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Twin Paradox


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I came to believe that due to the increased mass of an object traveling at near light speed, its gravity greatly increases. Due to its increased gravity, it bends the fabric of space time greatly, meaning that light takes longer to travel through the distortion of space time. Because the speed of light is constant to all observers, time must, must slow down to accommodate for the increased traveling distance of light. Due to my lack of understanding, I have somehow contorted rudimentary theories into a completely erroneous mess.


Yes, you have. :p


First, leave gravity out of this. While it is true that a clock higher over the surface of a planet will run faster than one lower in the gravitational field, it takes really, really large masses to make any noticiable effects. The masses of ordinary objects would do anything noticeable even if you multiply them by 100. (Corollary: what exactly do you mean by 'mass'? It is not a trivial question; if you define mass correctly it is invariant -- that is, independent of velocity.) In other words, unless you are talking black holes, gravity will rarely bother you. To be a little more precise, ignore gravity when the mass M and distance R satisfy GM/(R(c^2)) << 1 with G = Newton's constant and c = speed of light. Gravity is the subject of General Relativity, which really is too large a topic for a message boards and requires some pretty high level math (at least differential geometry).


I would like somebody to set me straight and explain to me why my former idea is incorrect and tell me the underlying principles that make Einstein's theory of special relativity and his length contraction equation the key to solving the twin paradox.


The twin paradox is not related to gravity and can be explained using special relativity alone. It is not that complicated; the problem is that it is difficult to draw diagrams on a message board...


Think of a set up where you have a light source on the floor and a mirror on the ceiling of a train which is moving with velocity v relative to the ground. Observer A is standing on the train. He sees the light move in straight lines, like so


__ Mirror


| c


\/ Source


Observer B is on the ground; he sees the train go by so the motion of the light looks like this to him (ignore the dots; this board doesn't like blank space):


................__ Mirror






Now, A observes the time thelight takes to go back and forth to be t = 2h/c where h is the height of the train (this is obvious).


Observer B, on the other hand, sees that the light is going in a diagonal but still at c. This is the key to the entire thing; if we were doing Galilean relativity, the light moving down the diagonal would be moving at a net speed of (c^2 + v^2)^1/2, but it doesn't; it always moves at c. Thus, the time taken can be obtained by just using the Pythagorean theorem. The vertical component of the light's velocity is must be (c^2 - v^2)^1/2 (because the horizontal component is v and the total must be c). Thus, the time the light takes to go back and forth is

t' = 2h/(c^2 - v^2)^1/2 so the ratio of the times t'/t = (1 - (v/c)^2)^-1/2. Note that this is always greater than one. This is how you get time dilation.


If you wanted length contraction, just look at the distance the light appears to travel to the observers.


The twin paradox is explained by time dilation; if you have one twin sit in place and the other go blast off on a rocket then turn around and come back, the traveling twin will be younger (time has been going slower for him). The paradox is that to the traveling twin it looks like the twin in place is moving. The resolution is that the traveler turns around in the middle of the journey; he is not actually in an inertial frame for the whole time.


EDIT: Here is a website which does this same explanation, but with better pictures:



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HAHA i believe i have a much firmer grasp than i did before very well explained. And i know know why this is the generally accepted solution to the twin paradox theory and i now believe this theory. However, for soem reason i can't shake the VERY sligh tpossibility that gravity can be used to solve the paradox although i know it ialmsot certainly doesnt becasue it would negate all of Einsteins work and equations. So what you have described makes the msot sense. You have succesfully aided my understanding the connection betwwen space and time [length contraction (which i furhter researched after u sent this) and time dialation].


Are you a physics major, professor, or a student who jsut took several courses in college?

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Are you a physics major, professor, or a student who jsut took several courses in college?


I'm a 4th year physics major. I've just started a semester of General Relativity (the kind that deals with non-negligible gravity) and it has a whole lot of math (as well as some more principles that contradict common sense). Fun stuff, but not for the casual viewer.

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