Thread: Twin's Paradox revisited.

I am not really sure if this is ATM or not. If not, please close the thread.Anyway,there are various interpretations of the twin's paradox and various so called explanations for the twin's paradox.I propose the following interpretation.O and O' are in the same inertial frame.O applies a constant acceleration and so does O' in opposite linear directions but for different time periods such that their relative motion is v.O apples the acceleration for the longer period.They agree to proceed with this relative v for a long period of agreed upon time.Then O applies a constant acceleration in the opposite direction of the prior acceleration for the shorter time period. O' applies a constant acceleration in the opposite direction of the prior acceleration for the longer time period such that they are forced back to the original inertial frame but at a much greater distance.They can then apply Einstein's clock synchronization procedure since they are now in the same inertial frame.What will the clocks reveal?Of course, SR claims they will both be dilated by the same value.

The "paradox" is entirely due to treating accelerating frames of reference in terms of "special relativity". Just draw a space-time diagram of the "paradox" and you can see that the accelerated twin ages lees and the acceleration is what causes it.

Originally Posted by bgbirdsey

Just draw a space-time diagram of the "paradox" and you can see that the accelerated twin ages lees and the acceleration is what causes it.

My idea of a space-time diagram is just a horizontal axis representing space, and a vertical axis representing time. The actual "drawing" of the diagram doesn't make it obvious that the accelerated twin ages less, but the "metric" that shows the accelerated twin ages less. So the gist of what your saying is correct.


dt = (1/c) sqrt((c dt)^2-dx^2)


A straight line in a space-time diagram represents unaccelerated motion.

If you've got a straight line between events (t1, x1) and (t2, x2) , then dt=(t2-t1) and dx=(x2-x1).

But if you're taking a circuitous path from (t1, x1) to (t2, x2) to (t3, x3) to (t4, x4) etc. then you have to add up the dt's for each part of the trip.

You'll find, counterintuitively, the longest path in time is the straight line.

But you are missing the point of the paradox by bringing up the "metric" (i.e. actually just a Lorentz transform, as the space-time is flat).

There are two "paradoxes" here:
(1) The 1st is the difference between the "every-day" Galilean and also prediction of no age difference, and the correct Relativistic result of an age difference.
(2) The inability to tell which twin is "moving" and which is "stationary" using a single set of inertial frames of reference.

With paradox 1, you just need to get over it, since there are experiments which prove that it happens.

With paradox 2, you need to do something like constructing the space-time diagram to see the difference between the two twins. In the typical interpretation of the diagram, the tangent line to a curve is related to gamma, and so to the time dilation. In reality, you could try to transform the diagram so that the tangential inertial frames of the accelerating twin look like vertical lines on the diagram, but there is really no self-consistent way of doing it since the GR metric actually comes into effect.

Originally Posted by bgbirdsey

But you are missing the point of the paradox by bringing up the "metric" (i.e. actually just a Lorentz transform, as the space-time is flat).

Don't confuse the "metric" with the Lorentz Transformation. The metric I gave above just tells how much one body ages as it follows a curved path. On the other hand, the Lorentz Transformation is a way of taking a map of all of the events in the universe, and then transforming the coordinates of those events from one inertial reference frame to another.

Originally Posted by bgbirdsey

In reality, you could try to transform the diagram so that the tangential inertial frames of the accelerating twin look like vertical lines on the diagram, but there is really no self-consistent way of doing it since the GR metric actually comes into effect.

Really, The GR metric is kind of a red herring for explaining this paradox. It gives a mathematical calculation for one quantity, time--aging of the accelerating twin, instead of a real physical explanation for a real physical question.

The question is "What does the accelerating twin see while he's accelerating."

Taking the curved path of the twin and making it vertical...? You say "there is no self-consistent way." I say that's just a really weird way of going about it. It's like saying "draw everything within a hundred meters of me as I walk around the city." Of course you can't put this on one diagram, because the view changes as you walk.

The "metric" explains nothing of how this view changes. The metric is inward-looking, only telling me what my watch says during my journey, or how far my feet move. Knowing what the twin's WATCH says over the course of his journey doesn't explain away the so called paradox. We want to know what he sees over the course of his journey. The best tool for this problem is the Lorentz Transformation, since it operates on all the events in space-time.

You just can't do this on a single static diagram. Instead, you need some way of animating the diagram, to perform the transformations on the fly, as the observer accelerates. Then you can get an idea of what the accelerating twin actually "sees."

In fact I've made some attempt to this effect some time ago. If it's alright to link here, it is at

Special Relativity Space-Time Applet

Originally Posted by bgbirdsey

Transforming to the rest frame of an observer is a very common practice (not an exotic one), and illustrating it on a spacetime diagram is completely normal. The minor complication of the fact that one observer is accelerating is immaterial to constructing a diagram that is "at rest" with respect to the accelerating observer.

I wanted to clarify this point... I cannot understand "at rest" with respect to an accelerating observer. Do you mean at just one point in time or over a period of time? Of course at any given time, any observer has exactly one velocity, and so it makes sense to draw the spacetime diagram for that one particular velocity. But I cannot imagine how to draw a spacetime diagram for an accelerating observer over a period of time, where his path is made straight. Do the other events in the diagram then follow a curve? How would you represent a path of contiguous events, when every event would be transformed into a curve? I think the whole thing would be a complete blur, and I've certainly never seen it done.

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The "Through Einstein's Eyes" project contained a quiz question that I found interesting.

Q: Accelerating towards an object can make it look like it is receding from us. Why?

A: Increasing aberration makes the objects appear to shrink

Maybe this is the best answer on the multiple-choice test, but if you'll humor me, I'll put it in my own words.

The actual fact is that the light you are seeing now is coming from events that happened at a certain point in space and time. When you accelerate toward those events, a Lorentz Transformation on those events will reveal that they ACTUALLY move farther away from the origin. It's not just an "apparent" shrinkage of the objects. The events that you are seeing immediately after the acceleration actually happened longer ago and further away, than the events you saw immediately before the acceleration.

This is also a key point behind explaining the twin paradox. When the accelerating twin goes out, he sees the length contracted distance behind him, and knows the "actual" position of the earth is further away than the image. When he turns around and comes back, the distance to the image increases, and he knows the "actual" position of the earth is closer than the image.

From that point on the "apparent" image is coming in superluminally.

For visualizing, the two twins see different things. The stay-at-home twin (A) sees (B) go out slowly and come back fast, but the outbound distance and inbound distances are equal. For (B) on the other hand, he sees (A) recede slowly, and come back fast. The outbound and inbound distances are the same. But the distance of the inbound trip is far far greater.

But the implications of this are bigger than that. Each time (B) accelerates, he's not just doing a Lorentz Transformation on the events associated with Earth. He's doing the transformation on every event in the universe, including the Big Bang. The accelerating twin ages slower--meaning that to him, the universe ages faster. Meaning, assuming that the universe was HOT at the beginning, and every particle underwent heavy acceleration, meaning the age of the universe should be much greater than the age of our galaxy, because our galaxy would be the "accelerating twin" of the twin paradox.

Meaning, there's actually a pretty simple explanation for cosmological "inflation."

But these ideas are probably not so obvious if you accept "Increasing aberration makes the objects appear to shrink" as a better explanation.

Originally Posted by bgbirdsey

The minor complication of the fact that one observer is accelerating is immaterial to constructing a diagram that is "at rest" with respect to the accelerating observer.

Let me be slightly more exact:

The minor complication of the fact that one observer is accelerating is immaterial to constructing a diagram that is instantaneously "at rest" with respect to the accelerating observer.

In fact you do understand "'at rest' with respect to an accelerating observer" since you experience it every day you live on the earth. The fact that you are actually slowly accelerating due to the rotation of the earth, the earth's orbit around the sun, etc. gives rise to several fictitious forces like the Coriolis effect which causes the rotation of high an low pressure weather systems, etc.

If you mean that you do not understand WHY one would want to do such a thing... you may be missing the point of the paradox. The typical paradox revolves around positing two inertial rest frames, and applying the rule that "moving clocks run slower". We cannot determine which twin is "moving" and which one is "stationary" from the fact that the two rest frames are moving relative to each other, so it is just as valid to say that either twin experiences time dilation.... which causes a root problem.

The only way out of this conundrum is to look at the effect of the acceleration (i.e. apply a series of infinitesimal boosts to the accelerating twin), and look at the accumulated Lorentz transform between the two frames. Then you do find that (in fact!) the accelerating twin experiences the distance that he is traveling shorten and the other twin growing old at an accelerated rate, and that (in fact!) the stationary twin sees the other twin shrink in his direction of motion and age more slowly. The two effects that you pointed out.

A better example of the dual explanations of the same event is how muons created in cosmic ray showers int the upper atmosphere can possibly make it to the surface of the earth. From the point of view of an observer on the earth, their moving clock is slow so their lifetime is extended by gamma, and from the rest frame of the muon the distance appears shorter so it can travel to the surface of the earth within its lifetime. HOWEVER we could look at this in a completely different way if we just ASSUMED that the muon was at rest and the earth was accelerated to relativistic speeds.... The effect is exactly the same, the muons make it to the surface of the earth, but the explanations are opposite. (Again, this is the root of the twin "paradox" and the reason that it is important to follow the acceleration of the "moving" twin._

Like all scientific "paradoxes" it is the limited or mis-application of the science that causes the apparent paradox.

The reason that one might attempt to construct a spacetime diagram that is instantaneously at rest with respect to to the accelerating twin is to try to reconstruct the apparent "paradox" (i.e. each twin sees the other moving, so both clocks must be slowed).

However, as I pointed out, one would determine that this is not really possible, and forget about interpretation where the twin left behind on the earth is actually the one with the slowed clock. The real reason is that the transformation of the earth-bound twin in the diagram may violate relativistic invariance, making the transformation not self-consistent.

As a simple example, think of relativistically transforming to a non-inertial frrame that is fixed to the surface of the earth. At some distance, say a star in a distant galaxy, would appear to move faster than the speed of light.... NOT consistent with special relativity at all.

But it IS possible to create space-time diagrams instantaneously at rest with the moving twin. I have two good examples.

(1) File:Lorentz transform of world line.gif - Wikipedia, the free encyclopedia

(which keeps track of events and follows a pre-defined path)

(2) Special Relativity Space-Time Applet

(which keeps track of events, paths and, objects, and allows the user to control the observer's acceleration.)

At some distance, say a star in a distant galaxy, would appear to move faster than the speed of light.... NOT consistent with special relativity at all

I agree that a star could appear to move faster than the speed of light, but I disagree that this is inconsistent with special relativity.

If a star appears to move faster than the speed of light because you are accelerating toward it, the star is not actually "moving" faster than the speed of light; you are simply changing to an inertial reference frame where the star is further away.