Twin Paradox (again(again(again )))

[yogi]

"And trying to set what is fundamentally an SR thought experiment in a scenario where gravity is not negligible does not help with that goal; it hinders it. Why can't you just put the two twins, or two trains, or whatever, far out in empty space away from all gravitating bodies? Then you could use an actual inertial frame to do the analysis, instead of using an obviously non-inertial frame and then handwaving about how "it doesn't really matter".

Peter, I use the Earth because in a practical experiment it is easy to use the Earth as an inertial frame even though it is not a perfect inertial fame - for a high speed particle or a high speed train, there is not a material difference due to the slight amount of curvature. GPS satellites lose time due to their relative velocity with respect to the non-rotating Earth centered reference frame and they gain time because of the potential energy difference between the Earth's surface and their orbit altitude. If you like to straighten out the tracks in space far removed from other influences - that's fine - do a one way trip and double the time dilation to get the round trip result - you will get almost the same answer - Perfect actually did the math some years back and posted the result of the Earth's curvature as insignificant - and that is the situation described in part IV of Einstein's 1905 paper. He makes the transition from a polygonal line to a curved line and gives the special relativity answer that is correct and that should have been the end of it. In my scenario, Gravitational acceleration is the same for both trains because they are both on the Earth's surface. ... a centrifuged clock does not increase the time dilation over what is measured if the clock were traveling at the same speed with respect to the Earth's surface- the time lost is simply due to the velocity - there is not an acceleration component to be added - the fact that there are G fields around is of no moment.

 

[PeterDonis]

for a high speed particle or a high speed train, there is not a material difference due to the slight amount of curvature.

The curvature is only "slight" if you restrict to a single local inertial frame. A train going from one of Earth's poles to its equator is not confined to a single local inertial frame.

Perfect actually did the math some years back and posted the result of the Earth's curvature as insignificant

Insignificant for what purpose? And over what range of the Earth's surface?

that is the situation described in part IV of Einstein's 1905 paper.

Einstein's 1905 paper assumed flat spacetime with no gravity. Yes, he used thought experiments with trains running on tracks, but he assumed the tracks were on a perfectly flat surface in a space with no gravity, not the actual curved Earth with gravity. He didn't spell all that out, but it's implicit in the math.

Gravitational acceleration is the same for both trains because they are both on the Earth's surface

No, this is not correct. First, the magnitude of the acceleration does vary over the Earth's surface; what is constant on the Earth's surface is the gravitational potential (i.e., clocks on the surface of the rotating Earth all go at the same rate), not its gradient, which is the acceleration. Second, even if we leave the variation in magnitude out, the direction of gravitational acceleration obviously varies over the Earth's surface. That alone is enough to invalidate modeling a significant portion of the Earth's surface as a single inertial frame.

 

[harrylin]

[..] in a practical experiment it is easy to use the Earth as an inertial frame even though it is not a perfect inertial fame - for a high speed particle or a high speed train, there is not a material difference due to the slight amount of curvature. GPS satellites lose time due to their relative velocity with respect to the non-rotating Earth centered reference frame and they gain time because of the potential energy difference between the Earth's surface and their orbit altitude. If you like to straighten out the tracks in space far removed from other influences - that's fine - do a one way trip and double the time dilation to get the round trip result - you will get almost the same answer - Perfect actually did the math some years back and posted the result of the Earth's curvature as insignificant

OK, however:

- and that is the situation described in part IV of Einstein's 1905 paper. He makes the transition from a polygonal line to a curved line and gives the special relativity answer that is correct and that should have been the end of it. [..].

Ehm no. See: #43

 

[yogi]

do not believe this. The elapsed time on any clock is the proper distance it has traveled and this can be calculated if the worldlines and metric are known. This is true in GR and SR. It resolves any possible paradoxes.

You have a very limited knowledge of the history of the twin paradox - as to the different ways time dilation is explained - Read Max Born's book and he will try to convince you that GR must be used - as did Feynman and a host of others. Time dilation is a reality, but their are many words written about it - whether acceleration is involved or has nothing to do with it ...I know there is no paradox - that is simply an identifying label. What is interesting is that the so called experts don't agree - if you think they do, you have not read many books or papers.

 

[Nugatory]

You have a very limited knowledge of the history of the twin paradox - as to the different ways time dilation is explained - Read Max Born's book and he will try to convince you that GR must be used - as did Feynman and a host of others. Time dilation is a reality, but their are many words written about it - whether acceleration is involved or has nothing to do with it ...I know there is no paradox - that is simply an identifying label. What is interesting is that the so called experts don't agree - if you think they do, you have not read many books or papers.

You are mistaken about the lack of expert agreement, perhaps because you have been sampling books and papers written at different times. It took some decades to hammer out the modern mathematical formalism of relativity, and if you look at stuff written before that process was complete you will certainly find disagreement - but all that tells us is that there was once disagreement.

Any residual disagreement you find today is just a residue of ambiguous English: If I use the mathematical machinery for arbitrary coordinate systems, which is required to solve problems in general relativity but optional for problems in special relativity, to solve a flat-spacetime problem do I say that I am using SR or GR? Fifty years ago you would find a fair number of people saying that I was using GR, especially because the mathematical methods in question would be familiar only to those who had already studied GR. Nowadays though... Not many people would sign up for the proposition that SR plus coordinate transformations is not still SR.

 

[PeterDonis]

The OP's question has been responded to sufficiently. Thread closed.