Does the Twin Paradox Break Symmetry in the Zig-Zag Scenario?

[ash64449]

I'm not sure exactly what "one's point of view" means

Well, for example each one sees other one's clock to run slower.( "We" can see that happen when we analyse same thing in a different frame,but that is not object's point of view)

his calculation leaves out a portion of the HT's worldline should count as pointing out his error "from his point of view".

yes. agreed.

EDIT: I don't understand how that attached file came from. I had problems while quoting your messages.

 

[ash64449]

If you're willing to accept that the TT's "point of view" can switch simultaneity conventions instantly, sure

Well, that's the most important part of my method of solution. His simultaneity convention changes at the turn around But definitely not the method your talking about i used.(the non-inertial chart you are talking about)

You also have to ignore the portions of the TT's trip where he is accelerating, not inertial--for any real scenario,

Yes. in fact i have ignored that. we can introduce situation in twin paradox in which acceleration is ignored but rest remain the same.

the only way to describe the TT as being at rest is to use non-inertial coordinates.

OK. It is a method. It is the only way if we there is a condition that i have to stick to the same co-ordinates the whole time.

But there is another method to make TT at rest. This involves switching between two inertial frames that helps TT to be at rest.

Note: the only reason why switching frame is needed is to describe one's point of view. But if that's not the case, Then there are chances you get error because we must stick to the same frame of reference for any calculation,except those calculation which describe one's point of view(for example, we can take a frame in which Earth and planet are moving towards the left and TT at rest initially and solve twin paradox but it does not solve the problem addressed by me " what is the error in the analysis of TT's point of view"?)

 

[PAllen]

Emboldened by Peter's essay on frames versus coordinates, I thought of producing a little essay on the relationship between my post #31 and Peter's #35 (along the way making terminology consistent between these posts). However, I think the a more important concept to address is the pervasive (and false) idea that it is reasonable for a non-inertial observer to assume distant positions and times correspond to those of an inertial observer momentarily co-located and stationary with respect to the non-inertial observer. This gets at some of the same issues as my prior plan, but with a more physical slant.

The first thing to ask is how does an inertial observer get from a local frame as described in #35 to a global, natural, coordinate system (which, in SR, is often called - sloppily - a global inertial frame, rather than an inertial coordinate system)? There are actually several ways, which all happen to produce the same result for the inertial observer. Simultaneity can be established with Einstein's simultaneity procedure. Distances can either be measured by two way light speed or rulers matched to a standard ruler, or by parallax. For inertial observers, it doesn't matter what (reasonable) procedure you choose, the result will be the same. One way to conceptualize the resulting coordinates is a grid of clocks connected by stiff springs that are never under tension or compression (in a moment it will be clear why specify this rather than rulers). The clocks, once synchronized, keep mutually consistent time (comparing them any time later shows they are in synch, accounting for light delays). Obviously, you don't actually construct such a thing, but you certainly would think reasonable coordinates would be equivalent to such a thing. A fundamental point is that the equivalence of different procedures produces a unique natural choice.

Then, you should ask, what are reasonable (global) coordinates for a non-inertial observer? The surprising answer (expanded on below), is that there aren't any! A little more precisely, there are many possible choices, but none of them satisfy the expectations you might have from the inertial case. Further, each fails in different ways, so there is none to prefer. As a result, there is simply no fixed meaning to be attached to a statement like "how far away is that distant object" for a non-inertial observe. The best you can do is say: "choosing to use procedure x (rather than y), the distance is blah". Even more surprisingly, while available procedures include radar ranging and parallax, one that is not available for all non-inertial observers is rulers!

Ok, so let's try by build physically motivated coordinates for a non-inertial observer. One might assume we can posit a grid of clocks connected by springs such that the springs don't stretch or compress. We allow that each clock is independently propelled as needed, to achieve this. We don't even worry, for now, about synchronizing the clocks. The question is, can we set up such a grid for a rocket that starts at rest and accelerates [edit: and turns], then coasts. The surprising answer is no, this is impossible, in principle in SR! The relevant theorem is Herglotz-Noether. Thus, the most basic concept of what a coordinate grid means is unachievable for a [edit: some ] non-inertial observer. There are different ways of understanding this, but one way is that trying to apply the constraint that the springs don't stretch or compress requires that some clocks move super-luminally (or worse, backwards in time). This same theorem implies that the concept of a long rigid ruler for a [edit: general] non-inertial observer does not exist, even conceptually.

On recognizing that it is impossible to build rigid grid coordinates for a non-inertial observer, what can you do? The two most common choices are radar coodinates or Fermi-Normal coordinates (developed for GR but useful for non-inertial observers in SR). At laboratory scales (or within a rocket), they are identical to plausible experimental precision. Radar coordinates are operationally defined and can be extended globally as long as the defining observer's world line is inertial before some event, and inertial after some event. The beginning and end inertial motions need not be the same. Fermi-Normal coordinates try to glue together momentarily co-moving inertial coordinates, to the extent that it is possible. As implied by Herglotz-Noether, this has a few unexpected consequences:

1) A world line of constant position, varying coordinate time, will not generally represent a timelike world line (this is a consequence of the impossibility of a rigid ruler per Herglotz-Noether).
2) To produce a valid coordinate chart (e.g. to avoid multiply labeling events), the coverage of the coordinates may be very limited. For a zigzag trajectory, they will be limited to a world tube around the origin.

 

[ash64449]

why not just construct a non-inertial coordinate chart in which the TT is at rest the whole time?

How to construct such non-inertial coordinate chart? is it the same as space-time diagram analysis?

 

[PeterDonis]

each one sees other one's clock to run slower.

It depends on what you mean by "see".

If you mean what each one actually sees, literally, with their eyes or a telescope (or a measuring device that detects Doppler shifts), then they will each see the other's clock running slower if they are moving away from each other, but they will each see the other's clock running faster if they are moving towards each other.

If you mean what each one calculates after correcting the above observations for light travel time, then the result depends on what kind of coordinates they are using. If each one is using inertial coordinates in which he is at rest, then yes, each one will "see" (calculate) the other's clock to be running slower. But if they decide to use some other coordinates, that might not be the case. There is no absolute fact of the matter about relative clock rates for spatially separated observers; the only absolute facts of the matter are about the elapsed times each one experiences on their own clocks when following particular paths through spacetime.

that's the most important part of my method of solution. His simultaneity convention changes at the turn around

But this "most important part" of your method is also the part that makes the TT's calculation of the HT's elapsed time wrong. The change in simultaneity convention at the turnaround is what causes the TT to leave out a portion of the HT's worldline in his calculations.

 

[ash64449]

But this "most important part" of your method is also the part that makes the TT's calculation of the HT's elapsed time wrong. The change in simultaneity convention at the turnaround is what causes the TT to leave out a portion of the HT's worldline in his calculations.

This is an image in the Wikipedia article "twin paradox".

You see that there is a portion left out(in between the red lines and blue lines), I think you are talking about this.

I think i will describe my solution to Twin paradox and that will help to find out what you and I was actually trying to say.

Now error in the analysis of TT's point of view in the calculation of HT's age is this- He ignored Relativity Of simultaneity that leads to non-synchronization of the clocks of Earth and the planet.

Now When TT started the journey, Both the TT's clock and HT's clocks read zero. But when TT left, From his point of view(use an inertial frame where both Earth and planet are moving to describe TT's point of view), Clocks of Earth and the planet are non-synchronous. This time Clock of the planet is at the back side of motion, Therefore planet clock is ahead than Earth's clock by 6.4 years(by calculation), Now when TT started the journey, Earth's clock was reading zero. Therefore Planet's clock should read 6.4 years.

Now HT's clock is running slower. TT takes 6 years to reach planet. By this time both the clocks( Earth clock and planet clock) will go ahead by 3.6 years.(6*0.6=3.6)

So Earth's clock will be reading 3.6 years and planet's clock will be reading 10 years when TT reaches planet.

Now for the return part of the journey, it is the Earth's clock that is at the rear and the planet's clock is at the back and hence their reading should differ by 6.4 years with Earth clock's time ahead. But during the start of the return journey,planet's clock is reading 10 years, therefore Earth's clock is reading 16.4 years( see the change in simultaneity convention, To describe TT's point of view we had to switch frames,(why? when describing his point of view,TT should be rest) which changes simultaneity convention)

Now TT takes another 6 years by which both clocks-earth's and the planet's clock go ahead by 3.6 years.

So by the time TT reaches earth, Earth's clock is reading 20 years of elapsed proper time and TT's clock 12 years of elapsed proper time.

Change is simultaneity convention at the turn around is key here.

 

[ash64449]

I see,,The portion in between 3.6 and 16.4 years is left out at the turn around from TT's point of view.
But that should not be worried. Why?

Because at the end from both's point of view calculation show that when they meet, HT shows 20 years and TT shows 12 years.

Since clocks show proper time, then both agree that HT has 20 years of elapsed proper time and TT 12 years of elapsed proper time and hence TT ages less than HT.

But really my method of analysis is about point of view analysis because original question asked for it- Point of view's are something which do not have absolute significance or no existence.

It is the space-time interval that has absolute significance since it doesn't change when we change frames. So that method(Peter's analysis) of analysis is better.

 

[PeterDonis]

The portion in between 3.6 and 16.4 years is left out at the turn around from TT's point of view.

Because at the end from both's point of view calculation show that when they meet, HT shows 20 years and TT shows 12 years.

These two statements do not seem consistent. If the first statement is true, then the calculation from the TT's point of view should be off by 16.4 - 3.6 = 12.8 years.

 

[PeterDonis]

Point of view's are something which do not have absolute significance or no existence.

Not only that, there is no unique definition of what an observer's "point of view" is; there are multiple ways of constructing one. Your analysis uses one particular way: adopt the simultaneity convention of the inertial frame in which the observer is at rest, and when the observer changes his state of motion, change simultaneity conventions instantly. But that is not the only possible way of doing it. As I said before, there is no absolute fact of the matter about relative clock rates for spatially separated observers; there is also no absolute fact of the matter about which spacelike separated events are simultaneous. It's a matter of convention, and there are multiple possible conventions.

 

[ash64449]

These two statements do not seem consistent. If the first statement is true, then the calculation from the TT's point of view should be off by 16.4 - 3.6 = 12.8 years.

OK. the first statement is false. Is there any false issue in my solution of the paradox?

 

[PeterDonis]

the first statement is false

Why? How does TT account for that portion of HT's worldline in his calculation?

 

[ash64449]

How does TT account for that portion of HT's worldline in his calculation

At the end of the trip, Elapsed proper time for HT as measured by TT is still 20 years and TT's elapsed proper time is 12 years. So why should we account for it?

 

[PeterDonis]

why should we account for it?

You should account for it because you are claiming to show how things look "from TT's viewpoint". If "TT's viewpoint" is correct, TT should be able to use his "viewpoint" to predict that HT's clock will read 20 years when they meet up again. But your analysis "from TT's viewpoint" does not show how TT makes that prediction.

 

[ash64449]

But your analysis "from TT's viewpoint" does not show how TT makes that prediction.

Didn't i do that? the last part of the post #66 explains that.

 

[ash64449]

That is at the turn around, the elapsed proper time jumped from 3.6 years to 16.4 years for the clock at the earth(from TT's point of view) but for planet the clock still reads 10 years. Therefore simultaneity convention changes.

 

[ash64449]

i used the word 'left out'. That was wrong. Never was elapsed proper time of HT left out from my explanation of TT's point of view.

 

[PeterDonis]

at the turn around, the elapsed proper time jumped from 3.6 years to 16.4 years for the clock at the earth(from TT's point of view)

Ok, so what if TT were moving towards HT, but then turned around to move away again? (For example, suppose we are analyzing a modified scenario in which TT goes out, comes part way back, turns around and goes out again, and then comes all the way back.) That would make HT's "elapsed proper time now" go backwards. Do you consider that a viable "point of view" for TT to take?

 

[PAllen]

Ok, so what if TT were moving towards HT, but then turned around to move away again? (For example, suppose we are analyzing a modified scenario in which TT goes out, comes part way back, turns around and goes out again, and then comes all the way back.) That would make HT's "elapsed proper time now" go backwards. Do you consider that a viable "point of view" for TT to take?

Especially given that what you visually see has no such anomaly, nor is there any such anomaly if you are continually exchanging messages with HT.

 

[ash64449]

For example, suppose we are analyzing a modified scenario in which TT goes out, comes part way back, turns around and goes out again, and then comes all the way back

Do you mean the scenario in which TT leaves HT, in the middle of the journey,TT rotates,reach planet and TT reach HT again?

 

[PeterDonis]

Do you mean the scenario in which TT leaves HT, in the middle of the journey,TT rotates,reach planet and TT reach HT again?

I don't think so, but I'm not sure what you are describing here.

The scenario I am talking about, which I don't think had been discussed in this thread before I brought it up, would be described using the HT's rest frame as follows: TT leaves and travels outbound to a distant location (planet, space station, whatever); then TT turns around and heads back towards HT; halfway back, TT turns around again and heads back to the planet/space station/whatever; then TT turns around and heads back all the way to HT and they meet up again.

In this scenario, when TT turns around the second time (when he's gone halfway back to HT, and turns around to head outward again), by your definition of the TT's "point of view", the HT's elapsed proper time will go backwards.

 

[PAllen]

The scenario I am talking about, which I don't think had been discussed in this thread before I brought it up, would be described using the HT's rest frame as follows: TT leaves and travels outbound to a distant location (planet, space station, whatever); then TT turns around and heads back towards HT; halfway back, TT turns around again and heads back to the planet/space station/whatever; then TT turns around and heads back all the way to HT and they meet up again.

In this scenario, when TT turns around the second time (when he's gone halfway back to HT, and turns around to head outward again), by your definition of the TT's "point of view", the HT's elapsed proper time will go backwards.

I mentioned the difficulties of zigzag paths for Fermi-Normal coordinates - that you have to restrict them to a world tube around the 'traveler'. For sudden zigzags, this word tube touches the traveler on one side or the other.

 

[PAllen]

No.it won't

EDIT: As i exactly calculated, it seems to me that it will.. How can this anomaly be addressed?

By admitting that the momentarily comoving inertial frame is not a reasonable model for a zig zag traveler's measurements because their past is different and any operational approach to setting up coordinates will be completely different.

To try to shorten my long essay some posts back, the basis of inertial coordinates is procedures that can be used to define them. If a zig zag traveler uses some reasonable operational approach to setting up coordinates, the problem does not arise. That is, nothing in their observations have this jump backwards in time, so if they use observationally based coordinates, there is no problem.

 

[PAllen]

A way to describe a criterion for reasonable observation based coordinates is that anything you see is considered simultaneous to some event in your past later than anything else you've already seen (and assigned). Basically, you are modeling light delay. Any model meeting this criterion will provide consistent coordinates for anything you can see.

 

[ash64449]

Do you consider that a viable "point of view" for TT to take

No.. This is something that TT cannot accept. To address this issue, A complicated situation arises in which i have to consider turn back at the half-way is not instantaneous.

Even that is not viable. Then another question will come as to why this one is not instantaneous while turn around at the planet is instantaneous?

Then even if i consider the turn around is not instantaneous at the half-way, i see after the process, HT should read the same time as that of the time when TT reached half-way and didn't turn.

So everything gets complicated and hence not viable.

Can you give the answer to your scenario? What will be the reading of the clocks when TT and HT meet up again according to your scenario according to my analysis?

 

[ash64449]

A way to describe a criterion for reasonable observation based coordinates is that anything you see is considered simultaneous to some event in your past later than anything else you've already seen (and assigned). Basically, you are modeling light delay. Any model meeting this criterion will provide consistent coordinates for anything you can see.

Can you show me what your telling by answering to the question as to what HT and TT will see their clocks reading when both meet again according to PeterDonis's scenario?

I think that is the best way to understand what you are telling.

 

[ash64449]

I think If i use HT's rest frame, TT's travel along the space-time is a zig-zag path and by using the relationship between HT's proper time and TT's proper time, i can see what their clocks will read at the end of the scenario.

 

[ash64449]

By calculation, I see HT should read 30 years while TT should read 18 years.

 

[ash64449]

Yeah. I think i found the solution. The solution is "go for it". Let us consider that TT's measure of HT's proper time went backwards.

When TT reaches mid-way and turns,

Clock at HT reads 11.8 while clock at the planet reads 18.2.

When TT reaches the planet, TT now predicts Clock at HT should read 13.6 years while planet should read 20 years.

Now when he makes another turn, Clock at the HT reads 26.4 years and clock at the planet reads 20 years.
When he reaches the earth, HT clock now reads 30 years. And his clock should read 18 years.This is all predicted by TT's point of view and i can see that Point of view method is not viable since at certain instants we need to think that proper time of HT should run backwards- which is not possible.

Anyway as i said earlier, Point of view is something that doesn't have real existence.

I think PeterDonis was trying to say this.

 

[PAllen]

Can you show me what your telling by answering to the question as to what HT and TT will see their clocks reading when both meet again according to PeterDonis's scenario?

I think that is the best way to understand what you are telling.

What they see when they meet is invariant - the same for any valid coordinates.

The most straightforward coordinates meeting the criteria I describe are called radar coordinates. TT simply uses Einstein simultaneity convention - send a signal out to HT, receive a signal back, assign the event of HT receipt to half way between TT send and TT receive. These are called rader coordinates.

 

[ash64449]

What they see when they meet is invariant - the same for any valid coordinates.

I can see. I used a different method "point of view" but still answer came same.