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The Twin Paradox and the Equivalence Principle 
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#19
Jun312, 09:02 AM

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it's a calculation. If in inertial coordinates, the invariant interval is given by: ds^{2} = dt^{2}  dx^{2}/c^{2} and you change coordinates to X,T, then it follows that in terms of X and T ds^{2} = (A^{2}  C^{2}/c^{2}) dT^{2} + 2(AB  CD/c^{2}) dX dT + (B^{2}  D^{2}/c^{2}) dX^{2} where A = ∂t/∂T, B = ∂t/∂X C = ∂x/∂T D = ∂x/∂X This is just calculus, there is no new insight about gravitation or the way it affects clocks required. 


#20
Jun312, 09:44 AM

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#21
Jun312, 11:03 AM

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It's actually like trying to compute the length of a path on the surface of the Earth using maps. If you look at a flat map, distances are completely distorted near the North Pole and South Pole (the North Pole itself is stretched out into a line, instead of being just a point). It's not very good for computing distances. If you look at maps for smaller regions, say 100 km x 100 km, then a flat map gives a good approximation to distances. So a way to compute the length of a very long route over the Earth is to start with a 100x100 map of the starting point. Compute the length up until the route passes out of the region. Then get another map, and figure out which point on the second map corresponds to the exit point on the first map. Keep adding up the distances until the route passes outside of the region for the second map. Then get a third map, etc. So you can compute the total distance as follows: Let P_{1} be the starting point. Find some map that covers P_{1}. Let P_{2} be the last point on the first map. Pick a second map that covers P_{2}. Let P_{3} be the last point on the second map. Etc. Let D_{n} be the distance between P_{n} and P_{n+1}, as measured using the nth map. Then D = sum over n of D_{n}. The same sort of thing works to find elapsed time for a complicated path through spacetime. 


#22
Jun312, 02:05 PM

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A nice explanation of the twin paradox using the GR point of view is in the old book "Relativity, Thermodynamics, and Cosmology" by Tolman.
If you go to Google Books at books.google.com/?hl=EN and search "Tolman Cosmolgy" you will find the book. The explanation starts on page 194. 


#23
Jun312, 02:38 PM

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Any valid coordinate chart you choose can "explain" it correctly, but the explanation will differ. In none of them will the other twin age backwards. 


#24
Jun312, 03:30 PM

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Here’s a link to a derivation of the rate of an arbitrarily accelerating clock as observed by an arbitrarily accelerating observer assuming that the clock and observer are always moving along the same line.
//sdrv.ms/JPxY5H The derivation is tedious, but the final result (equation 18) is quite simple. The rate of ticking of the clock according to the observer is just a product of a time dilation factor due to relative speed (u) and a factor due to the acceleration felt by the observer (g). (How u and g are defined for the accelerating clock and observer is discussed in the derivation.) The factor due to the acceleration is interesting, it has the appearance of “gravitational time dilation” due to a uniform gravitational field of strength g. 


#25
Jun312, 04:01 PM

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#26
Jun712, 01:44 PM

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To be able to talk about how old a twin is "right now" requires a choice about a time coordinate that covers both twins. For an inertial frame, there is a unique "best" time coordinate, but for an accelerated twin, there is no unique choice of a time coordinate. 


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Jun712, 02:18 PM

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#28
Jun712, 05:57 PM

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#29
Jun712, 08:50 PM

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Yes, of course your sister always has some definite age, just as her wristwatch will always record some definite time since you and she parted. However, different observers on different rocket ships moving at different speeds will have different ideas about what that age is; and that's not especially remarkable because they all have different ideas of what's "right then". They're checking her watch and her age at different times so of course they get different answers. None of this has anything to do with what your sister experiences. She's happily living her life back at home, celebrating a birthday once a year and keeping track of her age in her home reference frame, unconcerned that you and the other rocket observers are seeing time pass at different rates. 


#30
Jun712, 08:55 PM

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#31
Jun812, 11:05 AM

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You might think that your sister's actual age is whatever age she has, according to the Earth's coordinate system. However, if you use a different coordinate system, you would compute a different age for your sister "right now" and you would never run into any contradiction. Suppose you are billions of miles away from your sister. While you are away, your sister celebrates her 20th birthdaycall that event e_{sister}and you celebrate your 20th birthdaycall that event e_{you}. Then the question is: is e_{sister} before, after, or at the same time as e_{you}? You can say that e_{sister} is definitely before e_{you} if it is possible to send a signal from the first to the second. If you receive a video transmission of her birthday party, and it arrives before you celebrate your birthday, then you know for sure that her birthday happened earlier. You can say that e_{sister} is definitely after e_{you} if it is possible to send a signal from the second to the first. If she receives a video transmission of your birthday party, and it arrives before she celebrates her birthday, then you know for sure that your birthday happened earlier. But if it is not possible to send a signal in either direction in timeyour video of birthday arrives after her birthday, and viceversethen the two events are called "spacelike separated". According to some coordinate systems, your sister's birthday was first. According to some other coordinate systems, your birthday was first. According to yet other coordinate systems, the two events happened at the same time. As far as the physics is concerned, there is no "right" answer; any one of those coordinate systems are just as good as any other. 


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Jun812, 03:25 PM

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#33
Jun812, 03:43 PM

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If the captain, the driver, and you are all using the same reference frame, as you likely are because you're all on the same ship, then you will all be talking about the same thing when you say "right then", so you all will have the same answer for the "How old is she and what is she doing right now?" question (see, I slipped those evil and treacherous words in too; otherwise the question would be well and thoroughly unanswerable). However, someone else in another spaceship could go zooming by at a high rate of speed relative to you, so is using a different reference frame. As he passes your ship, you call out "How old is my sister right now?", and he shouts his answer back to you. It could be different from the answer that you got from your ship's crew  and they'd both be right. 


#34
Jun812, 04:08 PM

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#35
Jun812, 04:32 PM

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Of course "scio me nihil scire " is in power here. Universe doesn't behave the way we want... Maybe.. Maybe not. This is relativity isnt? There is many scenarios of "what would happend with twins if..." I would like to ask... What Would happend if we invert the problem. Make v and t. If we world is relative and indeed it could be. Because we dont know is it relative. Scientis who believe in relativite often search for universal formula. But about inverted twins paradox. If we can manipulate the time or... Dependance of time... We may invert ot that the time will back... . If something moves really fast with velocity c we have a distance 0km/s to 300000km/s. the c velocity is the biggest known velocity right? Relqtion between twins is like 1 to 7 (about years) . But c... Is not the biggest value in relative world. If the twins move to opposite directions what than? They are moving (in relation to each other) with velocity 2c... So 600 000 is value that we should consider.
Ah god I ve started other topis. What was i... Ah invertet paradox. I dare to think that everything is relative. We have electron and pozyton... Neutrino and antineutrino... So if we make one brothers living time longer in the relation to other.. We may invert this proces and make time stop and move back. Because everything is possible and we are still litle insects. Ps. Sory if my english is not as well as should be. 


#36
Jun812, 07:29 PM

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No nononono; it is you who has not understood what you are being told.
You are on your rocket ship and so it the captain and the navigator ... so you all agree about what age she is "right now" just like you all agree on what the ship's time is. YOu just have to decide what you mean by "right now". If you didn't know how to work out her age yourself, then you maybe ask them to do it for you right? Then they would be justified in asking which coordinate system you wanted the calculation done for: the ship's, her's, whatever? Each are equally "valid" certainly for the purposes of "imagining what she's doing right now". Usually  to work what you are doing "right now" I just look at you ... if you are a long wayaway I may want to factor in the time it takes light to travel to me. That the sort of thing you mean? So you imagine looking at her with a telescope and you see that she is 20yo but you are 3ly away so she must be 23? We can ask: how old are you when you made the observation? 


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