Twin Paradox (again(again(again )))

  • #26
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It IS possible to work the problem from Diana's "perspective". You can indeed ask the question: "At each instant of Diana's life during the trip, what does she conclude about Artemis' current age at that instant?". You can get that answer (for each instant "t" in Diana's life) by using the "co-moving inertial frame" at that instant. The Lorentz equations can be used to relate that "co-moving inertial frame" and Artemis' inertial frame ... i.e., the Lorentz equations can tell you what Artemis' current age is when Diana's age is "t", ACCORDING TO DIANA. That capitalized phrase is needed, because Artemis will NOT agree ... his conclusions about the relationship between their ages during (most of) the trip are quite different from Diana's conclusions. The whole process for carrying out the above procedure is well described in the webpage

https://sites.google.com/site/cadoequation/cado-reference-frame

The short answer is that, according to Diana, Artemis ages slowly during Diana's constant-speed outbound and inbound portions of her trip, but during her accelerated turnaround, Artemis ages VERY quickly. And the TOTAL of those three components of Artemis' aging is such that Artemis' and Diana's "perspectives" both agree at the end of the trip (but they generally disagree during the trip).


I know nothing about the CADO reference frame methodology. But your explanation is spot on. For some observers (accelerating ones) an inertially moving clock runs FASTER. That is a consequence of Special Relativity.

Reference:
Basic Relativity
by Richard A Mould
Copyright 1994
Chapter 8 "Uniform Acceleration"
 
  • #27
PeterDonis
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The whole process for carrying out the above procedure is well described in the webpage

https://sites.google.com/site/cadoequation/cado-reference-frame
It should be noted that this is only one way of constructing a reference frame for an object that undergoes non-uniform motion. Other ways will give different relationships between Diana's and Artemis's aging during the trip; the only thing they must all agree on is their respective elapsed proper time when they meet up again.

For some observers (accelerating ones) an inertially moving clock runs FASTER. That is a consequence of Special Relativity.
As above, this is true if we construct the non-inertial frame for the accelerating observer a certain way. There are other ways of doing so that can give different answers. The only thing they must all agree on is invariants such as the elapsed proper time along a given worldline between two given events.
 
  • #28
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That is simply not true (unless, perhaps, you're using an unreasonably restrictive definition of "the laws of SR").[..].
That the laws of SR relate to inertial reference frames is consistent with that the laws of classical mechanics relate to inertial reference systems*. Of course, that does not prevent us from mapping predictions of classical mechanics and SR to accelerating reference systems. If you find that "unreasonable", then so be it. :oldeyes:

*"Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good"
- http://www.fourmilab.ch/etexts/einstein/specrel/www/
 
  • #29
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I know nothing about the CADO reference frame methodology.
[...]
The "CADO reference frame", or "CADO frame", is just the "co-moving-inertial-frames reference frame". There are several advantages for using a different name for the same thing. First of all, it's a shorter, simpler name. And it makes it clearer that it is a SINGLE reference frame for the accelerating observer ... sometimes, when people see that longer name used, with the plural inertial frames, they conclude that the accelerating observer can't have his own, single, reference frame for his whole trip (or for his whole life). It is his "own" reference frame only in the sense that he is always located at its spatial origin. People often use the term "rest frame" to describe that situation, but that term can be misleading, because the CADO frame does not imply that the accelerating observer denies that he is the one who is accelerating ... i.e., the CADO frame is purely a special relativity construction, without needing to invent any fictitious gravitational fields.

The terminology (and the variable names) used in the CADO frame also try to make mistakes less likely in its application. And, instead of directly using the Lorentz equations to determine the current age of the distant inertial person (according to the accelerating observer), at each instant of the accelerating observer's life, the CADO frame uses the "CADO equation" (which is derived from the Lorentz equations, and gets the same result). The CADO equation is quicker and easier to use, and it makes mistakes less likely.
 
  • #30
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the current age of the distant inertial person (according to the accelerating observer)
Once again, this phrasing is somewhat misleading, since there is no such thing as "the" current age of the distant inertial person; it depends on the choice of coordinates. The CADO method is one way of setting up coordinates for an accelerating observer, but it is not the only one.
 
  • #31
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"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 earths 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 earths 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 earths 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 earths 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.
 
  • #32
PeterDonis
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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 earths 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 earths 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.
 
  • #33
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[..] 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 earths 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 earths 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
 
  • #34
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do not believe this. The elapsed time on any clock is the proper distance it has travelled 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.
 
  • #35
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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.
 
  • #36
PeterDonis
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The OP's question has been responded to sufficiently. Thread closed.
 

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