Here are two very similar scenarios but they seem to have contradictory results.
Scenario 1:
There is an inertial reference frame F0. Two identical twins have identical clocks. The two twins and clocks have a relative velocity V along the x-axis relative to each other. Relative to frame F0, they have equal speeds with one twin traveling in the negative x direction and other twin traveling in the positive x direction. When they pass each other their clocks are set to zero and the twins are the identical age. When each of their clocks reads T seconds, they decelerate at identical rates over a very short period of time until they each have zero velocity relative to each other and relative to F0. Both twins now at rest in F0 are the identical age and both of their clocks are in sync with the identical times as measured in frame F0.

Scenario 2:
There is an inertial reference frame F0. Two identical twins have identical clocks. They have a relative velocity V along the x-axis relative to each other but twin 1 has zero velocity relative to F0 and twin 2 has a velocity V relative to frame F0. When the twins pass each other their clocks are set to zero and the twins are the identical age. When the clock of twin 2 reads T seconds, that twin decelerates over a very short period of time until he has zero velocity relative to twin 1 and zero velocity relative to F0. Now the twins are of different ages and their clocks show different times as measured in frame F0.

In the two scenarios, the only thing different was the accelerations of the twins. The relative velocity between the twins in both scenarios were identical. Obviously in the first scenario, the accelerations, being identical rates did not create a difference in ages of the twins and did not cause their clocks to get out of sync and end up with different times in F0. Many who will try to explain the difference in outcomes of the two scenario will say in the second scenario the acceleration of twin 2 resulted in the age difference between the twins and resulted in their clocks no longer being in sync in frame F0. But there are two arguments that show that the acceleration of twin 2 cannot explain the age difference of the twins in scenario 2, with one of the arguments easy to demonstrate experimentally.

The first argument is simple. I'm going to use large numbers in this explanation but the same logic applies with smaller numbers. Let's say that in scenario 2, that when twin 2 decelerates so that he has zero velocity relative to F0, his age is one year different than the age of twin 1. Observers in F0 say that the deceleration had no significant affect on the age of twin 2 - there was already about a one year age difference between the two twins when twin 2 started his deceleration and their clocks were already out of sync and showed a one year difference in times. They also note that if the deceleration occurred 1 second after the twins initially meet, there would not be a one year difference in the ages of the two twins or of their clocks. So the deceleration of twin 2 had virtually no effect on the age difference of the twins.

The second argument we can actually do experimentally and verify that the acceleration has virtually zero effect on the age difference of the twins in scenario 2. Many physicists teach that relativistic effects can only be observed with relativistic velocities. But as we all know after studying the Lorentz transform equations that relativistic effects can be easily observed with everyday velocities if relativistic lengths are involved. So again, let the twins have a 1 year age difference when twin 2 starts his deceleration. But let the relative velocity between the two twins be say 3 meters per second. Granted those numbers might not be possible if the universe is finite, but if it isn't we can test for ourselves how the deceleration from 3 meters per second affects the time shown on twin 2's clock. I think everyone who tries to change the velocity of a clock by 3 meters per second say using 1 g that the time shown on the clock would not show a significant change and could not explain a one year difference in the age of the twins or the time shown on their clocks.

So why in scenario 1 do the clocks of the two twins remain in sync and show identical times no matter what the separation between the clocks is when they decelerate into frame F0 in scenario 1 while in scenario 2 the two clocks show vastly different times and depend on the separation distance between the twins when twin 2 starts his deceleration?

David Seppala
Bastrop TX

On Monday, August 9, 2021 at 9:31:04 AM UTC-7, sep...@yahoo.com wrote:
> [more of your misconceptions]

Aren't you forgetting something? Your misconceptions were being cleared up in the other thread, and instead of trying to reach clarity and understanding, you abandon that thread and start a new one, going back to square one. You will never be able to make any progress if you continually run away from the explanations of your misconceptions.

> Here are two very similar scenarios but they seem to have contradictory results.

There are no contradictory results in special relativity. You have tried 800 times to find contradictory results from applying a simple linear transformation of coordinates, and 800 times you have failed, because it is mathematically impossible for a simple linear transformation of coordinates to yield any logical contradiction. Let's look at your 801st attempt:

> Scenario 1:
> There is an inertial reference frame F0. Two identical twins have identical clocks. The two twins and clocks have a relative velocity V along the x-axis relative to each other. Relative to frame F0, they have equal speeds with one twin traveling in the negative x direction and other twin traveling in the positive x direction. When they pass each other their clocks are set to zero and the twins are the identical age. When each of their clocks reads T seconds, they decelerate at identical rates over a very short period of time until they each have zero velocity relative to each other and relative to F0. Both twins now at rest in F0 are the identical age and both of their clocks are in sync with the identical times as measured in frame F0.

You are stipulating that the twins are identical in proper age when they cross paths, but you need to keep in mind that this wouldn't happen by accident, you would need to ensure that they both had the same elapsed proper time from their moment of birth until the moment when they cross paths, i.e., they need to have histories of motion their entire lives to yield this equality. You can't just set their proper ages equal to each other like you can set clocks equal to each other.

> Scenario 2:
> There is an inertial reference frame F0. Two identical twins have identical clocks. They have a relative velocity V along the x-axis relative to each other but twin 1 has zero velocity relative to F0 and twin 2 has a velocity V relative to frame F0. When the twins pass each other their clocks are set to zero and the twins are the identical age. When the clock of twin 2 reads T seconds, that twin decelerates over a very short period of time until he has zero velocity relative to twin 1 and zero velocity relative to F0. Now the twins are of different ages and their clocks show different times as measured in frame F0.

Right. Nothing contradictory about any of this. (Again with the caution about equating the twins proper ages when they cross paths, which is even more acute in this example, since it is asymmetrical, so you would need to have carefully pre-compensated their motions and proper ages to arrange for this above scenario to be realized... possible in principle but not easy.)

> Many who will try to explain the difference in outcomes of the two scenario
> will say....

There is not point in constructing stupid straw men about what "many will say". What special relativity says is that the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2]. And inertial coordinate systems are related by x'=(x-vt)g, t'=(t-vx/c^2)g. That is all.. Do you understand that this accounts for both of your scenarios, and all other scenarios as well?

> There are two arguments that show that the acceleration of twin 2 cannot explain the age
> difference of the twins in scenario 2, with one of the arguments easy to demonstrate experimentally.

You are wasting your time arguing against straw men. If you want to criticize special relativity you need to address special relativity, not some nitwit straw men.

> Why in scenario 1 do the clocks of the two twins remain in sync and show identical
> times no matter what the separation between the clocks is when they decelerate into
> frame F0 in scenario 1...

Already explained above. The metric of spacetime is Minkowskian, which signifies that the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2], and of course inertial coordinate systems are related by x'=(x-vt)g, t'=(t-vx/c^2)g.

> Why in scenario 2 the two clocks show vastly different times and depend on the
> separation distance between the twins when twin 2 starts his deceleration?

Again, the metric of spacetime is Minkowskian, meaning the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2], and of course inertial coordinate systems are related by x'=(x-vt)g, t'=(t-vx/c^2)g.

Special Relativity: 801 ...... Barnpole Dave: 0

On Monday, August 9, 2021 at 9:31:04 AM UTC-7, sep...@yahoo.com wrote:
> Here are two very similar scenarios but they seem to have contradictory results.
> Scenario 1:
> There is an inertial reference frame F0. Two identical twins have identical clocks. The two twins and clocks have a relative velocity V along the x-axis relative to each other. Relative to frame F0, they have equal speeds with one twin traveling in the negative x direction and other twin traveling in the positive x direction. When they pass each other their clocks are set to zero and the twins are the identical age. When each of their clocks reads T seconds, they decelerate at identical rates over a very short period of time until they each have zero velocity relative to each other and relative to F0. Both twins now at rest in F0 are the identical age and both of their clocks are in sync with the identical times as measured in frame F0.
>
> Scenario 2:
> There is an inertial reference frame F0. Two identical twins have identical clocks. They have a relative velocity V along the x-axis relative to each other but twin 1 has zero velocity relative to F0 and twin 2 has a velocity V relative to frame F0. When the twins pass each other their clocks are set to zero and the twins are the identical age. When the clock of twin 2 reads T seconds, that twin decelerates over a very short period of time until he has zero velocity relative to twin 1 and zero velocity relative to F0. Now the twins are of different ages and their clocks show different times as measured in frame F0.
>
> In the two scenarios, the only thing different was the accelerations of the twins. The relative velocity between the twins in both scenarios were identical. Obviously in the first scenario, the accelerations, being identical rates did not create a difference in ages of the twins and did not cause their clocks to get out of sync and end up with different times in F0. Many who will try to explain the difference in outcomes of the two scenario will say in the second scenario the acceleration of twin 2 resulted in the age difference between the twins and resulted in their clocks no longer being in sync in frame F0. But there are two arguments that show that the acceleration of twin 2 cannot explain the age difference of the twins in scenario 2, with one of the arguments easy to demonstrate experimentally.
>
> The first argument is simple. I'm going to use large numbers in this explanation but the same logic applies with smaller numbers. Let's say that in scenario 2, that when twin 2 decelerates so that he has zero velocity relative to F0, his age is one year different than the age of twin 1. Observers in F0 say that the deceleration had no significant affect on the age of twin 2 - there was already about a one year age difference between the two twins when twin 2 started his deceleration and their clocks were already out of sync and showed a one year difference in times. They also note that if the deceleration occurred 1 second after the twins initially meet, there would not be a one year difference in the ages of the two twins or of their clocks. So the deceleration of twin 2 had virtually no effect on the age difference of the twins.
>
> The second argument we can actually do experimentally and verify that the acceleration has virtually zero effect on the age difference of the twins in scenario 2. Many physicists teach that relativistic effects can only be observed with relativistic velocities. But as we all know after studying the Lorentz transform equations that relativistic effects can be easily observed with everyday velocities if relativistic lengths are involved. So again, let the twins have a 1 year age difference when twin 2 starts his deceleration. But let the relative velocity between the two twins be say 3 meters per second. Granted those numbers might not be possible if the universe is finite, but if it isn't we can test for ourselves how the deceleration from 3 meters per second affects the time shown on twin 2's clock. I think everyone who tries to change the velocity of a clock by 3 meters per second say using 1 g that the time shown on the clock would not show a significant change and could not explain a one year difference in the age of the twins or the time shown on their clocks.
>
> So why in scenario 1 do the clocks of the two twins remain in sync and show identical times no matter what the separation between the clocks is when they decelerate into frame F0 in scenario 1 while in scenario 2 the two clocks show vastly different times and depend on the separation distance between the twins when twin 2 starts his deceleration?
>
> David Seppala
> Bastrop TX

If both twins see the other's clock going slow
would they not have the same clock?

On Monday, August 9, 2021 at 9:31:04 AM UTC-7, sep...@yahoo.com wrote:
> Many who will try to explain the difference in outcomes of the two scenario will say....

There's not point in constructing stupid straw men about what "many will say". What special relativity says is that the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2]. That is all. Is there something about this you don't understand?

> There are two arguments that show that the acceleration of twin 2 cannot explain the age
> difference of the twins in scenario 2, with one of the arguments easy to demonstrate experimentally.

You're wasting your time arguing against straw men. If you want to criticize special relativity you need to address special relativity, not some nitwit straw men.

> Why in scenario 1 do the clocks of the two twins remain in sync and show identical
> times no matter what the separation between the clocks is when they decelerate into
> frame F0 in scenario 1...

Already explained above. The metric of spacetime is Minkowskian, which signifies that the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2].

> Why in scenario 2 the two clocks show vastly different times and depend on the
> separation distance between the twins when twin 2 starts his deceleration?

Again, the metric of spacetime is Minkowskian, meaning the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2].

special relativity: 801 ....... Barnpole Dave: 0

On Monday, August 9, 2021 at 12:48:34 PM UTC-5, Al Coe wrote:
> On Monday, August 9, 2021 at 9:31:04 AM UTC-7, sep...@yahoo.com wrote:
> > Many who will try to explain the difference in outcomes of the two scenario will say....
>
> There's not point in constructing stupid straw men about what "many will say". What special relativity says is that the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2]. That is all. Is there something about this you don't understand?
>
> > There are two arguments that show that the acceleration of twin 2 cannot explain the age
> > difference of the twins in scenario 2, with one of the arguments easy to demonstrate experimentally.
> You're wasting your time arguing against straw men. If you want to criticize special relativity you need to address special relativity, not some nitwit straw men.
>
> > Why in scenario 1 do the clocks of the two twins remain in sync and show identical
> > times no matter what the separation between the clocks is when they decelerate into
> > frame F0 in scenario 1...
>
> Already explained above. The metric of spacetime is Minkowskian, which signifies that the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2].
>
> > Why in scenario 2 the two clocks show vastly different times and depend on the
> > separation distance between the twins when twin 2 starts his deceleration?
> Again, the metric of spacetime is Minkowskian, meaning the elapsed proper time along a linear path that moves a spatial distance dx in time dt for *any* given system of inertial coordinates x,t is sqrt[dt^2 - (dx/c)^2].
>
> special relativity: 801 ....... Barnpole Dave: 0

Al
Why are the two twins the same age at the end in scenario 1 and have an age difference of 1 year at the end of scenario 2. Explain why the acceleration differences at the end of each scenario account for the difference in ages of the twins. That was the only difference in the two scenarios.

David Seppala
Bastrop TX

On Monday, August 9, 2021 at 12:40:35 PM UTC-7, sep...@yahoo.com wrote:
> Why are the two twins the same age at the end in scenario 1 and have an
> age difference of 1 year at the end of scenario 2.

Again, in Scenario 1, in terms of the system S0 of inertial coordinates, you have specified that each twin goes a spatial distance of +dx or -dx in the span of time dt, so the elapsed proper times both equal sqrt[(dt)^2 - (dx/c)^2]. In Scenario 2, in terms of the same system S0, you have specified that one twin traverses dx=0 in the time dt, and the other traverses some dx > 0 in the time dt, so the elapsed proper times are dt and sqrt[dt^2 - (dx/c)^2] respectively.

> Explain why the acceleration...

If you can find the word "acceleration" in the above answer to your question, I will explain it.

special relativity: 802 ...... Barnpole Dave: 0

On Monday, August 9, 2021 at 2:59:43 PM UTC-5, Al Coe wrote:
> On Monday, August 9, 2021 at 12:40:35 PM UTC-7, sep...@yahoo.com wrote:
> > Why are the two twins the same age at the end in scenario 1 and have an
> > age difference of 1 year at the end of scenario 2.
> Again, in Scenario 1, in terms of the system S0 of inertial coordinates, you have specified that each twin goes a spatial distance of +dx or -dx in the span of time dt, so the elapsed proper times both equal sqrt[(dt)^2 - (dx/c)^2]. In Scenario 2, in terms of the same system S0, you have specified that one twin traverses dx=0 in the time dt, and the other traverses some dx > 0 in the time dt, so the elapsed proper times are dt and sqrt[dt^2 - (dx/c)^2] respectively.
>
> > Explain why the acceleration...
>
> If you can find the word "acceleration" in the above answer to your question, I will explain it.
>
> special relativity: 802 ...... Barnpole Dave: 0

You use the term sqrt[dt^2 - (dx/c)^2] so explain why the acceleration term is a function of the extended time and not just the duration of the acceleration.
David Seppala
Bastrop TX

On Monday, August 9, 2021 at 2:40:15 PM UTC-7, sep...@yahoo.com wrote:
> > > Why are the two twins the same age at the end in scenario 1 and have an
> > > age difference of 1 year at the end of scenario 2.
> > Again, in Scenario 1, in terms of the system S0 of inertial coordinates, you have specified that each twin goes a spatial distance of +dx or -dx in the span of time dt, so the elapsed proper times both equal sqrt[(dt)^2 - (dx/c)^2]. In Scenario 2, in terms of the same system S0, you have specified that one twin traverses dx=0 in the time dt, and the other traverses some dx > 0 in the time dt, so the elapsed proper times are dt and sqrt[dt^2 - (dx/c)^2] respectively.
> >
> > > Explain why the acceleration...
> >
> > If you can find the word "acceleration" in the above answer to your question, I will explain it.
> >
> You use the term sqrt[dt^2 - (dx/c)^2] so explain why the acceleration term is a function of
> the extended time and not just the duration of the acceleration.

There is no acceleration term. Can you state which term you think is an "acceleration term"?

Special relativity: 803 ...... Barnpole Dave: 0

On Monday, August 9, 2021 at 4:56:13 PM UTC-5, Al Coe wrote:
> On Monday, August 9, 2021 at 2:40:15 PM UTC-7, sep...@yahoo.com wrote:
> > > > Why are the two twins the same age at the end in scenario 1 and have an
> > > > age difference of 1 year at the end of scenario 2.
> > > Again, in Scenario 1, in terms of the system S0 of inertial coordinates, you have specified that each twin goes a spatial distance of +dx or -dx in the span of time dt, so the elapsed proper times both equal sqrt[(dt)^2 - (dx/c)^2]. In Scenario 2, in terms of the same system S0, you have specified that one twin traverses dx=0 in the time dt, and the other traverses some dx > 0 in the time dt, so the elapsed proper times are dt and sqrt[dt^2 - (dx/c)^2] respectively.
> > >
> > > > Explain why the acceleration...
> > >
> > > If you can find the word "acceleration" in the above answer to your question, I will explain it.
> > >
> > You use the term sqrt[dt^2 - (dx/c)^2] so explain why the acceleration term is a function of
> > the extended time and not just the duration of the acceleration.
> There is no acceleration term. Can you state which term you think is an "acceleration term"?
>
> Special relativity: 803 ...... Barnpole Dave: 0
Thanks,
David Seppala
Bastrop TX

sepp623@yahoo.com wrote:
> On Monday, August 9, 2021 at 4:56:13 PM UTC-5, Al Coe wrote:
>> On Monday, August 9, 2021 at 2:40:15 PM UTC-7, sep...@yahoo.com wrote:
>>>>> Why are the two twins the same age at the end in scenario 1 and have an
>>>>> age difference of 1 year at the end of scenario 2.
>>>> Again, in Scenario 1, in terms of the system S0 of inertial coordinates, you have specified that each twin goes a spatial distance of +dx or -dx in the span of time dt, so the elapsed proper times both equal sqrt[(dt)^2 - (dx/c)^2]. In Scenario 2, in terms of the same system S0, you have specified that one twin traverses dx=0 in the time dt, and the other traverses some dx > 0 in the time dt, so the elapsed proper times are dt and sqrt[dt^2 - (dx/c)^2] respectively.
>>>>
>>>>> Explain why the acceleration...
>>>>
>>>> If you can find the word "acceleration" in the above answer to your question, I will explain it.
>>>>
>>> You use the term sqrt[dt^2 - (dx/c)^2] so explain why the acceleration term is a function of
>>> the extended time and not just the duration of the acceleration.
>> There is no acceleration term. Can you state which term you think is an "acceleration term"?
>>
>> Special relativity: 803 ...... Barnpole Dave: 0
> Thanks,
> David Seppala
> Bastrop TX

David, you've on this stupid loop for at least 20 years. Why don't you
want to stop and teach yourself physics, including the basic of SR?

Seriously, what is you point?

On 10-Aug-21 2:31 am, sepp623@yahoo.com wrote:
> Here are two very similar scenarios but they seem to have contradictory results.

What happens when you express this in coordinate terms, and apply the
Lorentz transform?

Sylvia.

On Monday, August 9, 2021 at 4:36:21 PM UTC-7, sep...@yahoo.com wrote:
>
> Thanks,
> David Seppala
> Bastrop TX

Seppaler,

You need to come to terms with the fact that you are mentally ill. Very ill.

On Monday, August 9, 2021 at 4:36:21 PM UTC-7, sep...@yahoo.com wrote:
> > > > > Why are the two twins the same age at the end in scenario 1 and have an
> > > > > age difference of 1 year at the end of scenario 2.
> > > > Again, in Scenario 1, in terms of the system S0 of inertial coordinates, you have specified that each twin goes a spatial distance of +dx or -dx in the span of time dt, so the elapsed proper times both equal sqrt[(dt)^2 - (dx/c)^2]. In Scenario 2, in terms of the same system S0, you have specified that one twin traverses dx=0 in the time dt, and the other traverses some dx > 0 in the time dt, so the elapsed proper times are dt and sqrt[dt^2 - (dx/c)^2] respectively.
> > > >
> > > > > Explain why the acceleration...
> > > >
> > > > If you can find the word "acceleration" in the above answer to your question, I will explain it.
> > > >
> > > You use the term sqrt[dt^2 - (dx/c)^2] so explain why the acceleration term is a function of
> > > the extended time and not just the duration of the acceleration.
> >
> > There is no acceleration term. Can you state which term you think is an "acceleration term"?
>
> Thanks,

This is why you never make any progress. You pose some ridiculously convoluted question, that confuses you because you misunderstand the relativity of simultaneity; you are given the detailed explanation; you ask a batch of follow-up questions, each of which is patiently and thoroughly answered; and then, when you can see that the explanation of special relativity is unobjectionable, you just run away.

Above you ask me to explain the "acceleration term", and I've pointed out multiple times that there is no acceleration term in the explanation. Quite reasonably, I ask you to tell me what you think is the "acceleration term" in sqrt[dt^2 - (dx/c)^2], and instead of answering (as I have patiently answered dozens of your questions), you just run away.... until next time. The 804th time is the charm.