Revisiting the Flaws of the Light Clock in Special and General Relativity

[yuiop]

I don't understand this. An inertial observer moving at half the velocity of the traveller could legitimately see the traveller's and the Earth twin's path lengths as being the same.

This is true on the travellers outward leg, when the twins are spatially separated. The third observer (C) with intermediate velocity says equal proper times have elapsed for the Home twin (A) and the traveling twin (B), when B arrives at the turnaround point, while from B's point of view, A has the longer path and the shortest elapsed proper time and A says B has the longest path and shortest elapsed proper time. Everyone has a different opinion and ambiguity reigns. Once B turns around and arrives back home, observers A, B and C will all agree that B has traveled a longer path through spacetime and than A and all will agree that B has less elapsed proper time than A. All ambiguity is removed once the clocks return to a common location and the differences are unambiguously accounted for in all frames in terms of differences in path lengths.

Whilst this thread has gotten quite big, the point I am actually trying to bring out (as you are probably aware), is how do we know the time dilation SR equations are valid if the light clock thought experiment can be applied equally to the "traveller" and the earth-bound twin, because there are no special frames of reference?

Happy to ignore acceleration completely.

As mentioned in other posts by Mentz, it is better to think in terms of proper time between events, which can be unambiguously defined, rather than in terms of relative time dilation of spatially separated clocks with relative motion, which is ambiguous. When two clocks have non-zero relative motion, there is by definition no single reference frame in which they are both at rest and no way to unambiguously define their relative instantaneous clock rates.

Even when B has turned around and is on his way home, it is still possible to find reference frames in which more time has elapsed for B than for A. It is not the turn around (or the acceleration involved) that removes the ambiguity. It is the arrival of the two clocks at common location that removes the ambiguity about the elapsed proper times, but if the twins do not come to rest wrt each other, they will still disagree about their instantaneous relative clock rates at the final passing event.

 

[gnomechompsky]

This is true on the travellers outward leg, when the twins are spatially separated. The third observer (C) with intermediate velocity says equal proper times have elapsed for the Home twin (A) and the traveling twin (B), when B arrives at the turnaround point, while from B's point of view, A has the longer path and the shortest elapsed proper time and A says B has the longest path and shortest elapsed proper time. Everyone has a different opinion and ambiguity reigns. Once B turns around and arrives back home, observers A, B and C will all agree that B has traveled a longer path through spacetime and than A and all will agree that B has less elapsed proper time than A.

I still don't think you are justifying this. If all movement is relative, and A,B and C were all in inertial frames of reference, why has B gone through the longer path when he returns to A? Is it not equally valid to say that A (Earth) moved away from B then returned to B?

 

[yuiop]

I still don't think you are justifying this. If all movement is relative, and A,B and C were all in inertial frames of reference, why has B gone through the longer path when he returns to A? Is it not equally valid to say that A (Earth) moved away from B then returned to B?

Your third observer C would certainly never see A (Earth) moving away from B and then returning to B. Observer B would feel the acceleration as he changed direction, while observer A never feels a change in direction, so the two situations are certainly not equivalent or symmetrical from that point of view. The observer that feels or meausres proper acceleration is the one that has really changed direction and acceleration is absolute in that sense. I can only suggest you get some graph paper and try drawing the the space versus time diagrams yourself. That is one of the best ways to get an intuitive understanding of what is happening.

Note: You said "If all movement is relative, and A,B and C were all in inertial frames of reference, why has B gone through the longer path when he returns to A? the important word is "were". If B has returned to A, B has not remained in an inertial frame.

 

[Passionflower]

I was talking about an old fashioned alternative method of clock synchronisation or comparison of elapsed clock times by "slow transport" of clocks. This is probably an unnecessary distraction in this thread and I probably shouldn't have mentioned it. If you are still curious about slow clock transport synchronization see http://en.wikipedia.org/wiki/One-way_speed_of_light#Slow_transport or google the terms.

I am familiar with the term but I still like to understand what you mean, do you have the calculations? Note that if two clocks run with a different rate and they want to meet halfway they will disagree about where halfway actually is, slow transport or not.

 

[CPL.Luke]

Mentz,

I think this is the time where you have to demonstrate how the observer in the spaceship would go about calculating the proper time of the observer on the earth, from their reference frame.

you are correct that the proper times will differ, however with just plane SR, you can't calculate the proper time of the Earth correctly (from the space ships perspective). If you attempt to do this in SR you will find that the earthbound twin has aged less than the spacebound twin.

EDIT: by valid frame I meant inertial frame, the space ships frame is non inertial at various points, thus in order to properly calculate the proper time you would have to sum over these areas, while you could do this for certain special examples (such as rindler space) the solution is not so simple as to look at the proper time.

Ignoring the non-inertial parts of the spacetime diagram causes the calculation to fail from the spaceships perspective.

 

[Austin0]

Let us say A remains at rest in frame A. All references to coordinate measurements will mean measurements made by observers at rest in frame A. B passes A at coordinate time zero at a coordinate velocity of +0.8c. After a coordinate time of 10 years, B passes C who is going in the opposite direction with a coordinate velocity of -0.8c. The coordinate distance between event (B passing A) = event(B,A) and event(B,C) is 8 lightyears. Other observers in different reference frames will disagree with the coordinate times, distance and velocities measured by A and will also disagree on what clock A reads at event(B,C), but all observers will agree that 6 years of proper time elapses on clock B between events (A,B) and (B,C). (Definitive statement 1.) Eventually clock C passes A at event(C,A). All observers agree that 6 years of proper time passes on clock C between events (B,C) and (C,A). (Definitive statement 2). All observers will agree that 20 years of proper time elapses on clock A between events (B,A) and (C,A). (Definitive statement 3). All observers agree that the combined elapsed proper time between the 3 events (B,A), (B,C) and (C,A) is 12 years (Definitive statement 4) and that this proper time interval is less that the proper time interval between events (B,A) and (C,A). (Definitive statement 5).

Here I am defining the proper time between any two timelike events (the invariant time interval) as being the time measured by an inertially moving ideal clock that is coincident at both events. I am also using the term "all observers" to mean inertial observers that are not necessarily at rest in frame A.

It would seem that you have demonstrated that frame C was actually moving relative to frame A as the inertial clock located in C and moving toward A had less elapsed proper time than inertial clock A moving toward clock C at the same relative v.

Explanation?

 

[gnomechompsky]

Your third observer C would certainly never see A (Earth) moving away from B and then returning to B.

Disagree. If C was at 1/2 the velocity of B (and also began his return to C at the same time) he would see both moving away from him (and then towards him) at the same speed.

Observer B would feel the acceleration as he changed direction, while observer A never feels a change in direction, so the two situations are certainly not equivalent or symmetrical from that point of view.

Ok, I can agree on this, but as I said in my first post, the only asymmetry in the Twins Paradox comes from acceleration.

However, what is it about the light clock thought experiment that makes it valid on B from A's frame of reference, but not valid on A from B's frame of reference? The acceleration doesn't matter, because B will still see A's light clock as "dilating", on both the outward and return journey. Is there another way to derive the time dilation equation without this paradox?

I can only suggest you get some graph paper and try drawing the the space versus time diagrams yourself. That is one of the best ways to get an intuitive understanding of what is happening.

I can't do this yet, because it would seem to be easily possible to reverse the space/time graph from the frame of reference for B and get the same result.

 

[Austin0]

Originally Posted by kev
Let us say A remains at rest in frame A. All references to coordinate measurements will mean measurements made by observers at rest in frame A. B passes A at coordinate time zero at a coordinate velocity of +0.8c. After a coordinate time of 10 years, B passes C who is going in the opposite direction with a coordinate velocity of -0.8c. The coordinate distance between event (B passing A) = event(B,A) and event(B,C) is 8 lightyears. Other observers in different reference frames will disagree with the coordinate times, distance and velocities measured by A and will also disagree on what clock A reads at event(B,C), but all observers will agree that 6 years of proper time elapses on clock B between events (A,B) and (B,C). (Definitive statement 1.) Eventually clock C passes A at event(C,A). All observers agree that 6 years of proper time passes on clock C between events (B,C) and (C,A). (Definitive statement 2). All observers will agree that 20 years of proper time elapses on clock A between events (B,A) and (C,A). (Definitive statement 3). All observers agree that the combined elapsed proper time between the 3 events (B,A), (B,C) and (C,A) is 12 years (Definitive statement 4) and that this proper time interval is less that the proper time interval between events (B,A) and (C,A). (Definitive statement 5).

Here I am defining the proper time between any two timelike events (the invariant time interval) as being the time measured by an inertially moving ideal clock that is coincident at both events. I am also using the term "all observers" to mean inertial observers that are not necessarily at rest in frame A.

Suppose there is a frame D such that A and C are traveling relative to it, at 0.5c and -0.5c
respectively. At event (B,C) D.. t'''=0 and the spatial interval between A and C is dx''' with observed clock times in A and C of t₀ and t''₀

As observed in D ...A and C meet at dx'''/2 with observed clock times in A and C of T and T''
In both frames dx'''/ 0.5 = dt , dt'' from this it would seem to follow that dt = T-t₀ =T''-t''₀= dt'' Wouldn't this be equal elapsed proper time observed in both frames between events (B,C) and (C,A) ?

Do you see some reason why this would not apply??

 

[Mentz114]

Mentz,
I think this is the time where you have to demonstrate how the observer in the spaceship would go about calculating the proper time of the observer on the earth, from their reference frame.

Given any worldline, one can calculate the elapsed time between two events on it. Furthermore, all observers will agree on this. So the elapsed time as defined above will be the same from all IFRs.

..., however with just plain SR, you can't calculate the proper time of the Earth correctly (from the space ships perspective). If you attempt to do this in SR you will find that the earthbound twin has aged less than the spacebound twin.

I don't agree with this. As I've said, the proper time is the same in all frames.

EDIT: by valid frame I meant inertial frame, the space ships frame is non inertial at various points, thus in order to properly calculate the proper time you would have to sum over these areas, while you could do this for certain special examples (such as rindler space) the solution is not so simple as to look at the proper time.

In the classic twins 'paradox', we don't need Rindler coords, and the resolution is to ignore the instantaneous velocity dependent time-dilation and use the proper time to calculate what the elapsed times are.

Ignoring the non-inertial parts of the spacetime diagram causes the calculation to fail from the spaceships perspective.

The proper time interval is found by integrating along the whole worldline. It doesn't matter if parts are non-inertial.

I don't understand why the differential ageing of the twins is seen as some sort of paradox. Usually the pro-paradox arguments are

1. symmetry - both twins see the other as moving.
That's irrelevant, it's the proper times that give the elapsed times
2. each sees the others clock running slowly
also irrelevant.

 

[gnomechompsky]

1. symmetry - both twins see the other as moving.
That's irrelevant, it's the proper times that give the elapsed times.

How do we derive the formula for proper time? Is it from our geometric understanding of the light clock thought experiment? We can't use proper time to resolve the Twin Paradox if we have not yet justified the premise on which it is based.

Is it possible to derive proper time without using the light clock?

 

[Mentz114]

How do we derive the formula for proper time? Is it from our geometric understanding of the light clock thought experiment? We can't use proper time to resolve the Twin Paradox if we have not yet justified the premise on which it is based.

Is it possible to derive proper time without using the light clock?

In Minkowski spacetime, the infinitesimal line element is given by this formula

$$c^2d\tau^2=c^2dt^2-dx^2-dy^2-dz^2$$

and this forms the basis of the calculation. It is the geometry of Minkowski spacetime, and a postulate of SR is that $\tau$ is the time measured by a clock on the worldline.

I don't see that the light clock comes into at all.

 

[gnomechompsky]

Well Minkowski must have derived the formula somehow. What experiment did he do to get this?

Also, it would seem that the formula above doesn't address the paradox because either side of the equation can equally be applied to either observer. Why is the twin on Earth not on a world line moving away from the Traveller?

 

[Mentz114]

Well Minkowski must have derived the formula somehow. What experiment did he do to get this?

Good question. There is experimental support for SR, but I think Minkowski just intuited the line element.

Also, it would seem that the formula above doesn't address the paradox because either side of the equation can equally be applied to either observer.

The formula applies to whatever segment of worldline you want it to, and it will give a number.

Why is the twin on Earth not on a world line moving away from the Traveller?

From the travellers point of view the Earth is moving away.

I don't know how you are with spacetime diagrams but here's two that show a traveller from the Earth frame, and the Earth from the travellers frame on his outbound trip.

Reading the propertimes off the graphs we find

earth frame : Earth elapsed time ~13
first leg frame : Earth elapsed time sqrt[(14.5*14.5)-(5.5*5.5)] ~ 13

The point is that applying the proper time formula, you get same elapsed time on a segment from any frame.

 

[gnomechompsky]

On those graphs why is Earth always on the straight line? Why can't we make a graph where the traveller is on the straight line and Earth reverses direction?

 

[Mentz114]

Elapsed time on a non-inertial worldline

This graph shows a traveller doing a smooth trip out and back. The worldline is given by

x =0.01 (t² - 1 )⁴

using numerical integration the elapsed time on the traveller's clock is 1.9998564 compared with 2.0000 on the Earth clock.

 

[gnomechompsky]

This graph shows a traveller doing a smooth trip out and back. The worldline is given by

x =0.01 (t² - 1 )⁴

using numerical integration the elapsed time on the traveller's clock is 1.9998564 compared with 2.0000 on the Earth clock.

I think you misunderstood my question. Referring back to the original two graphs, why can we not just switch the reference frames so it is Earth that is moving away and then doing a return journey?

Same with the graph above, the blue curved line could equally be Earth from the frame of reference of the Traveller.

 

[CPL.Luke]

Mentz you are assuming that it is minkowski space for the traveler, if that was the case then the equations you show would have both parties disagreeing on who was younger.

The problem is solved from the travelers frame when you realize that the traveler passes through rindler space each time he accelerates, the Earth is always in minkowski space.

To illustrate this, in post 50 you performed the calculation from the Earth's perspective, can you perform it from the travelers perspective? your solution is correct, but for the wrong reasons.

 

[Mentz114]

I think you misunderstood my question. Referring back to the original two graphs, why can we not just switch the reference frames so it is Earth that is moving away and then doing a return journey?

Because finding rocket motors powerful enough to accelerate the Earth is a problem. But if you could do it, then the Earth's worldline would be curved and its proper time shorter.

Mentz you are assuming that it is minkowski space for the traveler, if that was the case then the equations you show would have both parties disagreeing on who was younger.

Proper time is invariant under Lorentz transformation, and any instant on the curved world line is connected to the Earth worldline by a Lorentz transformation.

The problem is solved from the travelers frame when you realize that the traveler passes through rindler space each time he accelerates, the Earth is always in minkowski space.

It is always Minkowski space. There's no 'Rindler' spacetime, only Rindler coords in Minkowski spacetime.

To illustrate this, in post 50 you performed the calculation from the Earth's perspective, can you perform it from the travelers perspective? your solution is correct, but for the wrong reasons.

Please explain why the calculation is wrong. Seriously, if you think that the elapsed times are frame dependent for the trip under question then demonstrate it.

 

[gnomechompsky]

Because finding rocket motors powerful enough to accelerate the Earth is a problem. But if you could do it, then the Earth's worldline would be curved and its proper time shorter.

Ah, so we are back to the point in my original post, it is the acceleration "felt" by the object that generates the asymmetry in the Twins Paradox.

However, even if we know that it is the Space traveller accelerating, according to the formulas for proper time and SR time dilation, acceleration is not a factor.

 

[Mentz114]

Mentz you are assuming that it is minkowski space for the traveler, if that was the case then the equations you show would have both parties disagreeing on who was younger.

To return to this, if you applied the rules, then any point on the curved worldline has a tangent at which we can assume a comoving inertial frame with velocity equal to the tangent. This is still connected to the Earth frame by Lorentz transformation so the calculation of the Earth proper time would give the same answer. Using this approach there is no disagreement.

Ah, so we are back to the point in my original post, it is the acceleration "felt" by the object that generates the asymmetry in the Twins Paradox.

However, even if we know that it is the Space traveller accelerating, according to the formulas for proper time and SR time dilation, acceleration is not a factor.

I've forgotten a long time ago what this thread was about

 

[Austin0]

Because finding rocket motors powerful enough to accelerate the Earth is a problem. But if you could do it, then the Earth's worldline would be curved and its proper time shorter.
Proper time is invariant under Lorentz transformation, and any instant on the curved world line is connected to the Earth worldline by a Lorentz transformation.
Please explain why the calculation is wrong. Seriously, if you think that the elapsed times are frame dependent for the trip under question then demonstrate it.

You and kev seem to have both ignored post #43 in which I posed a possible counter for kev's demonstration of the problem (with only inertial frames). And a different conclusion wrt elapsed proper time.
If there is some flaw in what I presented I would be glad to learn of it.

You have consistently failed to address the OP's real question.
Disregarding acceleration as a factor (which you have in fact included above) but simply operating with the essential kinematic assumption of SR, what would prevent a perfectly symmetrical Minkowski diagram of the Earth from the perspective of the accelerated frame??
A mirror image??
A plot of the spacetime locations of the Earth relative to the accelerated frame.
I.e. The Earth having the same curved segments, inertial segments etc.
If this was done you would then have totally reciprocal and mutually exclusive elapsed proper times as derived from integrated worldlines between events , no??
As far as I have seen the OP has a relevant question and having read a great many twin threads it appears that there is no real consensus as to the proper resolution and whether acceleration is a crucial criteria or not. Everyone agrees there is no real paradox and agrees on the outcome but there still seems to be questions worth pursuing regarding both acceleration and simultaneity. IMHO

 

[matheinste]

You and kev seem to have both ignored post #43 in which I posed a possible counter for kev's demonstration of the problem (with only inertial frames). And a different conclusion wrt elapsed proper time.
If there is some flaw in what I presented I would be glad to learn of it.

You have consistently failed to address the OP's real question.
Disregarding acceleration as a factor (which you have in fact included above) but simply operating with the essential kinematic assumption of SR, what would prevent a perfectly symmetrical Minkowski diagram of the Earth from the perspective of the accelerated frame??
A mirror image??
A plot of the spacetime locations of the Earth relative to the accelerated frame.
I.e. The Earth having the same curved segments, inertial segments etc.
If this was done you would then have totally reciprocal and mutually exclusive elapsed proper times as derived from integrated worldlines between events , no??
As far as I have seen the OP has a relevant question and having read a great many twin threads it appears that there is no real consensus as to the proper resolution and whether acceleration is a crucial criteria or not. Everyone agrees there is no real paradox and agrees on the outcome but there still seems to be questions worth pursuing regarding both acceleration and simultaneity. IMHO

No matter how you dress it up, or what scenario you propose, in SR, the proper time along a worldline, that is the time measuired by a clock that is present at all points along the worldline, is a measure of the spacetime distance along that worldline. There s no ambiguity and no lack of consensus on this point. Whether the Earth accelerates or the rocket accelerates or both accelerate, whether symmetrically or not, the proper time along the wordlines can measured by a clock traveling along the wordline or calculated by any observer, and compared. There will no disagreement about the reuslts.

As for the effect of acceleration on clocks, the clock hypothesis assumes ther is no effect and this is borne out by laboratory performed experiments to a very high degree. Of course for diiferential time intervals, which of couse require curved wordlines to describe them geometrically, we need acceleration to produce them, but it this curvature of the spactime path which leads to the different proper times, or not, along two wordlines. The acceleration is the cause of this curvature but has no direct effects on clocks. The compared readings on the clocks are in effect, loosely speaking, a measure of the relative curvature of the wordlines.

Matheinste

 

[Mentz114]

I go along with Matheinste's post above which summarises the situation very well. I wish the 'twin paradox' would go away ( although it's fun sometimes talking about it).

 

[yossell]

I go along with Matheinste's post above which summarises the situation very well. I wish the 'twin paradox' would go away ( although it's fun sometimes talking about it).

Oh - I found understanding why the twins paradox is not a paradox an essential part of my coming to understand relativity. The barn paradox too. I think they should be as much a part of a relativity course as the Lorentz equations.

 

[Mentz114]

Oh - I found understanding why the twins paradox is not a paradox an essential part of my coming to understand relativity. The barn paradox too. I think they should be as much a part of a relativity course as the Lorentz equations.

Yes, you're right. The barn and pole came up as an exam question once for me. It seemed to me that I understood SR much better when I fathomed space-time geometry, which answers most questions. When in doubt, draw a diagram.

 

[Passionflower]

The compared readings on the clocks are in effect, loosely speaking, a measure of the relative curvature of the wordlines.

That is incorrect, curvature or relative curvature has nothing to do with it. If it did then the clock hypothesis would be false.

Think of the twin paradox where the traveling twin follows the path of two edges of a triangle while the at home twin has a straight line. In Minkowski spacetime the total length (which is equal to the elapsed proper time) of the edges of the triangle is smaller than the straight line between them, which of course opposite in Euclidean space, a reversal of the triangle inequality.

 

[matheinste]

That is incorrect, curvature or relative curvature has nothing to do with it. If it did then the clock hypothesis would be false.

Think of the twin paradox where the traveling twin follows the path of two edges of a triangle while the at home twin has a straight line. In Minkowski spacetime the total length (which is equal to the elapsed proper time) of the edges of the triangle is smaller than the straight line between them, which of course opposite in Euclidean space, a reversal of the triangle inequality.

Any depiction of realistic acceleration requires a curved path to represent it in a spacetime diagram. I do however see the point you are trying to make and perhaps my choice of words was not appropriate.

Thanks for the help but I am pretty much at home with the properties of the geometry required for the spacetime metric.

Matheinste.

 

[phyti]

For simplicity the times are referenced as A or B followed by the year.
Twin B leaves twin A moving at .8c, reverses direction at B12, and returns.
Fig. 1 shows A's view of B's trip. The axis of simultaneity (gray) for B is (A7.2, B12) outbound, and (B12, A32.8) inbound. The instantaneous jump from A7.2 to A32.8 is due to excluding any period of acceleration for B to transfer from the outbound to the inbound frame of reference. The ratio of B-time to A-time is 24/40 = .60.

Fig. 2 is B's view using the Einstein simultaneity convention. The discontinuous motion of A at A4 reflects the switching of frames without acceleration. The extreme distortion of times and locations, using this convention, is noted with A4 simultaneous with B-36, 36 years before they parted! The ratio of A-time to B-time is 4/6.7 = 36/60 = .60 for both path segments.

At this point the slow clock rate is reciprocal.

Fig. 3 is B's view using a horizontal axis of simultaneity, i.e. a translation of positions, and A moving at -.8c. Since the initial conditions place A in the 'chosen' static frame, B must be moving, therefore if he had the means to measure the 1-way light speed, it would not be c the absolute speed through space, but the speed relative to him, c-v and c+v (magenta). The ratio of A-time to B-time is 4/2.4 = 36/21.6 = 1.67, in agreement with the result for the closed path. The time dilation is now asymmetrical as calculated by both A and B. The extreme space and time shifts are also removed.

An example of relative speeds.
The question: How much time is required for a car moving at 60 mph, to overtake a car moving at 50 mph with a 1 mile lead?
The answer: distance/(v1-v2) = 1/(60-50) = 1/10 = .1 hr = 6 min.
It's the relative or closing speed that determines the answer. Neither car would expect the other to approach at 60 mph. If the lead car used 60 for the chase car rate, the initial separation would have been 60*.1 hr = 6 miles, not 1 mile. The absolute car speed is relative to the ground. The relative car speed is relative to the other car. They are two different types of relations. If light replaces the chase car, its speed c is relative to space, defined as an invisible but fixed frame of reference, and its relative speed as c-v, with v the speed of the object being chased. The fact that relative light speed is different from c, doesn't contradict its absolute speed, no more than the 10 mph closing speed alters the 60 mph chase car speed.

Fig. 4 shows a more realistic case with a short period of acceleration for B transitioning between frames. B would explain the curved portion of A's motion as resulting from an equivalent g-field during his acceleration. This also provides an asymmetrical view with 40 A events to 24 B events.

The simultaneity definition is the source of the 'paradox' where unequal path lengths are defined as equal, for the purpose of preserving constant light speed.
https://www.physicsforums.com/attachments/27694

 

[Austin0]

You and kev seem to have both ignored post #43 in which I posed a possible counter for kev's demonstration of the problem (with only inertial frames). And a different conclusion wrt elapsed proper time.
If there is some flaw in what I presented I would be glad to learn of it.

You have consistently failed to address the OP's real question.
Disregarding acceleration as a factor (which you have in fact included above) but simply operating with the essential kinematic assumption of SR, what would prevent a perfectly symmetrical Minkowski diagram of the Earth from the perspective of the accelerated frame??
A mirror image??
A plot of the spacetime locations of the Earth relative to the accelerated frame.
I.e. The Earth having the same curved segments, inertial segments etc.
If this was done you would then have totally reciprocal and mutually exclusive elapsed proper times as derived from integrated worldlines between events , no??
As far as I have seen the OP has a relevant question and having read a great many twin threads it appears that there is no real consensus as to the proper resolution and whether acceleration is a crucial criteria or not. Everyone agrees there is no real paradox and agrees on the outcome but there still seems to be questions worth pursuing regarding both acceleration and simultaneity. IMHO

No matter how you dress it up, or what scenario you propose, in SR, the proper time along a worldline, that is the time measuired by a clock that is present at all points along the worldline, is a measure of the spacetime distance along that worldline. There s no ambiguity and no lack of consensus on this point. Whether the Earth accelerates or the rocket accelerates or both accelerate, whether symmetrically or not, the proper time along the wordlines can measured by a clock traveling along the wordline or calculated by any observer, and compared. There will no disagreement about the reuslts.

As for the effect of acceleration on clocks, the clock hypothesis assumes ther is no effect and this is borne out by laboratory performed experiments to a very high degree. Of course for diiferential time intervals, which of couse require curved wordlines to describe them geometrically, we need acceleration to produce them, but it this curvature of the spactime path which leads to the different proper times, or not, along two wordlines. The acceleration is the cause of this curvature but has no direct effects on clocks. The compared readings on the clocks are in effect, loosely speaking, a measure of the relative curvature of the wordlines.

Matheinste

Although everything you have said here is true it does not actually address anything I said.
There is no disagreement regarding the fundamental interpretation of worldlines and integrated path lengths as proper time. I certainly have no question in this regard and from what I have read from the OP that is not his point or question either.
As far as the clock hypothesis goes, I have myself on more than one occasion cited it as a counter to the acceleration argument with regard to non-reciprocal dilation.
So all of your post is another straw man argument aimed at questions never asked and positions never stated.
In the meantime totally ignoring explicit questions and arguments.
kev made , in his own words "difinitive " statements wrt his own scenario with only inertial frames.
Part of this was the assertion that his conclusions would be agreed upon by all inertial observers.
I offered a simple and clear example of one possible inertial frame that seemed to disagree with this definitive statement.
I could very well be wrong. If that is so then it should be simple for you to show me this and demonstrate your case.

I offered the simple premise that simply based on kinematics and coordinate systems it is easily possible to plot the Earth's time/space positions relative to the accelerated system with the accelerated system at rest. AS per the OP's repeated question. If this is done the application of the same integration of the Earth's worldline , the same interpretation of proper time , would produce the reciprocal but opposite conclusion.
This is not done for other reasons. I.e. an acclerated frame is not valid, acceleration is real etc etc etc.
I am not saying these are not valid reasons only that to simply say proper time is the intergration of the worldline simply begs the question.
I certainly have no answer as to the best resolution to the twin question or non-reciprocal dilation , only the belief that it represents a meaningful question

However, I can make a definitive statement about the elapsed proper time between any two timelike events.

Let us say A remains at rest in frame A. All references to coordinate measurements will mean measurements made by observers at rest in frame A. B passes A at coordinate time zero at a coordinate velocity of +0.8c. After a coordinate time of 10 years, B passes C who is going in the opposite direction with a coordinate velocity of -0.8c. The coordinate distance between event (B passing A) = event(B,A) and event(B,C) is 8 lightyears. Other observers in different reference frames will disagree with the coordinate times, distance and velocities measured by A and will also disagree on what clock A reads at event(B,C), but all observers will agree that 6 years of proper time elapses on clock B between events (A,B) and (B,C). (Definitive statement 1.) Eventually clock C passes A at event(C,A). All observers agree that 6 years of proper time passes on clock C between events (B,C) and (C,A). (Definitive statement 2). All observers will agree that 20 years of proper time elapses on clock A between events (B,A) and (C,A). (Definitive statement 3). All observers agree that the combined elapsed proper time between the 3 events (B,A), (B,C) and (C,A) is 12 years (Definitive statement 4) and that this proper time interval is less that the proper time interval between events (B,A) and (C,A). (Definitive statement 5).

Suppose there is a frame D such that A and C are traveling relative to it, at 0.5c and -0.5c
respectively. At event (B,C) D.. t'''=0 and the spatial interval between A and C is dx''' with observed clock times in A and C of t₀ and t''₀

As observed in D ...A and C meet at dx'''/2 with observed clock times in A and C of T and T''
For both frames dx'''*0.5/ 0.5c = dt , dt'' from this it would seem to follow that dt = T-t₀ =T''-t''₀= dt'' Wouldn't this be equal elapsed proper time observed in both frames between events (B,C) and (C,A) ?

Do you see some reason why this would not apply??

 

[matheinste]

As far as I have seen the OP has a relevant question and having read a great many twin threads it appears that there is no real consensus as to the proper resolution and whether acceleration is a crucial criteria or not. Everyone agrees there is no real paradox and agrees on the outcome but there still seems to be questions worth pursuing regarding both acceleration and simultaneity. IMHO

My apologies if my reply was irrelevant. I was commenting on the above part of your post.
I'm afraid when I come accoss the words "Twin paradox" in the same paragraph as "no real consensus as to the proper resolution" I cannot help myself, its an automatic reaction. Most regular posters here have developed a mechanism for overcoming such reactions. I must try harder to do the same

Matheinste.

 

[yuiop]

I offered the simple premise that simply based on kinematics and coordinate systems it is easily possible to plot the Earth's time/space positions relative to the accelerated system with the accelerated system at rest. AS per the OP's repeated question. If this is done the application of the same integration of the Earth's worldline , the same interpretation of proper time , would produce the reciprocal but opposite conclusion.

I will probably end up having to draw some diagrams for this explanation, but this is my initial attempt using only words.

In the space time diagram of the stay on Earth twin (ignoring gravity) the Earth twins wordline is vertical and the traveling twin has the longer path corresponding to the other two sides of a triangle. In this space time diagram, all horizontal lines are lines of simultaneity according to the Earth twin and in this diagram longer paths unambiguously represent paths with the shortest proper time interval when clocks share an initial starting location and share a final end location.

Now if we try and draw the space time diagram of the traveling twin, it is no longer true that all lines of simultaneity according to the traveling twin are horizontal and parallel. Some event E1 on the traveling twins worldline that the traveling twin considers to happen simultaneously with event E1' on the Earth twins worldline, can also be simultaneous with some other later event E2' on the Earth twins worldline. Because lines of simultaneity in the traveling twins space time diagram causes ambiguous interpretation of simultaneous events in his own frame, it is difficult to draw any definitive conclusions from the traveling twins space time diagram and you can certainly cannot conclude that that a longer worldline in the spacetime diagram of a non-inertial observer is the path of least proper time. This is only true for spacetime diagrams of inertial observers and the traveling twin does not qualify as an inertial observer.

 

[yossell]

My apologies if my reply was irrelevant. I was commenting on the above part of your post.
I'm afraid when I come accoss the words "Twin paradox" in the same paragraph as "no real consensus as to the proper resolution" I cannot help myself, its an automatic reaction.

I don't even know what consensus is, so I'm not in a position to comment. There is of course great consensus about the resolution of paradox, but I'm sure Austin0 never meant to question this.

 

[Austin0]

I will probably end up having to draw some diagrams for this explanation, but this is my initial attempt using only words.

In the space time diagram of the stay on Earth twin (ignoring gravity) the Earth twins wordline is vertical and the traveling twin has the longer path corresponding to the other two sides of a triangle. In this space time diagram, all horizontal lines are lines of simultaneity according to the Earth twin and in this diagram longer paths unambiguously represent paths with the shortest proper time interval when clocks share an initial starting location and share a final end location.

Now if we try and draw the space time diagram of the traveling twin, it is no longer true that all lines of simultaneity according to the traveling twin are horizontal and parallel. Some event E1 on the traveling twins worldline that the traveling twin considers to happen simultaneously with event E1' on the Earth twins worldline, can also be simultaneous with some other later event E2' on the Earth twins worldline. Because lines of simultaneity in the traveling twins space time diagram causes ambiguous interpretation of simultaneous events in his own frame, it is difficult to draw any definitive conclusions from the traveling twins space time diagram and you can certainly cannot conclude that that a longer worldline in the spacetime diagram of a non-inertial observer is the path of least proper time. This is only true for spacetime diagrams of inertial observers and the traveling twin does not qualify as an inertial observer.[/QUOTE]

It is completely unneccessary to draw diagrams. Particulary since you would attempt to draw them with the assumptions derived from drawings with the Earth as the rest frame.
You are missing the point. If you remove the physics i.e. the fact that accelerometers tell us the space twin is accelerating. Based on the purely kinematic assumption of SR that motion is relative there is no reason why the lines of simultaneity in a diagram with the space twin at rest would not be parallel. WIth the Earth lines with varying degrees of slope.
With the Earth worldline with curved segments. Etc
I am in no way suggesting this is a productive or valid course to follow only that in the end , this course is essentially eliminated on the basis of your last sentence , which in fact I had made explicit in previous posts in this thread and don't disagree with. ANd this very point is also explicitly the question of the OP.
You have in this thread attempted to eliminate the physics, acceleration, as a determining factor for a preferred frame i.e. the Earth and show it with only inertial frames involved..
You have still ignored my post relevant and counter to your demonstration, why is this??

 

[Austin0]

I don't even know what consensus is, so I'm not in a position to comment. There is of course great consensus about the resolution of paradox, but I'm sure Austin0 never meant to question this.

Nope not me. At least not the consensus that there is in fact no paradox.
But there are resolutions that do not directly involve acceleration as a crucial determinent.
And a certain lack of consistency between various valid and logical resolutions IMHO

 

[yossell]

Whoooossshhhhh!