posted
I am, for my own amusement and in order to procrastinate, trying to obtain a layman's understanding of special and general relativity theory. Perhaps not surprisingly, this is turning out somewhat confusing and rather than spending half an hour googling for answers, I thought it would prove both more amusing and more time consuming if I asked knowledgable Hatrackers to help me.
Some time ago, on the other side of Hatrack, King of Men pointed out that the paradoxical thing about the twin paradox was not so much that one twin turned out younger than the other when they reunited. Rather, since special relativity postulates that it is impossible to determine who is moving in relation to the other the (seeming) paradox is that it is the travelling twin who is older when it could just as well been the "stationary" twin who had been moving. He answered himself by stating that what can be determined, also in a relativistic universe, is that the travelling twin had been accelerating.
From this scant evidence and my own vague recollections I concluded that actual time dilation (as opposed to perceived relativistic time dilation where two observers passing each other at relativistic speed both see the other as moving in slow-motion) is a function of acceleration rather than speed (question no. 1: is this correct?). This would seem to square with general relativity which postulates that acceleration and gravity is indistinguishable, both expressing themselves as a curvation in space-time. And time dilates in a gravity well.
My problem is that when I read about experiments such as sending an atomic clock around the world in a Boeing I get the impression that the measured difference in time relative to a stationary clock is a result of its speed rather than it having undergone different accelerations. Also, when calculating time dilation in e.g. the twin paradox experiment focus is on one twin having obtained e.g. 80 % of c rather than on the acceleration needed to achieve this speed. This would make sense if time dilation is linear in relation to acceleration, i.e. accelerating to 80 % of c using 1 G space ship would result in same net time dilation as obtaining the same speed using the ultra-modern 100 G space ship. But if that were true, the twin who stayed on earth would age approximately at the same rate since he lives at a constant 1 G acceleration during the whole time the other twin is travelling. The only difference would be the travelling twin having to escape earth's gravity well, which would not account for much. Am I missing something?
I have more questions, but if someone would explain the above to me, it would be a start.
posted
I've only got about five minutes, but I'll see what I can do later this week. They are interesting questions, I can't think of the answer offhand.
Posts: 10645 | Registered: Jul 2004
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The paradox lies in the question "Why is the traveling brother younger?" Special relativity tells us that an observed clock, traveling at a high speed past an observer, appears to run more slowly....Since relativity says that there is no absolute motion, wouldn’t the brother traveling to the star also see his brother’s clock on the earth move more slowly? If this were the case, wouldn’t they both be the same age?Posts: 1256 | Registered: May 2005
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posted
Thanks for digging up that link, camus. From this:
quote:When the paradox is addressed, it is usually done so only briefly, by saying that the one who feels the acceleration is the one who is younger at the end of the trip. Hence, the brother who travels to the star is younger. While the result is correct, the explanation is misleading. Because of these types of incomplete explanations, to many partially informed people, the accelerations appear to be the issue. Therefore, it is believed that the general theory of relativity is required to explain the paradox. Of course, this conclusion is based on yet another mistake, since we don't need general relativity to handle accelerations. The paradox can be unraveled by special relativity alone, and the accelerations incurred by the traveler are incidental. An explanation follows.
it would appear that I was wrong (although it's nice to be thought of as "partially informed" ). Unfortunately, I do not completely understand the explanation. I can follow the offered example, and the graph is nice, but the why of it all escapes me. The crux of the matter seems to be this:
quote:The traveler uses the length-contraction equation of special relativity to measure distance. So the star six light-years away to the homebody appears to be only 4.8 light-years away to the traveler at a speed of 0.6c. Therefore, to the traveler, the trip to the star takes only eight years (4.8/0.6), whereas the homebody calculates it taking 10 years (6.0/0.6).
So the trip is actually shorter at 0.6c, or at least appears to be shorter. Huh. I still think there ought to be an answer to the "why" of this beyond "because". And the acceleration seems to fit the bill. The article says that the traveler leaves earth reference frame (and returns) whereas the homebody stays put. But the means of leaving is acceleration and it seems counter-intuitive that gravity is only "incidental". Perhaps this is one of those things that can only be understood mathemagically.
There doesn't seem to be very much interest in this topic, but here is one other thing I've been wondering about. I've been trying to visualize the geometry of space-time using the common representation exemplified in this wikipedia article. I've been thinking sharper curve = higher gravity = greater time dilation. But on a physics forum I happened to come across someone, who seemed to know what he was talking about, wrote that although you would be weightless in the middle of the earth, time dilation still applies. If this is correct, would it mean that opposing gravity forces doesn't create a "flat" space-time geometry, but rather one that curves in all directions? And/or that time dilation is somehow unrelated to perceived acceleration?
Posts: 896 | Registered: Feb 2001
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posted
Remember, the curvature is in 4 dimensions, not the 2/3 dimensions you'll see represented in images. What do you mean by "time dilation" in the context of being in the middle of the earth? Massive objects do create "wells" in space time, they do not flatten out. From the center of the earth, to get back to the surface, you are going "uphill", no matter which direction you take from the center. "Flat" spacetime can occur only in the absence of gravitational effects (which essentially means nowhere, but empty space between galactic superclusters probably is a close approximation).
-- As for the length-contraction, if the person is traveling at a constant .6c in a vacuum, then there is NO acceleration (aside from the amount needed to get up to that speed, turn around, and then slow down to stop at earth). So I don't think acceleration can explain it. It has to do with the fact that even though the person travelling is going .6c, they still observe light from the destination star at going at a constant velocity c. Yet the person on the earth also sees that light as going at c. That seems paradoxical! How can someone going so fast toward the light not see the light as moving faster? The solution is either time-dilation (time slows down for the traveler) or length contraction (the distance the light being observed by the traveller must be shorter). These phenomena are two sides of the same coin. Unfortunately I don't know the theories well enough to know when one ought to use length-contraction vs. time-dilation.
Ultimately, a lot of the solutions are because the equations work out that way... And experimental evidence appears to corroborate the equations solutions
quote:It has to do with the fact that even though the person travelling is going .6c, they still observe light from the destination star at going at a constant velocity c. Yet the person on the earth also sees that light as going at c. That seems paradoxical! How can someone going so fast toward the light not see the light as moving faster? The solution is either time-dilation (time slows down for the traveler) or length contraction (the distance the light being observed by the traveller must be shorter).
It appears to me to be a combination of both. The time dilation that is experienced by the traveler affects his measurement of distance (d=v*t). Since "t" for the traveler is different than it is for the homebody, "d" is also different.
Here is a link that has a pretty helpful description of Length Contraction. There are even pictures and animations to help visualize the concept.
quote:Perhaps this is one of those things that can only be understood mathemagically.
Hey, I like that term "mathemagically." It pretty much sums up my grasp of the higher mathematics that are involved in these types of calculations.
Posts: 1256 | Registered: May 2005
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posted
I went back and reread all of the above posts and I'm now pretty sure that I either over simplified the original question or addressed a different question altogether.
So the original question, if I understand correctly, is, if it is impossible to determine which body is stationary and which body is moving, then what determines actual time dilation?
My (rather uneducated) guess is that since acceleration can be determined, the acceleration determines which body is actually moving. The fact that the traveler has to both accelerate and then later decelerate when he arrives back at earth would imply that he was the one that was actually moving. So the speed is actually what determines time dilation, but it is the acceleration that determines which body has the speed. Again, I'm probably over simplifying something here, so hopefully someone more knowledgeable will speak up.
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posted
Speed determines time dilation, not acceleration. As to why...
Take a photon, bounce it back and forth between two reflectors which are 1 light second apart. The time from impact on one to the other is 1 second. This is our clock. Now move the plates at some velocity in one direction. The photon cannot move either faster or slower than it originally was; it's a photon that moves at the speed of light c. However it must now travel a longer distance to get from one reflector to the other, so it takes more time. Our second is now longer.
The above is a crude explanation which glosses over gravitational effects, quantum stress, and medium (the stuff you are traveling through). But it is valid. What determines who is moving is relative to the speed of light, which doesn't change except in relation to what it's traveling through. Which segues nicely to:
Bokonon mentioned that a person traveling at .6c sees the same light as the person not moving. That's not actually true. You still see, but you see different light. At all times, our sun is emitting a very large spectrum of electromagnetic radiation (light), some of which we can see, some of which we can't. Let's pretend for a moment that the sun emits all wavelengths equally, making our sun a black-body radiator. If I'm standing still, I see between .7 and .4 micron (um) waves. If I start moving away from the sun, my eyes can still only detect between .7 and .4 um, but the waves appear longer than they are. So I will be seeing waves which are actually, for example, between .6 and .3 um. If I move towards the sun, it goes the other way, I will see between .8 and .5 um. This is called a "shift" or a "doppler effect" depending on whether the observer or the light source is moving.
Something similar happens with sound waves. If you stand near a highway and listen to a truck as it passes, it sounds higher pitched when it's coming towards you than it does when it's moving away. The waves haven't changed, they are just reaching you faster or slower.
Posts: 354 | Registered: Jan 2006
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posted
I thought that the Universe was laid out like a stack of pancakes made of octahedrons, with a big cylinder-shaped hole in the middle. In that case, all time would be relative to the center point, would it not? In other words, depending on the time of day and time of year, time would flow a little differently in different places around the planet, depending on the orientation to the galactic center.
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Bunch of us used to make flyers of TimeCube stuff to put around the cafeteria at college. Watching people's reactions was the funniest thing EVER!
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quote:What do you mean by "time dilation" in the context of being in the middle of the earth? Massive objects do create "wells" in space time, they do not flatten out.
Yes, I'm aware of this. My thinking goes as follows: earth creates a gravity well. The geometry it creates in space-time translates, or is perceived, as a 1G acceleration on the surface. As per Newton, this "force" decrease in proportion to the square of the distance if you leave the surface upwards. But what if you dig down? The further down you are, the less the acceleration towards the centre is felt until you float apparently weightless in the middle of the earth. Would this not indicate that the geometry of space-time progressively "flattens" when moving below the surface?
From the wikipedia link in my post above:
quote:Gravitational time dilation: Clocks will run slower at lower gravitational potentials (deeper within a gravity well). Confirmed by the Haefele-Keating experiment and GPS.
If gravitational time dilation applies (equally, or stronger) also in the centre of the earth where no perceived acceleration is felt, then my assumption of the shape of space-time cannot be correct. Somehow the mass of Earth distorts space-time also when interferring equally from all direction. I was wondering how this distortion could be imagined.
Camus,
quote:So the original question, if I understand correctly, is, if it is impossible to determine which body is stationary and which body is moving, then what determines actual time dilation?
Yes, this was my original problem. And I have more or less arrived as the same (uneducated) solution as you. But I'm still not happy with it .
Raventhief,
quote:Now move the plates at some velocity in one direction. The photon cannot move either faster or slower than it originally was; it's a photon that moves at the speed of light c. However it must now travel a longer distance to get from one reflector to the other, so it takes more time. Our second is now longer.
Well, the second where the light travels at the same direction as the plates move is longer; the second back is shorter. But I think this example is useful, I just have to ponder it a bit longer. Thanks, anyway. I did not see the relevance of the Doppler effect at first (although it is mentioned in Camus' first link), but I am beginning to get it. The Doppler effect helps to define the common reference frame from which the traveler departs (and return to) during his trip.
I like how the stars are actually closer if you travel by relativistic speed. It's annoying of you plan on going back, but it does mean that a star that from earth appears to be 200 light-years away is actually reachable within a human life time.
Posts: 896 | Registered: Feb 2001
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posted
Tristan, here is a description of an experiment that helps explain why the clock is slower in the gravity well. Essentially 2 (at least) factors affect time-dilation: velocity and gravitational effects. It's all due to the different effects outlined by special and general relativity, I think. Apparently the gravitational effects of being "higher" in a well are larger than the effects of being at a higher velocity:
quote: * Hefele and Keating, in 1971, flew caesium clocks east and west around the Earth in commercial airliners, to compare the elapsed time to that for a clock that remained at the US Naval Observatory. Two opposite effects came in to play. The clocks were expected to age quicker (show a larger elapsed time) than the reference clock, since they were in a higher gravitational potential for most of the trip (c.f. Pound, Rebka). The clocks were expected to age slower because of the speed of the travel. The gravitational effect was the larger, and the clocks suffered a net gain in elapsed time. To within experimental error, the net gain was consistent with the difference between the predicted gravitational gain and the predicted velocity time loss. In 2005, the National Physical Laboratory in the United Kingdom, report their limited replication of this experiment. The NPL experiment differed from the original in that the ceasium clocks were sent on a shorter trip (London-Washingon D. C. return), but the clocks were more accurate. The reported results are within 4% of the predictions of relativity.
posted
Tristan: sorry, I wasn't clear. The plates and motion are oriented in the same direction. So if I hold up my two hands as the plates so the photon bounces from my left to right, then I move forwards. No matter the direction of travel, the dilation is the same, we just have to reorient our clock. As to why... I'm not quite sure, and I don't think anyone is.
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). Unfortunately, I do not completely understand the explanation. I can follow the offered example, and the graph is nice, but the why of it all escapes me. The crux of the matter seems to be this:
