which he is always at the spatial origin) during the whole trip" than the one you are assuming and they can have the home twin getting older at different rates. Don't assume that the non-inertial reference frame that Brian Greene promotes is the only way to do it. In other words, there is more than one key to understanding the traveler's perspective in 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.