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Thread: The Twin Paradox

  1. #1 The Twin Paradox 
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    Ok I give and bow to superior knowledge



    Can someone explain to me how the Twin Paradox is supposed to work, It essentially says 'if one twin goes of at near light speed into space, and returns 10 years later, he will be younger than his earthbound twin'

    I know what it says I just don't get why I've searched the internet and all I can find are refferances to People who have given examples and various equations that have no basis, they just are

    perfect example was wikipedia :-
    Consider a space ship traveling from Earth to the nearest star system outside of our solar system: a distance d = 4.45 light years away, at a speed v = 0.866c (i.e., 86.6 percent of the speed of light). The Earth-based mission control reasons about the journey this way (for convenience in this thought experiment the ship is assumed to immediately attain its full speed upon departure): the round trip will take t = 2d / v = 10.28 years in Earth time (i.e. everybody on earth will be 10.28 years older when the ship returns). The amount of time as measured on the ship's clocks and the aging of the travelers during their trip will be reduced by the factor \epsilon = \sqrt{1 - v^2/c^2}, the reciprocal of the Lorentz factor. In this case \epsilon = 0.500 \, and the travelers will have aged only 0.500×10.28 = 5.14 years when they return.

    why is the reciprocal of the Lorentz factor used? there is no logic behind it other than someone saying, well it is


    I also found mentions of Special Relativity and Time Dilation, if someone could explain what they are and how the fit into this that'd be great too



    I suspect my problem comes down to something like this
    'Scientists tend to overcompress, to make their arguments difficult to follow by leaving out too many steps. They do this because they have a hard time writing and they would like to get it over with as soon as possible.... Six weeks of work are subsumed into the word “obviously.” '—Sidney Coleman


    and I fully expect to get an answer that leaves me none the wiser but hey! I'll ask anyway



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    Booms


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  3. #2 Re: The Twin Paradox 
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    Quote Originally Posted by Booms
    Ok I give and bow to superior knowledge



    Can someone explain to me how the Twin Paradox is supposed to work, It essentially says 'if one twin goes of at near light speed into space, and returns 10 years later, he will be younger than his earthbound twin'

    I know what it says I just don't get why I've searched the internet and all I can find are refferances to People who have given examples and various equations that have no basis, they just are

    perfect example was wikipedia :-
    Consider a space ship traveling from Earth to the nearest star system outside of our solar system: a distance d = 4.45 light years away, at a speed v = 0.866c (i.e., 86.6 percent of the speed of light). The Earth-based mission control reasons about the journey this way (for convenience in this thought experiment the ship is assumed to immediately attain its full speed upon departure): the round trip will take t = 2d / v = 10.28 years in Earth time (i.e. everybody on earth will be 10.28 years older when the ship returns). The amount of time as measured on the ship's clocks and the aging of the travelers during their trip will be reduced by the factor \epsilon = \sqrt{1 - v^2/c^2}, the reciprocal of the Lorentz factor. In this case \epsilon = 0.500 \, and the travelers will have aged only 0.500×10.28 = 5.14 years when they return.

    why is the reciprocal of the Lorentz factor used? there is no logic behind it other than someone saying, well it is


    I also found mentions of Special Relativity and Time Dilation, if someone could explain what they are and how the fit into this that'd be great too



    I suspect my problem comes down to something like this
    'Scientists tend to overcompress, to make their arguments difficult to follow by leaving out too many steps. They do this because they have a hard time writing and they would like to get it over with as soon as possible.... Six weeks of work are subsumed into the word “obviously.” '—Sidney Coleman


    and I fully expect to get an answer that leaves me none the wiser but hey! I'll ask anyway



    Thanks
    Booms
    I assume that you are aware of the phenomena of "time dilation" that is part of special relativity. If not this Wiki article is pretty good.

    http://en.wikipedia.org/wiki/Special_relativity

    The Twin "Paradox" goes like this.

    You start with two twins on Earth. One twin takes off in a space ship which reaches a high speed, nearly the speed of light, relative to his Earth-bound twin. He reaches a distant star, turns around and returns to Earth, also at nearly the speed of light.

    The high speed relative to the Earth-bound twin results in the Earth-bound twin perceiving the clock of the traveling twin as being slow relative to his own clock --the traveling twin has aged less. This is correct and has been demonstrated experimentally using aircraft and atomic clocks.

    Now the "paradox". The traveling twin also sees the Earth-bound twin as accelerating away from him and traveling at nearly the speed of light. So applying special relativity (incorrectly) it is concluded that the traveling twin also sees the Earth-bound twin as aging less than he does. So on return each finds the other to be younger.

    There are a couple of ways to resolve this issue. One is to note that the traveling twin is accelerating and that therefore the proper theory is not special relativity, but rather general relativity. General relativity shows the traveling twin to have aged less.

    Another resolution is obtained by recognizing that special relativity applies only in an inertial reference frame. The Earth-bound twin is in an inertial reference frame. The traveling twin is not -- he feels the force of acceleration on leaving Earth, on decelerating and turning around at the distant star, and again at his return to Earth. So, in this case there is only one reference frame in which special relativity can be applied, and that is the reference frame of the stay-at-home twin.


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    Dr Rocket's explanation is certainly wrong. He simply repeated explanations that people once accepted. It is wrong because it is avoiding the key issue behind the paradox (and the truth). Lets do the experiment again, but slightly differently which shows that acceleration is not the reason.

    Lets say there are 2 identical twins, twin 1 and twin 2. Twin 1 is placed in spaceship 1 and Twin 2 is placed in spaceship 2. Both Ships are next to eachother. Space ship 1 and 2 then both accelerate to .9c but then Spaceship 1 immediately deceleartes back to rest (the same velocity before he started accelarating). Then when arrives at rest, he immediately accelarates again back up to .9c, then when he reaches this, he once again immediately deccelerates back down and remains at rest and stays in that position. Now spaceship 2 continues at a constant velocity of .9c from what his clocks show as being a year. He then immediately decerates back down to his original speed befire his started his journey. Then he immediately accelerates up to .9c back in the direction of spaceship 1. When his clocks show that another year has passed, he immediately deccelerates back down to his original speed before he started the journey. Spaceship 1 and 2 are now both next to eachother. Both have also accelerated twice and deccelerated twice in the exact same manner. Twin 1 and Twin 2 now meet up with eachother and they realise that Twin 1 has aged MUCH more than twin 2.

    According to Special relativity, each twin sees the other twin as traveling, so each should see the other aging more slowly. How can an absolute effect (one twin really does age less) result from a relative motion? As I showed before, this cannot be explained by general relativity either because both twins also accelarated in the same way.

    This shows that time is not relative, but is ABSOLUTE. Relativity is all about illusion. The effects of relativity are still perfectly true, but you must realise that the person who is travelling is being decieved, because his frequency slows down as he travels faster. The aether also exists, it has never been proven not to exist. Everything behaves in an absolute way.
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  5. #4  
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    hmm, perhaps I wrote my question wrongly
    Or perhaps I'm just currently too dense to understand something this blatently complex



    I know what the Twin Paradox is and, thanks to Dr Rocket's link I vaguely understand what Time Dilation is

    what I need to know is why, why would twin 1 age much more than twin 2 simply because of a change of speed?
    If someone can explain why the twin paradox works, or why a moving clock will become out of sync with a stationary clock, without refering to Pre-written equations, or explain the logic behind the equations that would be great
    for example is the formula for determining Time Dilation, however it gives no reasoning behind this, why is this the formula? why not a different formula?


    the whole thing seems to just be assumed as the correct answer, everywhere I look I get an answer similar to "why does the earth spin" replied with "because it does/can" when what I want is an answer explaining the Gravitational pull of the sun and the internal momentum generated by the centre of the planet etc etc


    Edit: that said I appreciate it's possible no-one fully understands it any more, and the only guy who could have explained it without assumptions is probably dead.
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    what I need to know is why, why would twin 1 age much more than twin 2 simply because of a change of speed?
    I'll probably be advised to read a few more books on special relativity after I answer this, but here goes. Note that this is a highly simplistic explanation.

    Special relativity begins with just two postulates:

    1) The speed of light is constant in all reference frames i.e. whether you are in motion or not.

    2) The laws of physics remain the same in all reference frames.

    I now introduce a definition of time as follows: the amount of time taken for light to reach you. This is reasonable: light contains information, with the help of which only can we determine a change in time. Obviously, if light comes to you more slowly, time will be seen to move more slowly, because the information required to differentiate between the past and the future is taking longer to get to you.

    Now suppose you are at rest. Light hits you constantly, meaning that to an observer next to you, also at rest, time will pass at the same rate you measure it to be. His clock wil tick at the same rate as yours.

    Suppose, however, that you are now moving with a certain velocity v. Light, since it always moves at the same speed, will take longer to reach you, as it now also has to cover the distance you've left behind as you're moving. Naturally, it will take light a longer time to reach you, making it seem as if time is moving more slowly. Since your clock is also moving at the same speed, it too is subject to this, and will tick slower than the observer on the ground, because it measures time itself to be moving slower.

    And that is how time dilation works. Again, this is a highly simplistic explanation, which should enable you to understand it quite clearly.
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  7. #6  
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    Quote Originally Posted by Waveman28
    Dr Rocket's explanation is certainly wrong. He simply repeated explanations that people once accepted. It is wrong because it is avoiding the key issue behind the paradox (and the truth). Lets do the experiment again, but slightly differently which shows that acceleration is not the reason.

    Lets say there are 2 identical twins, twin 1 and twin 2. Twin 1 is placed in spaceship 1 and Twin 2 is placed in spaceship 2. Both Ships are next to eachother. Space ship 1 and 2 then both accelerate to .9c but then Spaceship 1 immediately deceleartes back to rest (the same velocity before he started accelarating). Then when arrives at rest, he immediately accelarates again back up to .9c, then when he reaches this, he once again immediately deccelerates back down and remains at rest and stays in that position. Now spaceship 2 continues at a constant velocity of .9c from what his clocks show as being a year. He then immediately decerates back down to his original speed befire his started his journey. Then he immediately accelerates up to .9c back in the direction of spaceship 1. When his clocks show that another year has passed, he immediately deccelerates back down to his original speed before he started the journey. Spaceship 1 and 2 are now both next to eachother. Both have also accelerated twice and deccelerated twice in the exact same manner. Twin 1 and Twin 2 now meet up with eachother and they realise that Twin 1 has aged MUCH more than twin 2.

    According to Special relativity, each twin sees the other twin as traveling, so each should see the other aging more slowly. How can an absolute effect (one twin really does age less) result from a relative motion? As I showed before, this cannot be explained by general relativity either because both twins also accelarated in the same way.

    This shows that time is not relative, but is ABSOLUTE. Relativity is all about illusion. The effects of relativity are still perfectly true, but you must realise that the person who is travelling is being decieved, because his frequency slows down as he travels faster. The aether also exists, it has never been proven not to exist. Everything behaves in an absolute way.
    All you have proven here is that You do not understand Relativity. If you did, you would know that it isn't just the magnitude and durations of the accelerations that count, but also the combination of direction and relative distance between the two twins at the time of the accelerations.
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  8. #7  
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    Quote Originally Posted by Janus
    Quote Originally Posted by Waveman28
    Dr Rocket's explanation is certainly wrong. He simply repeated explanations that people once accepted. It is wrong because it is avoiding the key issue behind the paradox (and the truth). Lets do the experiment again, but slightly differently which shows that acceleration is not the reason.

    Lets say there are 2 identical twins, twin 1 and twin 2. Twin 1 is placed in spaceship 1 and Twin 2 is placed in spaceship 2. Both Ships are next to eachother. Space ship 1 and 2 then both accelerate to .9c but then Spaceship 1 immediately deceleartes back to rest (the same velocity before he started accelarating). Then when arrives at rest, he immediately accelarates again back up to .9c, then when he reaches this, he once again immediately deccelerates back down and remains at rest and stays in that position. Now spaceship 2 continues at a constant velocity of .9c from what his clocks show as being a year. He then immediately decerates back down to his original speed befire his started his journey. Then he immediately accelerates up to .9c back in the direction of spaceship 1. When his clocks show that another year has passed, he immediately deccelerates back down to his original speed before he started the journey. Spaceship 1 and 2 are now both next to eachother. Both have also accelerated twice and deccelerated twice in the exact same manner. Twin 1 and Twin 2 now meet up with eachother and they realise that Twin 1 has aged MUCH more than twin 2.

    According to Special relativity, each twin sees the other twin as traveling, so each should see the other aging more slowly. How can an absolute effect (one twin really does age less) result from a relative motion? As I showed before, this cannot be explained by general relativity either because both twins also accelarated in the same way.

    This shows that time is not relative, but is ABSOLUTE. Relativity is all about illusion. The effects of relativity are still perfectly true, but you must realise that the person who is travelling is being decieved, because his frequency slows down as he travels faster. The aether also exists, it has never been proven not to exist. Everything behaves in an absolute way.
    All you have proven here is that You do not understand Relativity. If you did, you would know that it isn't just the magnitude and durations of the accelerations that count, but also the combination of direction and relative distance between the two twins at the time of the accelerations.
    Au contraire. He demonstrated ignorance that goes far beyond just relativity. He has pretty demonstrated ignorance of all of physics (what is "his frequency"), basic logic (the lack of proof for the non-existence of something is not proof of its existence), and one must surmise the language of the subject , mathematics, as well.

    A truly simply example of the ignorance is the statement that each twin has accelerated and decelerated in the same way. That is simply preposterous. One twin feels the effect of acceleration as he is pushed backward in his seat on launch, feels pulled forward on deceleration, and is pushed radially on turning at speed. The other feels nothing. These effects are not theoretical and can be easily measured with an accelerometer. The two reference frames are NOT equivalent in any sense. The fact that one twin feels the forces of acceleration is proof that the reference frame of that twin is not inertial, and special relativity applies in inertial reference frames and only in inertial reference frames.

    No, one must salute the demonstration of such wide-ranging and deep ignorance. In this day and age of ready access to information, it is a stupendous feat to have remained so ignorant.
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    To be fair, I wouldn't be entirely satisfied with your explanation of the twin paradox either. Not that I'm saying the other poster is right (I didn't really read it). Nor am I saying that you're wrong.

    I think the easiest way to see what's going on is to think in terms of the proper times of the two twins' trajectories in Minkowski space. Both twins start and finish at the same points A and B in spacetime. However, the stationary twin follows a straight path while the other observer follows a curved path.

    Now, in Minkowski space, the shortest distance between two points is NOT necessarily a straight line. Indeed, the "length" of the curved path (i.e., the proper time) will end up being shorter than the length of the straight path.

    But what about from the non-stationary frame of reference? The point is that the notion of "straight path" and "curved path" is Lorentz invariant: A straight path remains straight in every reference frame, and the same can be said for a curved path.

    More to the point, proper time is a Lorentz invariant concept.

    Again, I don't claim to be saying anything wildly different from what's already been said, but I haven't seen anyone in this thread say it in just this way, so hence my post.
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  10. #9  
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    Quote Originally Posted by salsaonline
    To be fair, I wouldn't be entirely satisfied with your explanation of the twin paradox either. Not that I'm saying the other poster is right (I didn't really read it). Nor am I saying that you're wrong.
    There are several ways to resolve the paradox.

    Within special relativity I think it is easiest to follow a couple of steps that are generall applicable to mathematical theorems. To wit:

    1) Verify that the hypotheses of the theorem are satisfied

    2) Then apply the theorem.

    In the case of special relativity step 1 entails using an inertial reference frame. In practice this means identifying a reference frame that is clearly inertial within the framework of the idealized problem as stated.

    In the case of the twin paradox it is clear that the reference frame of the stationary twin is inertial and that of the "traveling twin" is not. Therefore the apparent symmetry of the problem is not a symmetry at all. There is no paradox and the issues of interpretation arise simply because one is applying special relativity in a situation in which it does not apply at all.

    Better explanations involve general relativity and curved space-times, as you noted. However, the twin "paradox" can be resolved without that level of sophistication.

    To me resolution by simply verifying the underlying assumptions of special relativity is the easiest. I don't quarrel with other, correct, resolutions. However, waveman28 got essentially nothing right.
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    Quote Originally Posted by DrRocket
    There are several ways to resolve the paradox.

    Within special relativity I think it is easiest to follow a couple of steps that are generall applicable to mathematical theorems. To wit:

    1) Verify that the hypotheses of the theorem are satisfied

    2) Then apply the theorem.

    In the case of special relativity step 1 entails using an inertial reference frame. In practice this means identifying a reference frame that is clearly inertial within the framework of the idealized problem as stated.

    In the case of the twin paradox it is clear that the reference frame of the stationary twin is inertial and that of the "traveling twin" is not. Therefore the apparent symmetry of the problem is not a symmetry at all.
    Put in those words, I like your explanation.

    Incidentally, I never said anything about GR. I was speaking about the length of curved paths in the Lorentz metric. So there's no curvature, hence no GR.
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    Why do you believe in "minowski space". This concept is absurd. We live in a 3 Dimensional world. How can you explain "space curvature" in terms of 3 dimensions? Dont worry, i have read all about current physics and obviously know alot about it, for instance when you asked your question i could have simply told you what Dr Rocket said, but all he is doing is simply repeating information in a text book. I however, use common sense and i doubt alot of the concepts that todays textbooks teach. The people that developed them (Poincare, Lorentz, Voigt, etc.) are just ordinary people like you and me and they were prone to being incorrect (which many of them were). My criticism of Students and teachers these days, is that they almost immediately accept everything they read in textbooks etc. and think that it is gospel. But, i know that it certainly is not.

    For instance, everyone at present seems to think that length contraction is due to space-time contraction. Why? Because Einstein simply said so. Why did he immediately jump to this conclusion? A much wiser explaination was given by a physicist earlier than Einstein called George Fitzgerald. He gave a much more logical explaination by saying that the actual object ITSELF undergoes a PHYSICAL contraction, not space. This is already infinitely more logical than Einsteins approach, because we know that the object itself contracts, so why the heck would we say space is instead? This theory was later developed by Lorentz and is called Lorentz Aether Theory.

    My point is, use your brain. Show doubt towards things that do not make sense. I did this, and it led me to discover that time, length and mass are all absolute, but our motion through the aether causes us to be misled, which makes us THINK that we are at rest. This is why we think that all frames are the same, they are not, but we still think they are.
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    Quote Originally Posted by Waveman28
    Why do you believe in "minowski space". This concept is absurd. We live in a 3 Dimensional world. How can you explain "space curvature" in terms of 3 dimensions? Dont worry, i have read all about current physics and obviously know alot about it, for instance when you asked your question i could have simply told you what Dr Rocket said, but all he is doing is simply repeating information in a text book. I however, use common sense and i doubt alot of the concepts that todays textbooks teach. The people that developed them (Poincare, Lorentz, Voigt, etc.) are just ordinary people like you and me and they were prone to being incorrect (which many of them were). My criticism of Students and teachers these days, is that they almost immediately accept everything they read in textbooks etc. and think that it is gospel. But, i know that it certainly is not.

    For instance, everyone at present seems to think that length contraction is due to space-time contraction. Why? Because Einstein simply said so. Why did he immediately jump to this conclusion? A much wiser explaination was given by a physicist earlier than Einstein called George Fitzgerald. He gave a much more logical explaination by saying that the actual object ITSELF undergoes a PHYSICAL contraction, not space. This is already infinitely more logical than Einsteins approach, because we know that the object itself contracts, so why the heck would we say space is instead? This theory was later developed by Lorentz and is called Lorentz Aether Theory.

    My point is, use your brain. Show doubt towards things that do not make sense. I did this, and it led me to discover that time, length and mass are all absolute, but our motion through the aether causes us to be misled, which makes us THINK that we are at rest. This is why we think that all frames are the same, they are not, but we still think they are.
    Well, I'll say this. Quantum electrodynamics is deeply wedded to the Minkowski space point of view. And the experimental predictions of quantum electrodynamics are accurate to something like 12 decimal places. When you have a theory that does a better job than that, you let me know.
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    I didn't know the paradox was that both twins would see the other as younger. I thought that because the spaceship was traveling at close to the speed of light it's clocks ticked slower and the stationary twin on earth time moved at earth speeds. So when the guy in the spaceship returned he aged less and the earthbound twin aged normally. Why would the earthbound twin age slower when he was not in rapid acceleration. Particles in particle accelerators going close to the speed of light decay slower, they have longer lives, because they are traveling ant such tremendous speed. So would then the paradox would be that from the particles perspective the accelerator tube is aging more slowly so the particle thinks it has more time to live?? I may have of confused myself. But in essence when the twin arrives back on earth the one that left and came back would indeed be older?
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    Quote Originally Posted by Waveman28
    Why do you believe in "minowski space". This concept is absurd. We live in a 3 Dimensional world. How can you explain "space curvature" in terms of 3 dimensions? Dont worry, i have read all about current physics and obviously know alot about it, for instance when you asked your question i could have simply told you what Dr Rocket said, but all he is doing is simply repeating information in a text book. I however, use common sense and i doubt alot of the concepts that todays textbooks teach. The people that developed them (Poincare, Lorentz, Voigt, etc.) are just ordinary people like you and me and they were prone to being incorrect (which many of them were). My criticism of Students and teachers these days, is that they almost immediately accept everything they read in textbooks etc. and think that it is gospel. But, i know that it certainly is not.

    For instance, everyone at present seems to think that length contraction is due to space-time contraction. Why? Because Einstein simply said so. Why did he immediately jump to this conclusion? A much wiser explaination was given by a physicist earlier than Einstein called George Fitzgerald. He gave a much more logical explaination by saying that the actual object ITSELF undergoes a PHYSICAL contraction, not space. This is already infinitely more logical than Einsteins approach, because we know that the object itself contracts, so why the heck would we say space is instead? This theory was later developed by Lorentz and is called Lorentz Aether Theory.

    My point is, use your brain. Show doubt towards things that do not make sense. I did this, and it led me to discover that time, length and mass are all absolute, but our motion through the aether causes us to be misled, which makes us THINK that we are at rest. This is why we think that all frames are the same, they are not, but we still think they are.
    That is quite possibly the single most arrogant post I have read in my epicly short 16 years (well 13 if you skip the years I couldn't read)
    Even disregarding the "I obviously know lots about it" (from reading books? I read tonnes and am still asking about the TP you seem to have done a William Mccormack and said absolutely nothing






    Quote Originally Posted by Liongold
    what I need to know is why, why would twin 1 age much more than twin 2 simply because of a change of speed?
    I'll probably be advised to read a few more books on special relativity after I answer this, but here goes. Note that this is a highly simplistic explanation.

    Special relativity begins with just two postulates:

    1) The speed of light is constant in all reference frames i.e. whether you are in motion or not.

    2) The laws of physics remain the same in all reference frames.

    I now introduce a definition of time as follows: the amount of time taken for light to reach you. This is reasonable: light contains information, with the help of which only can we determine a change in time. Obviously, if light comes to you more slowly, time will be seen to move more slowly, because the information required to differentiate between the past and the future is taking longer to get to you.

    Now suppose you are at rest. Light hits you constantly, meaning that to an observer next to you, also at rest, time will pass at the same rate you measure it to be. His clock wil tick at the same rate as yours.

    Suppose, however, that you are now moving with a certain velocity v. Light, since it always moves at the same speed, will take longer to reach you, as it now also has to cover the distance you've left behind as you're moving. Naturally, it will take light a longer time to reach you, making it seem as if time is moving more slowly. Since your clock is also moving at the same speed, it too is subject to this, and will tick slower than the observer on the ground, because it measures time itself to be moving slower.

    And that is how time dilation works. Again, this is a highly simplistic explanation, which should enable you to understand it quite clearly.
    THANK YOU!
    I understand!, evidently I'm a complete moron since the only explanation that made complete sense to me is this 'highly simplistic' version

    Of course this understanding has annoyingly raised several other questions, no doubt born of my extremely childlike "But why?" mentality, and I shall find these answers elsewhere, but thank you for explaining it




    Also thanks to the others who tried to explain it too me and sorry for undoubtable ticking you off to no end with my incapability.
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    Hello. I'm new here and I don't know anything about anything, but I'm here to learn.

    A question about this. If time moves slower for the one twin, when he returns, is his brother biologically older?
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    Yes he biologically older. So it the earth and everything on it.
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    Quote Originally Posted by Wildstar
    Yes he biologically older. So it the earth and everything on it.
    But shouldn't his biological clock tick at its normal rate?
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    Quote Originally Posted by eroteme
    Quote Originally Posted by Wildstar
    Yes he biologically older. So it the earth and everything on it.
    But shouldn't his biological clock tick at its normal rate?
    His biological clock does tick at its normal rate. In his reference frame. Special relativity is basically a theory of time and space. Time is dependent on the reference frame in which it is measured. That is the whole point. Different observers see different versions of "time".
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  20. #19  
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    Here is a analogous phenomenon which helps see why there is no paradox in the "twins paradox".

    Imagine two twins standing together on a flat plane. They begin walking at slightly different directions. As far as twin A is concerned twin B is "falling behind" and likewise as far as twin B is concerned twin A is "falling behind".

    This is because each twin is defining a different "forward direction".

    Now to make the analogy more precise suppose that the twins do not actually watch each other walking but rather backtracks later and count footsteps. Twin A will see that for a given amount of his own forward motion twin B has made more footsteps to achieve a given forward component of position and vis versa.
    (Imagine each twin has no depth perception and can only see the the component of the spacing of the other's footsteps parallel to his own path.)

    Taking the footsteps as analogues of clock ticks we then see that each twin sees the other's "clock" running faster. (The opposite of what happens in special relativity).

    The reason we see an opposite effect here is that we are describing different directions in Euclidean space. To change from one Twin's frame to the other's requires a normal rotation. In special relativity due to the indefinite metric of space-time to change from one twin's time arrow to the other's we must preform a pseudo-rotation i.e. a hyperbolic "rotation". (This hyperbolic rotation business is very non-intuitive but if you work the math and trust your numbers you can get a good sense of it with careful selective use of the analogy with standard rotations.)

    Where for example we see the forward (as defined by one twin) component to the distance between footsteps of the other twin proportional to the cosine of the angle between them...

    ...in space-time we will see the duration between clock ticks (as defined by one twin) proportional to the hyperbolic cosine of the pseudo-angle between them.
    (Recall that hyperbolic cosine gets bigger than one as we move away from cosh(0)=1 while regular cosine gets smaller away from cos(0)=1)

    But other than this reversal of sign the phenomena of footstep spacing and slowing clocks is exactly analogous. No paradox just different meanings to each twin's forward direction (either in space or in time). Or as they say...

    "It's all relative!"
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