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Twin Paradox (thorough explanation needed) 
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#37
Apr1211, 04:16 PM

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So we have two invariances: 1) 20 lightyears is explicitly defined as the distance in the frame of reference of the colony and Earth, the spaceman measures 17.32 lightyears: it doesn't matter if we consider the spaceman stationary or the planets, the spaceman will always think the distance is 17.32 lightyears, the people on the planets will always think it's 20 lightyears. 2) The starting and ending points are in the frame of reference of the colony and Earth: it doesn't matter if we consider the spaceman stationary or the planets, everyone will agree these two points are in the frame of reference of the planets. One or both of these lead to the situation not being entirely invertable, preventing a paradox. 


#38
Apr1211, 04:19 PM

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edit: I see you added a bit to your post: 


#39
Apr1211, 05:11 PM

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#40
Apr1211, 05:20 PM

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#41
Apr1211, 05:31 PM

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#42
Apr1211, 05:33 PM

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#43
Apr1211, 05:43 PM

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The first shows things according to the space ship while traveling from Earth to planet, the green line is the Earth worldline, the blue line the ship's and the red line the planet's. The yellow lines represent light signals. The years start at zero. The second one shows things from the Earth rest frame (and consequently the frame the ship ends up in after decelerated at the planet. Just imagine that the blue line merges with the red line after they meet) Notice how the signals from Earth are closed more closely together after deceleration than before and how the light signal sent from Earth at the 20 yr mark not only travels further but had to have left earlier according to the ship after deceleration than it does before acceleration. 


#44
Apr1211, 05:54 PM

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Also, you seem to be avoiding my question about what happens when we simply add a second spaceship behind the first one, without changing anything else. Do you agree the scenario becomes totally invertible with this addition? If so, it seems that any statement about one observer's view being "vindicated" must be invertible as well. 


#45
Apr1211, 05:56 PM

P: 52

In order to measure the distance between the earth and the colony, while moving, the ship would have to be in a frame of reference from which the earth, the colony and the distance between them can be treated as a single unit. None of the observers will know which clocks are running slow until the ship lands and their clocks can be compared. The ship on earth measures the distance to the colony at 10 LY. The ship takes off and half way to the colony measures the remaining distance, it turns out to be 5 LY. From within the moving frame of reference you cannot observe the time dilation, length contraction or relativistic mass of the object in motion. Time is experienced as proceeding at a constant and normal rate.., until you land on the colony or back on earth and find your clocks don't match. 


#46
Apr1211, 05:56 PM

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#47
Apr1211, 06:00 PM

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#48
Apr1211, 06:03 PM

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The two travelers can follow convoluted paths. You can compute the spatial length of each path by integrating along the worldline. When the travelers rejoin you can compare the spatial lengths: Of course, the principle of relativity of inertial motion is prime. I mention the principle of relativity of inertial motion because it may appear as if that principle prevents you from assessing which traveller traveled the longest spatial distance. But that is not the case. A thought experiment: Two spaceships, equipped with very accurate clocks and very accurate accelerometers, and the crews of the ships are well aware of the principles of special relativity. They part from each other, and they agree where and when they will rejoin. Since both crews are good at special relativity calculations they can at all times figure out how much proper time has elapsed for themselves compared to time on a ship that just stays put. So they can agree where and when they will rejoin; it's just that they need to figure in special relativity effects. The most basic form of moving is to not accelerate at all. Just coasting along. Then your motion is just moving in time. A simple case is that the two spaceship crews agree beforehand that they will rejoin at the point in space and time where a ship would be after one week of proper time of that ship, if it would not accelerate at all. The onboard accelerometers allow something analogous to dead reckoning. At all times the accelerometer readings allow you to reconstruct your velocity relative to the point of departure. Your trip will have stages with different acceleration, in different directions. Two ships can plot different courses, and return to the common point of departure. If the navigators do their math well then two spaceships are able to rejoin, using only the accelerometer reading based "dead reckoning". When the two spaceships have rejoined you can compare the course plots, and see which one has traveled a longer spatial distance. Note especially that what the comparison yields is a difference. You can figure out who has traveled a longer distance, and how much longer. You make no statements about how much distance has been traveled by each respective ship. By the principle of relativity of inertial motion you cannot make any such statement. But you are not prevented from saying something about how the spatial distances traveled compare. Special relativity states that in such a scenario the reckoned difference in spatial distance traveled and difference in amount of elapsed proper time follows a law. Special relativity does not explain why things happen that way. All that special relativity does is describe that that is what will happen. 


#49
Apr1211, 06:32 PM

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#50
Apr1211, 06:37 PM

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#51
Apr1211, 06:55 PM

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Length contraction and time dilation as measured for an object in motion, given a stationary observer and an observer in motion, looks the same to both observers. Both think the other is length contracted and running slow. Once they meet up and compare clocks they find that one clock is slow and the other unaffected. The velocity of the object/observer in motion results in length contraction and time dilation. The time dilation is supported by experiment and observation. Length contraction has no such confirmation that I am aware of but its close connection to time dilation suggests that it too is real for the object in motion, while it is in motion. 


#52
Apr1211, 07:06 PM

P: 96

So if we have two ships in the same frame of reference who drifted apart at nonrelativistic speeds and then have them fly past a planet at relativistic speed (but at rest in relation to each other) the people on the planet will notice the crew of the second ship (which are clones of the crew on the first ship) has aged faster than them. The traditional twin paradox then is actually the same as my thought experiment run twice, because the turning point is some point defined by Earth, stationary to Earth, it's not a planet, but that doesn't matter. 


#53
Apr1211, 07:43 PM

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#54
Apr1211, 07:47 PM

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