Two twins are at rest in F0 at x=0. At time t=0 the two twins accelerate to V0 wrt to F0 and then decelerate back to zero velocity relative to F0 in identical rockets. One rocket moved one twin a distance of L0 in the positive x direction in T0 seconds (as measured by observers in F0), while the other identical rocket moved the other twin a distance of L0 in the negative x direction in T0 seconds (as measured by observers in F0). The twins are now separated a distance of 2L0 as measured in F0 and each twin has zero relative velocity wrt to F0.
Then both twins accelerate in the positive x direction in identical rockets starting their accelerations simultaneously as measured by observers in F0. Both twins accelerate for an equal amount of time and then come to rest in another inertial reference frame F1. In inertial reference frame F1, the twins are separated by a distance of L1 as measured by observers in F1 and have zero velocity wrt F1.
The twins then decide to see each other, so as measured by observers in F1, the twins simultaneously accelerate and decelerate to the midpoint between them, a distance of L1 in identical times as measured by observers in F1.
The question I have is after the final leg of this journey, where the twins are at the same point in space, each with zero relative velocity wrt to F1 and to each other, are the twins the same age? In terms of F1, once the twins did the first leg of the journey and they had a separation of 2L0, the twins no longer were the same age. From the measurements by observers in F1, the twins each aged an identical amount during the last two legs of the journey, so therefore observers in F1 would say the twins do not have the same age when they meet in F1.
From the viewpoint of the twins, they both started at the same point in space and time, they both accelerated and decelerated in an identical amount of time during the first leg of the journey, they both accelerated an equal amount of time during the second leg of the journey, and they both accelerated and decelerated an equal amount of time during the last leg of the journey, and all accelerations where simultaneous from the twin's point of view, therefore they should have the identical ages when they are at rest at the same point in space where they meet in F1.
Which answer is correct?
Thanks,
David Seppala
Bastrop tX

sepp623@yahoo.com wrote:

> accelerations where simultaneous from the twin's point of view,
> therefore they should have the identical ages when they are at
> rest at the same point in space where they meet in F1.
> Which answer is correct?
> Thanks,
> David Seppala Bastrop tX

kiss my ass, you lost. When the arrival is another places you have to
calculate to that point, start to end.

On Wednesday, September 29, 2021 at 7:22:54 AM UTC-7, sep...@yahoo.com wrote:
> The question I have is after the final leg of this journey, where the twins are at the
> same point in space, each with zero relative velocity wrt to F1 and to each other,
> are the twins the same age?

Of course not.

> In terms of F1, once the twins did the first leg of the journey and they had a
> separation of 2L0, the twins no longer were the same age.

Right, at equal values of the time coordinate of F1 the proper ages of the twins are different.

> From the measurements by observers in F1, the twins each aged an identical
> amount during the last two legs of the journey, so therefore observers in F1
> would say the twins do not have the same age when they meet in F1.

Well, everyone (regardless of their state of motion) says the twins have aged different amounts between their separation and their re-uniting. The elapsed proper time along any given path is invariant, not coordinate dependent.

> From the viewpoint of the twins...

That phrase is both ambiguous and superfluous, and should be omitted. It's ambiguous, because you need to specify a system (or systems) of coordinates, which is not done by simply referring to individuals. The phrase "from the viewpoint of X" in the popular literature really just refers to an inertial coordinate system in which X is continuously at rest, but the twins in this scenario are not continuously at rest in any system of inertial coordinates, so you need to explicitly specify what system (or systems) of coordinates you are referring to. But it is superfluous because you can delete that phrase without loss of meaning, as follows:

> [However] they both started at the same point in space and time, they both
> accelerated and decelerated in an identical amount of [proper] time during the
> first leg of the journey, they both accelerated an equal amount of [proper] time
> during the second leg of the journey...

Note that I inserted the word "proper", to correctly convey what you are trying to say. Up to this point you are okay, but then you go off the rails here:

> ...and they both accelerated and decelerated an equal amount of [proper] time
> during the last leg of the journey...

You omitted one segment of proper time for the right-hand twin, who dwells at rest in F1 for an extra amount of proper time (longer than the other twin does) before initiating the last leg of his journey. This is an objective fact, and everyone agrees on this.

> ... and all accelerations were simultaneous from the twin's point of view...

To be clear, the first accelerations and decelerations were simultaneous in terms of F0, and the last accelerations were simultaneous in terms of F1.

> therefore they should have the identical ages when they ... meet...

No, the right-hand twin dwells for an extra period of proper time after coming to rest in F1 and before initiating the final leg of his journey. This extra dwell time is precisely the amount by which the twins differ in age when they re-unite.

> Which answer is correct?

Special relativity does not yield two different answers, it yields only one, which is that the twins have different ages when re-united, as explained above. You just made a math mistake by omitting the extra dwell segment from the right-hand path.

Special Relativity: 836 ..... Barnpole Dave: 0

On Wednesday, September 29, 2021 at 11:05:20 AM UTC-5, Al Coe wrote:
> On Wednesday, September 29, 2021 at 7:22:54 AM UTC-7, sep...@yahoo.com wrote:
> > The question I have is after the final leg of this journey, where the twins are at the
> > same point in space, each with zero relative velocity wrt to F1 and to each other,
> > are the twins the same age?
> Of course not.
> > In terms of F1, once the twins did the first leg of the journey and they had a
> > separation of 2L0, the twins no longer were the same age.
> Right, at equal values of the time coordinate of F1 the proper ages of the twins are different.
> > From the measurements by observers in F1, the twins each aged an identical
> > amount during the last two legs of the journey, so therefore observers in F1
> > would say the twins do not have the same age when they meet in F1.
> Well, everyone (regardless of their state of motion) says the twins have aged different amounts between their separation and their re-uniting. The elapsed proper time along any given path is invariant, not coordinate dependent.
>
> > From the viewpoint of the twins...
>
> That phrase is both ambiguous and superfluous, and should be omitted. It's ambiguous, because you need to specify a system (or systems) of coordinates, which is not done by simply referring to individuals. The phrase "from the viewpoint of X" in the popular literature really just refers to an inertial coordinate system in which X is continuously at rest, but the twins in this scenario are not continuously at rest in any system of inertial coordinates, so you need to explicitly specify what system (or systems) of coordinates you are referring to. But it is superfluous because you can delete that phrase without loss of meaning, as follows:
>
> > [However] they both started at the same point in space and time, they both
> > accelerated and decelerated in an identical amount of [proper] time during the
> > first leg of the journey, they both accelerated an equal amount of [proper] time
> > during the second leg of the journey...
>
> Note that I inserted the word "proper", to correctly convey what you are trying to say. Up to this point you are okay, but then you go off the rails here:
>
> > ...and they both accelerated and decelerated an equal amount of [proper] time
> > during the last leg of the journey...
>
> You omitted one segment of proper time for the right-hand twin, who dwells at rest in F1 for an extra amount of proper time (longer than the other twin does) before initiating the last leg of his journey. This is an objective fact, and everyone agrees on this.
>
> > ... and all accelerations were simultaneous from the twin's point of view...
>
> To be clear, the first accelerations and decelerations were simultaneous in terms of F0, and the last accelerations were simultaneous in terms of F1..
>
> > therefore they should have the identical ages when they ... meet...
>
> No, the right-hand twin dwells for an extra period of proper time after coming to rest in F1 and before initiating the final leg of his journey. This extra dwell time is precisely the amount by which the twins differ in age when they re-unite.
>
> > Which answer is correct?
>
> Special relativity does not yield two different answers, it yields only one, which is that the twins have different ages when re-united, as explained above. You just made a math mistake by omitting the extra dwell segment from the right-hand path.
>
> Special Relativity: 836 ..... Barnpole Dave: 0

Let's say one twin doesn't have any access to any clocks. While in F0, the one twin tells the other, I know you don't have any clocks and you tend to dawdle so I am going to make sure we have lunch together some time after we reach F1. So here's what I'm going to do. There is a mirror at the midpoint between you and I here while we are at rest in F0. So I'm going to send a light pulse to you and have part of that pulse reflect off the mirror back to me. When you receive that pulse, start accelerating into F1. I'll start accelerating when I receive the reflected pulse. That way I'll know that we both started accelerating simultaneously. I also observed that the way I planned our accelerations into F1, at the midway point between where you and I end up when we have zero velocity wrt to F1, there is a mirror there as well. When I am at rest in F1, I will send a light pulse to you and have part of that pulse reflect off the mirror back to me. When you receive that pulse, start your acceleration toward the mirror, and when I receive the reflection off the mirror I'll start my acceleration toward the mirror, that way I will know that we will meet at the mirror (or midway point) in F1 simultaneously. In which step of this procedure does the twin sending the pulses determine that the other twin spent "extra dwell time" in F1 before both the accelerations in F1 started?
David Seppala
Bastrop TX

On Wednesday, September 29, 2021 at 11:29:19 AM UTC-7, sep...@yahoo.com wrote:
> Let's say one twin doesn't have any access to any clocks.

There was no mention of clocks in the scenario, so there was already no supposition of clocks.

> While [at rest] in [terms of F0 the one twin tells the other, I am going to make sure we have
> lunch together some time after we reach [a state of rest in terms of] F1.

> There is a mirror at the midpoint between you and I here while we are at rest
> in F0. So I'm going to send a light pulse to you and have part of that pulse reflect
> off the mirror back to me. When you receive that pulse, start accelerating into F1.
> I'll start accelerating when I receive the reflected pulse. That way I'll know that we
> both started accelerating simultaneously.

And the other twin rolls his eyes and says "Are you completely brain-dead? It has already been stipulated that we begin our initial acceleration simultaneously in terms of F0. We know what simultaneity means in terms of any given system of coordinates, so your Rube Goldberg contraption is not adding anything to the specification of the scenario. The clear and simple explanation of your math error in analyzing this scenario was given in the previous post."

> I also observed that the way I planned our accelerations into F1, at the midway point
> between where you and I end up when we have zero velocity wrt to F1, there is a
> mirror there as well. When I am at rest in F1, I will send a light pulse to you and have
> part of that pulse reflect off the mirror back to me. When you receive that pulse, start
> your acceleration toward the mirror, and when I receive the reflection off the mirror I'll
> start my acceleration toward the mirror, that way I will know that we will meet at the
> mirror (or midway point) in F1 simultaneously.

And the other twin rolls his eyes again, and says "You really are brain-dead, aren't you? I realize we are twins, but you must have been dropped on your head at some point. It was already stipulated that we begin the final accelerations simultaneously in terms of F1, so your Rube Goldberg contraption is not adding anything to the specification of the scenario. The clear and simple explanation of your math error in analyzing this scenario was given in the previous post."

> In which step of this procedure does the twin sending the pulses determine that the
> other twin spent "extra dwell time" in F1 before both the accelerations in F1 started?

Neither of those procedures is designed to measure or evaluate the elapsed proper time of either twin. Those Rube Goldberg contraptions merely establish, for two separate worldlines, pairs of events that share the same time coordinate of two different systems of inertial coordinates.

To understand what is happening, and the math mistake you are making, just draw a little plot of the position versus time of the two paths. [pause for you to do that] Now do you understand?

Special Relativity: 837 .... Barnpole Dave: 0

If the twins are separating from one another at the same
relative speed they ought to share the same time.

Mitchell Raemsch

On Wednesday, September 29, 2021 at 2:11:39 PM UTC-5, Al Coe wrote:
> On Wednesday, September 29, 2021 at 11:29:19 AM UTC-7, sep...@yahoo.com wrote:
> > Let's say one twin doesn't have any access to any clocks.
> There was no mention of clocks in the scenario, so there was already no supposition of clocks.
>
> > While [at rest] in [terms of F0 the one twin tells the other, I am going to make sure we have
> > lunch together some time after we reach [a state of rest in terms of] F1.
> > There is a mirror at the midpoint between you and I here while we are at rest
> > in F0. So I'm going to send a light pulse to you and have part of that pulse reflect
> > off the mirror back to me. When you receive that pulse, start accelerating into F1.
> > I'll start accelerating when I receive the reflected pulse. That way I'll know that we
> > both started accelerating simultaneously.
> And the other twin rolls his eyes and says "Are you completely brain-dead? It has already been stipulated that we begin our initial acceleration simultaneously in terms of F0. We know what simultaneity means in terms of any given system of coordinates, so your Rube Goldberg contraption is not adding anything to the specification of the scenario. The clear and simple explanation of your math error in analyzing this scenario was given in the previous post."
> > I also observed that the way I planned our accelerations into F1, at the midway point
> > between where you and I end up when we have zero velocity wrt to F1, there is a
> > mirror there as well. When I am at rest in F1, I will send a light pulse to you and have
> > part of that pulse reflect off the mirror back to me. When you receive that pulse, start
> > your acceleration toward the mirror, and when I receive the reflection off the mirror I'll
> > start my acceleration toward the mirror, that way I will know that we will meet at the
> > mirror (or midway point) in F1 simultaneously.
> And the other twin rolls his eyes again, and says "You really are brain-dead, aren't you? I realize we are twins, but you must have been dropped on your head at some point. It was already stipulated that we begin the final accelerations simultaneously in terms of F1, so your Rube Goldberg contraption is not adding anything to the specification of the scenario. The clear and simple explanation of your math error in analyzing this scenario was given in the previous post."
> > In which step of this procedure does the twin sending the pulses determine that the
> > other twin spent "extra dwell time" in F1 before both the accelerations in F1 started?
> Neither of those procedures is designed to measure or evaluate the elapsed proper time of either twin. Those Rube Goldberg contraptions merely establish, for two separate worldlines, pairs of events that share the same time coordinate of two different systems of inertial coordinates.
>
> To understand what is happening, and the math mistake you are making, just draw a little plot of the position versus time of the two paths. [pause for you to do that] Now do you understand?
>
> Special Relativity: 837 .... Barnpole Dave: 0

Okay, so what you are telling the twin is that this is "pure math" - you cannot determine when your twin spent "extra dwell time" in F1 by making any physical measurements. It has to be pure math. You and your twin did not leave inertial reference frame F0 simultaneously if you used the light pulses to synchronize your accelerations. You can only use physical measurement methods to determine times only if you start at rest in an inertial reference system, accelerate, and then both return to rest in the same inertial reference frame. In all other cases you must use "pure math".
So for example, if you and your twin are the identical ages, and are traveling side by side in identical rockets and and your rockets are separated by 10 meters perpendicular to the direction of travel, you ages will remain the same throughout the trip, but if both of you simultaneously land on the equator of a planet that is rotating perpendicular to your direction of travel as soon as you land one of you will be older then the other. You might ask who will be older? The answer of course is that it depends on the direction of rotation of the planet.
Is it because your twin accelerated in any different way than you did? The answer is of course not - its just pure math!

David Seppala
Bastrop TX

On Wednesday, September 29, 2021 at 1:04:57 PM UTC-7, sep...@yahoo.com wrote:
> Okay, so what you are telling the twin is that this is "pure math" - you cannot
> determine when your twin spent "extra dwell time" in F1 by making any physical
> measurements.

No, your statement is insane, and has no relation to what I explained to you. Again, Lorentz invariance is not pure math, it is physics, but you are trying to show that Lorentz invariance entails a logical inconsistency, by assuming Lorentz invariance and deducing a contradiction, so the questions you are asking are purely mathematical, and the mistakes you are making are purely mathematical.

You didn't ask me how to empirically determine the dwell time, you asked me which of your two brain-dead operations determined the dwell time, and I informed you that neither of them did. You jump from this to claiming that I've said there is no way to determine the dwell time. Your thought processes are highly corrupted.

> You and your twin did not [accelerate from rest in terms of] inertial reference frame F0
> simultaneously if you used the light pulses to synchronize your accelerations.

Huh? Your statement is insane, and contradicts everything that has been said (not just by me, but by you as well) in the previous messages. Again, the meaning of simultaneity for a given system of coordinates is well known and not at issue. Your Rube Goldberg contraptions add nothing to the specification of the scenario.

> You can only use physical measurement methods to determine times only if you
> start at rest in an inertial reference system, accelerate, and then both return to rest
> in the same inertial reference frame. In all other cases you must use "pure math".

That statement, too, is utterly insane, and bears no relation to what you've been told. Please re-read (or read) the previous message.

> So for example, if you and your twin are the identical ages, and are traveling side by
> side in identical rockets and your rockets are separated by 10 meters perpendicular
> to the direction of travel, you ages will remain the same throughout the trip, but if
> both of you simultaneously land on the equator of a planet that is rotating
> perpendicular to your direction of travel as soon as you land one of you will be
> older then the other.

As always, you mis-spoke, neglecting the relativity of simultaneity. Remember, at all times the proper ages of separate twins are unequal at equal values of the time coordinates of some systems of coordinates, so it is incorrect to refer to "the" comparison of ages on separate worldlines as if there is a unique mapping from one to the other. You are committing the Fontenot Fallacy, which is the erroneous belief that there is a unique simultaneity associated with separate objects in arbitrary states of motion. There is not. One must always specify the coordinates, and this entails no contradiction or ambiguity at all. The reason, within your brain, you arrive at what you think are contradictions is that you are making mistakes of simple math and logic.

Look, since your mental incapacity makes it impossible for you to draw a simple plot of position versus time, try just writing down the coordinates for the scenario you specified... without making any math mistakes.

For clarity, make all the accelerations abrupt impulses, with all the relevant speeds being +-v. There are then four events, e0 being the common origin, and e1 and e2 being the ends of the first leg at F0 time T. Then assume both instantly accelerate first to rest in F0 and then to rest in F1, and then as soon as possible they both accelerate simultaneously from rest in F1 to the speed +-v toward the midpoint in terms of F1. The last legs are therefore initiated at e1 and e3, and the twins re-unite at e4. Thus in terms of F0 we have x0=0, t0=0, and x1=-vT, t1=T, and x2=vT, t2=T, and x3=vT[(1+v^2)/(1-v^2)], t3=T[1+v^2)/(1-v^2)], and finally they re-unite at x4=vT[1+v^2)/(1-v^2)], t4=2T/(1-v^2). The extra dwell time for the right-hand twin is from e2 to e3.

So, what part of this don't you understand?

On Wednesday, September 29, 2021 at 4:26:04 PM UTC-5, Al Coe wrote:
> On Wednesday, September 29, 2021 at 1:04:57 PM UTC-7, sep...@yahoo.com wrote:
> > Okay, so what you are telling the twin is that this is "pure math" - you cannot
> > determine when your twin spent "extra dwell time" in F1 by making any physical
> > measurements.
> No, your statement is insane, and has no relation to what I explained to you. Again, Lorentz invariance is not pure math, it is physics, but you are trying to show that Lorentz invariance entails a logical inconsistency, by assuming Lorentz invariance and deducing a contradiction, so the questions you are asking are purely mathematical, and the mistakes you are making are purely mathematical.
>
> You didn't ask me how to empirically determine the dwell time, you asked me which of your two brain-dead operations determined the dwell time, and I informed you that neither of them did. You jump from this to claiming that I've said there is no way to determine the dwell time. Your thought processes are highly corrupted.
>
> > You and your twin did not [accelerate from rest in terms of] inertial reference frame F0
> > simultaneously if you used the light pulses to synchronize your accelerations.
> Huh? Your statement is insane, and contradicts everything that has been said (not just by me, but by you as well) in the previous messages. Again, the meaning of simultaneity for a given system of coordinates is well known and not at issue. Your Rube Goldberg contraptions add nothing to the specification of the scenario.
> > You can only use physical measurement methods to determine times only if you
> > start at rest in an inertial reference system, accelerate, and then both return to rest
> > in the same inertial reference frame. In all other cases you must use "pure math".
> That statement, too, is utterly insane, and bears no relation to what you've been told. Please re-read (or read) the previous message.
> > So for example, if you and your twin are the identical ages, and are traveling side by
> > side in identical rockets and your rockets are separated by 10 meters perpendicular
> > to the direction of travel, you ages will remain the same throughout the trip, but if
> > both of you simultaneously land on the equator of a planet that is rotating
> > perpendicular to your direction of travel as soon as you land one of you will be
> > older then the other.
> As always, you mis-spoke, neglecting the relativity of simultaneity. Remember, at all times the proper ages of separate twins are unequal at equal values of the time coordinates of some systems of coordinates, so it is incorrect to refer to "the" comparison of ages on separate worldlines as if there is a unique mapping from one to the other. You are committing the Fontenot Fallacy, which is the erroneous belief that there is a unique simultaneity associated with separate objects in arbitrary states of motion. There is not. One must always specify the coordinates, and this entails no contradiction or ambiguity at all. The reason, within your brain, you arrive at what you think are contradictions is that you are making mistakes of simple math and logic.
>
> Look, since your mental incapacity makes it impossible for you to draw a simple plot of position versus time, try just writing down the coordinates for the scenario you specified... without making any math mistakes.
>
> For clarity, make all the accelerations abrupt impulses, with all the relevant speeds being +-v. There are then four events, e0 being the common origin, and e1 and e2 being the ends of the first leg at F0 time T. Then assume both instantly accelerate first to rest in F0 and then to rest in F1, and then as soon as possible they both accelerate simultaneously from rest in F1 to the speed +-v toward the midpoint in terms of F1. The last legs are therefore initiated at e1 and e3, and the twins re-unite at e4. Thus in terms of F0 we have x0=0, t0=0, and x1=-vT, t1=T, and x2=vT, t2=T, and x3=vT[(1+v^2)/(1-v^2)], t3=T[1+v^2)/(1-v^2)], and finally they re-unite at x4=vT[1+v^2)/(1-v^2)], t4=2T/(1-v^2). The extra dwell time for the right-hand twin is from e2 to e3.
>
> So, what part of this don't you understand?

You stated:
"You didn't ask me how to empirically determine the dwell time, you asked me which of your two brain-dead operations determined the dwell time, and I informed you that neither of them did."

Instead of repeatedly doing the "pure math", please explain how one twin empirically determines the extra dwell time of the other twin.
David Seppala
Bastrop TX

On Wednesday, September 29, 2021 at 2:44:24 PM UTC-7, sep...@yahoo.com wrote:
> Instead of repeatedly doing the "pure math"...

The mathematical explanation explicitly falsifies your mathematical claim that special relativity entails a contradiction. Each time this is explained, you run away.

> Please explain how one twin empirically determines the extra dwell time of the other twin.

There are literally infinitely many ways of doing that. The canonical way would be to construct a grid of standard rulers and clocks at rest and inertially synchronized in any desired state of inertial motion. For example, we can construct one grid at rest in F0, and another at rest in F1, and as the twins move we can record their positions and times, and verify that they do indeed fulfill all the conditions you stipulated, and this experimental data shows that the right hand twin dwelled from e2 to e3 before beginning its last leg, and that e1 is simultaneous with e3 in terms of F1, and that e1 is simultaneous with e2 in terms of F0. And so on. All of these are empirical, measurable, objective facts. Do you understand this?

On Wednesday, September 29, 2021 at 5:25:53 PM UTC-5, Al Coe wrote:
> On Wednesday, September 29, 2021 at 2:44:24 PM UTC-7, sep...@yahoo.com wrote:
> > Instead of repeatedly doing the "pure math"...
>
> The mathematical explanation explicitly falsifies your mathematical claim that special relativity entails a contradiction. Each time this is explained, you run away.
>
> > Please explain how one twin empirically determines the extra dwell time of the other twin.
>
> There are literally infinitely many ways of doing that. The canonical way would be to construct a grid of standard rulers and clocks at rest and inertially synchronized in any desired state of inertial motion. For example, we can construct one grid at rest in F0, and another at rest in F1, and as the twins move we can record their positions and times, and verify that they do indeed fulfill all the conditions you stipulated, and this experimental data shows that the right hand twin dwelled from e2 to e3 before beginning its last leg, and that e1 is simultaneous with e3 in terms of F1, and that e1 is simultaneous with e2 in terms of F0. And so on. All of these are empirical, measurable, objective facts. Do you understand this?

You didn't show how one twin determines that the other twin spent more dwell time before accelerating to the midpoint between the twins in F1.
David Seppala
Bastrop TX

On Wednesday, September 29, 2021 at 3:48:16 PM UTC-7, sep...@yahoo.com wrote:
> You didn't show how one twin determines that the other twin spent more dwell
> time before accelerating to the midpoint between the twins in F1.

Yes I did. Again, the positions and times of each twin in terms of any system of coordinates can be measured and recorded by the respective grids of standard clocks and rulers. What better way of measuring the positions and times could there be?

At this point it isn't even clear what you are disputing. Are you claiming that the twins will be the same age when they re-unite? Or are you agreeing that they will be different ages, and the difference will be exactly in agreement with the prediction of special relativity, and exactly equal to the extra dwell time that I've described, but that there is something about the description of the intermediate events that you think is wrong? Or that one or the other of the twins will be unaware of the facts? Or are you claiming that F0 and F1 are not related by a Lorentz transformation? Or are you claiming something entirely different? You seem to be just raving incoherently, without being able to express any specific objection. You ask how we measure times and places.... I explain how we measure times and places... and you say I haven't told you how we measure times and places. Can you clarify your objection?

On Wednesday, September 29, 2021 at 3:48:16 PM UTC-7, sep...@yahoo.com wrote:
> You didn't show how one twin determines that the other twin spent more dwell
> time before accelerating to the midpoint between the twins in F1.

Yes I did. As I said, there are infinitely many ways... for example, the positions and times of each twin in terms of any system of coordinates can be measured and recorded by standard grids of standard rulers and clocks at rest and inertially synchronized in any given frame. What better way of measuring the positions and times could there be?

Another method would be for the left hand twin to use dead reckoning based on accelerometer readings to determine his own F0 speed continually, and by looking at light from a standard light source held by the other twin (whose initial position is known) he could measure the F0 speed of the other twin, and hence integrate the position for each event along the other twin's path.

These two methods -- and infinitely many others -- all give the same results. Do you understand this?

At this point it isn't even clear what you are disputing. Are you claiming that the twins will be the same age when they re-unite? Or are you saying there is something about the description of the intermediate events that you think is wrong? Or are you claiming that the trajectories of the twins are unmeasurable? Or that the twins would have no way of being aware of the objective facts? Or are you claiming that F0 and F1 are not related by a Lorentz transformation? Or... something else?

I'm honestly puzzled by your statements. You seem to be just complaining incoherently, without being able to express any specific objection. For example, you ask how we measure times and places, and when I explain how we measure times and places, you say I haven't told you how we measure times and places. Your statements seem bizarre and irrational. Can you clarify your objection?

On Thursday, 30 September 2021 at 06:54:05 UTC+2, Al Coe wrote:
> On Wednesday, September 29, 2021 at 3:48:16 PM UTC-7, sep...@yahoo.com wrote:
> > You didn't show how one twin determines that the other twin spent more dwell
> > time before accelerating to the midpoint between the twins in F1.
> Yes I did. As I said, there are infinitely many ways... for example, the positions and times of each twin in terms of any system of coordinates can be measured and recorded by standard grids of standard rulers and clocks at rest and inertially synchronized in any given frame. What better way of measuring the positions and times could there be?
>
> Another method would be for the left hand twin to use dead reckoning based on accelerometer readings to determine his own F0 speed continually, and by looking at light from a standard light source held by the other twin (whose initial position is known) he could measure the F0 speed of the other twin, and hence integrate the position for each event along the other twin's path.

In the meantime in the real world, however, GPS clocks measure t'=t, just
like all serious clocks always did.

Den 29.09.2021 16:22, skrev sepp623@yahoo.com:
> Two twins are at rest in F0 at x=0. At time t=0 the two twins accelerate to V0 wrt to F0 and then decelerate back to zero velocity relative to F0 in identical rockets. One rocket moved one twin a distance of L0 in the positive x direction in T0 seconds (as measured by observers in F0), while the other identical rocket moved the other twin a distance of L0 in the negative x direction in T0 seconds (as measured by observers in F0). The twins are now separated a distance of 2L0 as measured in F0 and each twin has zero relative velocity wrt to F0.
> Then both twins accelerate in the positive x direction in identical rockets starting their accelerations simultaneously as measured by observers in F0. Both twins accelerate for an equal amount of time and then come to rest in another inertial reference frame F1. In inertial reference frame F1, the twins are separated by a distance of L1 as measured by observers in F1 and have zero velocity wrt F1.
> The twins then decide to see each other, so as measured by observers in F1, the twins simultaneously accelerate and decelerate to the midpoint between them, a distance of L1 in identical times as measured by observers in F1.
> The question I have is after the final leg of this journey, where the twins are at the same point in space, each with zero relative velocity wrt to F1 and to each other, are the twins the same age? In terms of F1, once the twins did the first leg of the journey and they had a separation of 2L0, the twins no longer were the same age. From the measurements by observers in F1, the twins each aged an identical amount during the last two legs of the journey, so therefore observers in F1 would say the twins do not have the same age when they meet in F1.
> From the viewpoint of the twins, they both started at the same point in space and time, they both accelerated and decelerated in an identical amount of time during the first leg of the journey, they both accelerated an equal amount of time during the second leg of the journey, and they both accelerated and decelerated an equal amount of time during the last leg of the journey, and all accelerations where simultaneous from the twin's point of view, therefore they should have the identical ages when they are at rest at the same point in space where they meet in F1.
> Which answer is correct?
> Thanks,
> David Seppala
> Bastrop tX
>

You can find the necessary equations to calculate
the answers to your questions here:

https://paulba.no/pdf/TwinsByMetric.pdf

You will have to define your scenario more precisely, though.

--
Paul

https://paulba.no/

On Thursday, 30 September 2021 at 12:26:42 UTC+2, Paul B. Andersen wrote:
> Den 29.09.2021 16:22, skrev sep...@yahoo.com:
> > Two twins are at rest in F0 at x=0. At time t=0 the two twins accelerate to V0 wrt to F0 and then decelerate back to zero velocity relative to F0 in identical rockets. One rocket moved one twin a distance of L0 in the positive x direction in T0 seconds (as measured by observers in F0), while the other identical rocket moved the other twin a distance of L0 in the negative x direction in T0 seconds (as measured by observers in F0). The twins are now separated a distance of 2L0 as measured in F0 and each twin has zero relative velocity wrt to F0.
> > Then both twins accelerate in the positive x direction in identical rockets starting their accelerations simultaneously as measured by observers in F0. Both twins accelerate for an equal amount of time and then come to rest in another inertial reference frame F1. In inertial reference frame F1, the twins are separated by a distance of L1 as measured by observers in F1 and have zero velocity wrt F1.
> > The twins then decide to see each other, so as measured by observers in F1, the twins simultaneously accelerate and decelerate to the midpoint between them, a distance of L1 in identical times as measured by observers in F1.
> > The question I have is after the final leg of this journey, where the twins are at the same point in space, each with zero relative velocity wrt to F1 and to each other, are the twins the same age? In terms of F1, once the twins did the first leg of the journey and they had a separation of 2L0, the twins no longer were the same age. From the measurements by observers in F1, the twins each aged an identical amount during the last two legs of the journey, so therefore observers in F1 would say the twins do not have the same age when they meet in F1.
> > From the viewpoint of the twins, they both started at the same point in space and time, they both accelerated and decelerated in an identical amount of time during the first leg of the journey, they both accelerated an equal amount of time during the second leg of the journey, and they both accelerated and decelerated an equal amount of time during the last leg of the journey, and all accelerations where simultaneous from the twin's point of view, therefore they should have the identical ages when they are at rest at the same point in space where they meet in F1.
> > Which answer is correct?
> > Thanks,
> > David Seppala
> > Bastrop tX
> >
> You can find the necessary equations to calculate
> the answers to your questions here:
>
> https://paulba.no/pdf/TwinsByMetric.pdf
>
> You will have to define your scenario more precisely, though.

In the meantime in the real world, of course, the clocks
of GPS will keep indicating t'=t, just like all serious clocks
always did.

On Wednesday, September 29, 2021 at 11:54:05 PM UTC-5, Al Coe wrote:
> On Wednesday, September 29, 2021 at 3:48:16 PM UTC-7, sep...@yahoo.com wrote:
> > You didn't show how one twin determines that the other twin spent more dwell
> > time before accelerating to the midpoint between the twins in F1.
> Yes I did. As I said, there are infinitely many ways... for example, the positions and times of each twin in terms of any system of coordinates can be measured and recorded by standard grids of standard rulers and clocks at rest and inertially synchronized in any given frame. What better way of measuring the positions and times could there be?
Okay, let's use a grid system of coordinates using standard rulers and clocks at rest in F0. When the twins are x=0, they have the same age. When each twin accelerates and decelerates in the identical fashion but in opposite directions, one twin ends up with zero velocity relative to F0 in T1 seconds at x = -L1, while the other twin ends up with zero velocity relative to F0 in T1 seconds at x = L1. When twin1 starts accelerating in the positive x direction at time T1, using the grid in F0, he reaches zero velocity relative to F1 in T2 seconds and travels a distance L2. When twin 2 starts accelerating in the positive x direction at time T1, using the grid in F0, twin 2 reaches zero velocity relative to F1 in T2 seconds. Using the F0 grid, both twins then start accelerating toward each other at time T1+T2.
How do the F0 measurements show that one twin dwelled a bit before starting his acceleration compared to the other twin?
David Seppala
Bastrop TX

Twins were the biggest botch of relativity.
Order is never a contradiction to itself...

Mitchell Raemsch

On Thursday, September 30, 2021 at 10:43:59 AM UTC-7, sep...@yahoo.com wrote:
> Let's use a grid system of coordinates using standard rulers and clocks at rest in F0.
> When the twins are x=0, they have the same age. When each twin accelerates and
> decelerates in the identical fashion but in opposite directions, one twin ends up with
> zero velocity relative to F0 in T1 seconds at x = -L1, while the other twin ends up with
> zero velocity relative to F0 in T1 seconds at x = L1. When twin1 starts accelerating in t
> he positive x direction at time T1, using the grid in F0, he reaches zero velocity relative
> to F1 in T2 seconds and travels a distance L2. When twin 2 starts accelerating in the
> positive x direction at time T1, using the grid in F0, twin 2 reaches zero velocity relative
> to F1 in T2 seconds.

Yes, so far that's consistent with the scenario you specified.

> Using the F0 grid, both twins then start accelerating toward each other at time T1+T2.

No, in your scenario you specified that the twins begin accelerating toward each other simultaneously in terms of F1, not simultaneously in terms of F0. That's why twin2 needs to wait until he is at the same t' coordinate as twin1 is when he starts his final acceleration. Remember?

> How do the F0 measurements show that one twin dwelled a bit before starting his
> acceleration compared to the other twin?

They show it by showing it. Duh. Twin2 reaches rest in terms of F1 at the F0-time T1+T2, and it must then (to comply with your specification) wait until the later value of the F0-time at which it has the same F1-time as twin1 when he is at F0-time T1+T2. This was all shown to you explicitly before, and you contemptuously disregarded it, remember?

Is there something about this you think is wrong or unclear?

On Thursday, September 30, 2021 at 11:19:04 AM UTC-7, Al Coe wrote:
> On Thursday, September 30, 2021 at 10:43:59 AM UTC-7, sep...@yahoo.com wrote:
> > Let's use a grid system of coordinates using standard rulers and clocks at rest in F0.
> > When the twins are x=0, they have the same age. When each twin accelerates and
> > decelerates in the identical fashion but in opposite directions, one twin ends up with
> > zero velocity relative to F0 in T1 seconds at x = -L1, while the other twin ends up with
> > zero velocity relative to F0 in T1 seconds at x = L1. When twin1 starts accelerating in t
> > he positive x direction at time T1, using the grid in F0, he reaches zero velocity relative
> > to F1 in T2 seconds and travels a distance L2. When twin 2 starts accelerating in the
> > positive x direction at time T1, using the grid in F0, twin 2 reaches zero velocity relative
> > to F1 in T2 seconds.
> Yes, so far that's consistent with the scenario you specified.
> > Using the F0 grid, both twins then start accelerating toward each other at time T1+T2.
> No, in your scenario you specified that the twins begin accelerating toward each other simultaneously in terms of F1, not simultaneously in terms of F0. That's why twin2 needs to wait until he is at the same t' coordinate as twin1 is when he starts his final acceleration. Remember?
> > How do the F0 measurements show that one twin dwelled a bit before starting his
> > acceleration compared to the other twin?
> They show it by showing it. Duh. Twin2 reaches rest in terms of F1 at the F0-time T1+T2, and it must then (to comply with your specification) wait until the later value of the F0-time at which it has the same F1-time as twin1 when he is at F0-time T1+T2. This was all shown to you explicitly before, and you contemptuously disregarded it, remember?
>
> Is there something about this you think is wrong or unclear?

If twins separate at the same relative speed
they ought to share the time.

Mitchell Raemsch

On Thursday, September 30, 2021 at 1:19:04 PM UTC-5, Al Coe wrote:
> On Thursday, September 30, 2021 at 10:43:59 AM UTC-7, sep...@yahoo.com wrote:
> > Let's use a grid system of coordinates using standard rulers and clocks at rest in F0.
> > When the twins are x=0, they have the same age. When each twin accelerates and
> > decelerates in the identical fashion but in opposite directions, one twin ends up with
> > zero velocity relative to F0 in T1 seconds at x = -L1, while the other twin ends up with
> > zero velocity relative to F0 in T1 seconds at x = L1. When twin1 starts accelerating in t
> > he positive x direction at time T1, using the grid in F0, he reaches zero velocity relative
> > to F1 in T2 seconds and travels a distance L2. When twin 2 starts accelerating in the
> > positive x direction at time T1, using the grid in F0, twin 2 reaches zero velocity relative
> > to F1 in T2 seconds.
> Yes, so far that's consistent with the scenario you specified.
> > Using the F0 grid, both twins then start accelerating toward each other at time T1+T2.
> No, in your scenario you specified that the twins begin accelerating toward each other simultaneously in terms of F1, not simultaneously in terms of F0. That's why twin2 needs to wait until he is at the same t' coordinate as twin1 is when he starts his final acceleration. Remember?
> > How do the F0 measurements show that one twin dwelled a bit before starting his
> > acceleration compared to the other twin?
> They show it by showing it. Duh. Twin2 reaches rest in terms of F1 at the F0-time T1+T2, and it must then (to comply with your specification) wait until the later value of the F0-time at which it has the same F1-time as twin1 when he is at F0-time T1+T2. This was all shown to you explicitly before, and you contemptuously disregarded it, remember?
>
> Is there something about this you think is wrong or unclear?

It is unclear to me why the twins, once they are both at rest in F1, say that one twin arrived there first. Didn't both twins say they arrived simultaneously at time T1+T2. Please explain how the twins came to that conclusion that one twin arrived first.
David Seppala
Bastrop TX

On Thursday, September 30, 2021 at 12:27:56 PM UTC-7, sep...@yahoo.com wrote:
> It is unclear to me why the twins, once they are both at rest in F1, say that one twin
> arrived there first. Didn't both twins say they arrived simultaneously at time T1+T2?

They arrived simultaneously in terms of F0, but not simultaneously in terms of F1. This is due to the relativity of simultaneity.

> Please explain how the twins came to that conclusion that one twin arrived first.

As explained before, we can construct a standard grid of standard rulers and clocks at rest and inertially synchronized in terms of any frame we like, including F0 and F1. By looking at the readings of these two coordinate systems, we find that the twins reached rest in F1 at equal values of the F0 time coordinate, but at unequal values of the F1-time coordinate. We read these results directly from the clocks in the respective grids. There is no ambiguity about this. In general, we find by comparing the positions and times in the two grids that the F0 coordinates are related to the F1 coordinates by a Lorentz transformation. (This corresponds to the physical fact that all forms of energy -- including kinetic energy -- have inertia.)

Do you think anything about this is wrong or unclear?

On Thursday, September 30, 2021 at 2:41:11 PM UTC-5, Al Coe wrote:
> On Thursday, September 30, 2021 at 12:27:56 PM UTC-7, sep...@yahoo.com wrote:
> > It is unclear to me why the twins, once they are both at rest in F1, say that one twin
> > arrived there first. Didn't both twins say they arrived simultaneously at time T1+T2?
>
> They arrived simultaneously in terms of F0, but not simultaneously in terms of F1. This is due to the relativity of simultaneity.
> > Please explain how the twins came to that conclusion that one twin arrived first.
> As explained before, we can construct a standard grid of standard rulers and clocks at rest and inertially synchronized in terms of any frame we like, including F0 and F1. By looking at the readings of these two coordinate systems, we find that the twins reached rest in F1 at equal values of the F0 time coordinate, but at unequal values of the F1-time coordinate. We read these results directly from the clocks in the respective grids. There is no ambiguity about this. In general, we find by comparing the positions and times in the two grids that the F0 coordinates are related to the F1 coordinates by a Lorentz transformation. (This corresponds to the physical fact that all forms of energy -- including kinetic energy -- have inertia.)
>
> Do you think anything about this is wrong or unclear?

Explain why, if the twins have zero velocity in any inertial reference system, why can't the twins use a mirror at the midpoint between them with one twin sending a pulse toward the other twin, and one twin getting part of the pulse reflected back to him after being reflected off the mirror to determine if two events are simultaneous from the twin's point of view. Please explain the restrictions the twins must follow before that method of determining a simultaneous event can be used by the twins.
David Seppala
Bastrop TX

On Thursday, September 30, 2021 at 12:54:28 PM UTC-7, sep...@yahoo.com wrote:
> Explain why, if the twins have zero velocity in any inertial reference system,
> why can't the twins use a mirror ... to determine if two events are simultaneous
> from the twin's point of view.

Again, there is no difficulty determining the times of separate events in terms of any specified system of coordinates. In the situation you just described, you are stipulating that both twins are mutually at rest in terms of the same system of coordinates throughout the process, so it's trivial. However, in the actual scenario that you specified we have one twin reaching rest in F1 while (at that same F1-time) the other twin is not yet at rest in F1. To carry out the mirror method of determining simultaneity in terms of the common rest frame of the twins they must have a common rest frame (duh), meaning they must be at rest in the same frame, which in your specified scenario they are not, until the left hand twin reaches rest in F1, which requires that the right hand twin dwell for a specific amount of time, since it reached rest in F1 at an earlier F1-time. That's why So all your ranting about that Rube Goldberg method is brain-dead and pointless. Do you understand this?

> Please explain the restrictions the twins must follow before that method of
> determining a simultaneous event can be used by the twins.

They must (throughout the process) both be at rest in a single frame, in which case they can determine simultaneity in that frame in that way. Of course, this isn't the only way of determining simultaneity in terms of F0, or in terms of F1, or any other. There is no restriction on the objects. We simply construct a standard grid of rulers and clocks, as explained previously, and then two events are simultaneous in terms of those coordinates if and only if they have the same time coordinate.

In the case under discussion, the twins are not both at rest in F1 in terms of F1-time until the right hand twin has dwelled for a specific amount of time after reaching rest in F1.

Do you understand now? Is anything about this still unclear to you?

On Thursday, September 30, 2021 at 3:16:52 PM UTC-5, Al Coe wrote:
> On Thursday, September 30, 2021 at 12:54:28 PM UTC-7, sep...@yahoo.com wrote:
> > Explain why, if the twins have zero velocity in any inertial reference system,
> > why can't the twins use a mirror ... to determine if two events are simultaneous
> > from the twin's point of view.
> Again, there is no difficulty determining the times of separate events in terms of any specified system of coordinates. In the situation you just described, you are stipulating that both twins are mutually at rest in terms of the same system of coordinates throughout the process, so it's trivial. However, in the actual scenario that you specified we have one twin reaching rest in F1 while (at that same F1-time) the other twin is not yet at rest in F1. To carry out the mirror method of determining simultaneity in terms of the common rest frame of the twins they must have a common rest frame (duh), meaning they must be at rest in the same frame, which in your specified scenario they are not, until the left hand twin reaches rest in F1, which requires that the right hand twin dwell for a specific amount of time, since it reached rest in F1 at an earlier F1-time. That's why So all your ranting about that Rube Goldberg method is brain-dead and pointless. Do you understand this?
> > Please explain the restrictions the twins must follow before that method of
> > determining a simultaneous event can be used by the twins.
> They must (throughout the process) both be at rest in a single frame, in which case they can determine simultaneity in that frame in that way. Of course, this isn't the only way of determining simultaneity in terms of F0, or in terms of F1, or any other. There is no restriction on the objects. We simply construct a standard grid of rulers and clocks, as explained previously, and then two events are simultaneous in terms of those coordinates if and only if they have the same time coordinate.
>
> In the case under discussion, the twins are not both at rest in F1 in terms of F1-time until the right hand twin has dwelled for a specific amount of time after reaching rest in F1.
>
> Do you understand now? Is anything about this still unclear to you?
Are both twins at rest in F1 in terms of the twins measurements we just discussed when their simultaneous times are T1+T2? Or do the twins now say, the accelerations were different and we did not reach zero velocity with respect to F1 simultaneously?
Thanks,
David Seppala
Bastrop TX