Physical_Anarchist said:
I'm no expert, so feel free to correct me here but the root of the problem arises from some strange property of light: it passes you by at the constant speed of 299 792 458 m/s. Even when you travel at 100 000 000 m/s relative to your friend, the light passes him by at 299 792 458 m/s and it also passes you by at 299 792 458 m/s. The answer then is time dilation.
Not alone, no. The fact that light is measured to travel at the same speed by all observers is a consequence of how Einstein proposed that each observer should define their own coordinate system--using a network of rulers and clocks which are at rest with respect to themselves, with the clocks synchronized using the "Einstein clock synchronization convention", which is based on each observer
assuming that light travels at the same speed in all directions in their coordinate system (so two clocks in an observer's system are defined to be synchronized if, when you set off a flash of light at their midpoint, they both read the same time at the moment the light from the flash reaches each one). The rationale for defining each observer's coordinate system this way becomes clear in retrospect, when you see that it is only when you define your coordinates this way that the laws of physics will be observed to work the same way in each observer's coordinate system (this is because the laws of physics have a property known as 'Lorentz-symmetry', the name based on the fact that the different coordinate systems described above will be related to each other by a set of equations known as the 'Lorentz transformation'). You're still free to define your coordinate systems in a different way, but then the laws of physics would take a different form in different observer's frames.
The Einstein clock synchronization is enough to insure that each observer will measure light to travel at a constant speed in all directions, as opposed to faster in some directions than others. But to explain why the
magnitude of this constant speed is the same for different observers, you also have to know that each observer will measure the rulers of those moving at velocity v relative to him to shrink by a factor of \sqrt{1 - v^2/c^2} and the ticks of clocks expand by a factor of 1/\sqrt{1 - v^2/c^2} (as long as the laws governing the rulers and the clocks have the property of Lorentz-symmetry, it's guaranteed this will happen). So, the fact that all observers measure the speed of light to be constant in all directions and the same from one observer's frame to another's is really a consequence of three things combined: time dilation, length contraction and "the relativity of simultaneity" (meaning that different observers will disagree whether a given pair of events happened 'at the same time' or not) which is a consequence of Einstein's clock synchronization convention. I posted a simple numerical example of how these three factors interact to insure a constant speed of light in
this thread, if you're interested.
Physical_Anarchist said:
It allows you to become slower even while you're traveling fast, so that you can see light pass you by at the same rate as before. Notice, however, that there is no acceleration in the problem, which means there shouldn't be in the paradox. Time dilation is a function of speed here, not acceleration.
Yes, in an inertial frame time dilation is
always a function of speed--if a clock is traveling at velocity v at a given moment, its rate of ticking will always be \sqrt{1 - v^2/c^2} times the rate of ticking of clocks at rest in that frame at that moment.
Physical_Anarchist said:
How do we then define the Twin Paradox without needlessly confusing the issue with accelerations? There's a number of ways we can do that.
1.Suppose that the twins are both astronauts. They each embark on a spaceship. They accelerate at the same rate, for the same predetermined length of time.
It is usually convenient in statements of the twin paradox to just assume the acceleration period is instantaneously brief, so that the twin switches from one velocity to another instantaneously.
Physical_Anarchist said:
Afterwards, one of them immediately engages his thrusters in reverse, in order to decelerate, and then to accelerate in the opposite direction, and finally decelerate again in order to stop at the point of origin. Meanwhile, the second one has stopped accelerating, so he is cruising at a uniform velocity. One year later, he does the same decelarating and accelerating maneouvres his bro did earlier. One year later, on final approach, he decelerates the same way his twin did and also ends up at the point of origin. Now they are together again, and the only difference is that one has been static for 2 years, while his bro has been in motion at a constant rate of speed. SR tells us that Time Dilation did occur in this scenario and that consequently one of them will be younger than the other. I challenge anyone to prove such a thing happened.
Simple, just analyze the problem from the point of view of the inertial frame of the spot where they both departed and later reunited (we can assume it's the earth, say). In this frame, one twin spent only a brief time moving at high velocity (suppose he instantaneously accelerated to 0.8c moving away from the earth, then after 0.01 years instaneously accelerated to 0.8c moving back towards it, then after another 0.1 years he reached Earth again and instantaneously accelerated so he was at rest on earth), while the other spent a whole year moving at high velocity. The first twin's clock was only ticking slow in this frame during the time he was moving relative to the earth, while the other twin's clock was ticking slow during the entire year, so the second twin's clock will have elapsed less time. Using the Lorentz transform, we could analyze this same situation from the point of view of any other inertial frame, and we'd always get the same answer to what the two clocks read when they reunited--I could show you the math if you want.
Physical_Anarchist said:
2.Another way we can frame the question is by having an alien with an atomick clock, onboard a spaceship in motion coincidentally synchronise it with an atomic clock situated somewhere along his flighpath. I am not suggesting causality, so there shouldn't be a problem with the fact that the 2 atomic clocks were reset at the same exact instant. The question then becomes: when the alien gets a snapshot of the second clock while flying by it, will the clocks still be synchronised?
Just to be clear, are there 2 different clocks in the alien's flightpath, as well as a third atomic clock on the alien's ship? And you're saying the alien's clock reads the same time as the first clock in his path at the moment he passes it, and then you want to know what will happen as he passes the second clock in his path and compares it with his own clock? In this case the answer will depend on which frame the two clocks were synchronized, because again, the "relativity of simultaneity" means that different frames disagree on whether two events (such as two different clocks ticking 12 o clock) happened at the same time or different times. If the two clocks are at rest with respect to each other and synchronized in their own rest frame, then in the alien's rest frame the first clock he passes will be ahead of the second one by a constant amount, and this explains why, even though both clocks are running slower than his, his clock still reads a smaller time than that of the second clock he passes (in the clocks' own frame, this is because the alien's clock was running slow). Again, I could show you a numerical example to explain why both frames make the same prediction about what the clocks read at the moment they pass despite disagreeing about which clock was running slow and whether or not the two clocks in his path were synchronized.
