The Twins Paradox

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LNT: not only that, but length changes too for objects moving with close to the speed of light That is, the faster they move, the shorter they appear.

Another paradox: let's say you have a car 5 meters in length, and it moves fast enough that its length for an observer on the ground is 3 meters. Now, let's assume that you have a 3.2 meters long garage, with very fast doors (one at one end, one at the other). So, theoretically, you should be able to enclose that car in the garage for a very short period of time. But, for a person in the car, its length is still 5 meters, so how will it fit in a garage 3.2 meters long?

welcome to the crazy world of relativity

Edit: HB

I guess the easy answer is that the photon is a wave, not a particle. Now, if a particle moves fast enough (close to th espeed of light) it will aquire the characterstic behaviour of a wave. So if you can make the train run fast enough ...

I am sure there is a more complete answer, though. Speed of the photon has to be the same for all observers because that's what coming out from the Maxwell equations governing the behaviour of electric and magnetic fields (and these equations are based on experimental measurements, so we know that they are right for our world). Now, it is an interesting question if one could build a theory where one does not have this constraint... but I suspect it would look much different from what we see.

[ September 22, 2005, 00:04: Message edited by: khaavern ]

 

Another good book on time dilation etc is Tau Zero by Poul Anderson

However the twin paradox has problems when you try to apply it to the real universe because we ignore the effects of certain forces and principles. Chief among these is gravity (on the macro level) which is assumed to be uniform but isn't; and quantum dynamics (on the micro level) which confuses most specific applications of physics. Einstein himself did not like Quantum Physics which partially led to his quote 'God does not play dice with the universe'; sadly he/she does and cheats.

 

@HB: Well, I can't tell you why (that's metaphysics - ask chev ) but I can tell you how velocities add (as I seem to be on a roll with the ASCII equation thing). The equation for velocity addition is as follows:

v(total) = v(1) + v(2) / {1 + [v(1)*v(2)/(c^2)]}

Which isn't very friendly-looking, I'll admit. Basically, the numerator is the normal, Newtonian Law of velocity addition - just add v(1) and v(2) to get the total velocity, which I have imaginatively denoted as v(total). The denominator adjusts our velocity addition law for speeds approaching that of light.

Now imagine the case of two photons travelling in opposite directions - in Newtonian physics, we expect the relative velocities to add to give 2c. If we enter that into the above equation, however, we find that the denominator exactly equals 2. Thus, our speeds still add to give c. By implication, you're not going to get any object travelling less than c observing something else travelling at speeds >c, even if that something else is a photon travelling in the opposite direction.

EDIT: khaavern - the photon isn't a wave or a particle; both waves and particles are convenient approximations of reality that apply in different situations. All we "really" have are "wavicles", as my old QM professor used to call them.

And I'd really, really steer clear of QM right now - if anyone thinks Relativity is odd and makes no sense, QM will make your head implode.

 

If an imploding head was moving away from me at .999 c....

 

The only thing you need to understand relativity is to think in an organized manner.

Allow me to demonstrate. I will prove that, contrary to accepted wisdom, time passes slower for the person on the ground

For this, let's consider a hypothetical spaceship, long enough that light takes one second from one end (A) to the other (B). Let's say that it moves from left to right (as in figure below) with respect to an observer on the ground, with a speed v = c/2:

<A--------B> >>>

now let's say that at time zero a lightbulb is turned on at point B. For an astronaut, it will take one second for the light to get from one end to the other.

However, an observer on the ground will see light starting at time 0 in point B, going with speed c towards A. However, A is also moving toward left (with speed c/2), so light will get there faster than 1 second (since the relative speed is 3/2 c, it should take approx. 2/3 seconds).

So, it took a shorter amount of time for light to get from B to A for the observer on the ground, than for the cosmonaut! Time passes slower on the ground, then (since speed of light is constant). Statement proved

But wait! Let's assume that there is a mirror in point A, which reflects light back towards point B. For a cosmonaut, it will take again one second for the light to reach B. For the observer on the ground, however, it will take about 2 seconds (since the relative speed of light and point B is c/2). So now time passes faster on the ground. Aaagh! running away

Magpie: you are right, of course

 

Hmm...

Would the observer from the ground see things like this (X marks spot of light hitting end of ship and my dots are not exact measurements of space)..


X....X............X....X...........X.....X...........X....X

and from the ship...

X.........X.........X..........X..........X..........X.........X

So the two observers would not agree on when or where the different events occured.

 

Bah. The time is not passing any faster or slower in that example. The observer on the ground will see the spaceship moving as well so will also see the light moving at 2/3 second.

 

No. The observer on the ground would measure the speed of light as c. The astro/cosmonaut would also measure the speed of light as c, but with the events occuring at different locations. And I have no idea if time dilation is involved. Nor do I really know what the hell I am talking about.

[ September 22, 2005, 02:07: Message edited by: Late-Night Thinker ]

 

LNT has some of it. The 'where' the events happen is as important as the 'when' they happen.

So, in relativity, does not quite make sense to compare intervals of time only. Let's look at the example with the spaceship: we have three events: (1) lightbulb turns on in point B, (2) light reaches point A, (3) light gets back to point B. The time inteval between events 1 and 2 is longer for the astronaut than for the man on the ground; the time interval btw events (1) and (3) is longer for the man on the ground. Of course, events (1) and (2) happen at different points in space with respect to the astronaut, while events (1) and (3) happen at same point in space (for him, obviously not for the man on the ground).

So then what do we mean by saying that time passes slower on the moving ship? Some stuff which happens at the same location on the ship takes longer according to the observer on the ground. So, for example, if the astronaut sits at a table and drinks his cofee in five minutes (after his watch), the man on the ground will say that more time (say, seven minutes) have passed. On the other hand, if the astronaut strolls to the other end of the ship during his cofee break the man on the ground might very well say that less than 5 minutes have passed (remember it's a pretty long ship ).

And finally, we can use the example with the spaceship to actually compute the factor by which time is slowed down . It goes like this:

The time astronaut measures between events (1) and (3) is 1 + 1 = 2 seconds (we said that the ship is long enought that it takes light one sec to get from one end to the other).

OTOH, the observer on the ground measures
1/(1 + v/c) + 1/(1-v/c) = 2/(1 - v^2/c^2) right?
(this would be 2/3 sec + 2 sec, if we take v = c/2).

Well, actually is not quite so, because if you remember, we said that things in movement appear shorter than things staying at rest. So if the spaceship is length L at rest, it will appear to have length r*L for the observer on the ground (where r is a factor smaller than 1). So then, the time measured on the ground will be
2 r / (1 - v^2/c^2)

So, how do we find out r? Simple, just use the principle of symmetry. That is, we can assume that the same factor involved in time dilatation is also involved in length contraction. So the time measured by the observer on the ground when the asstronaut measures 2 sec should be 2/r sec. (divide by r rather than multiply, because is longer). So then we can write
2/r = 2 r / (1 - v^2/c^2) ==> 1/r^2 = 1/(1 - v^2/c^2)

which is the formula which The Magpie wrote down earlier (and Lorenz derived in early nineties).

 

I think it is fascinating that in reality --whatever that means-- there is really no single universe; events happen at different places and at different times for every single observer. But yet they are all related in some fashion which I cannot fathom. For example, I have seen pictures of the grand structures of the universe. You know the one, it looks somewhat like if you placed a small explosive in a sphere of sand and then took a picture a microsecond after the 'boom'.

How did they determine this?

Is it only how the universe appears to us at our location? Or is it the average location of all galaxies based upon a summation of viewpoints as the number of observers approaches infinity? I just imagine the universe would appear quite different from a galaxy racing away at the edge of the visible universe. And we would both be correct.

There is no absolute when or where for anything. That is a mind bender. But yet we can determine how another observer would view things.

So is it mathematically possible to determine the "true" when/where based upon the average when/where as the number of observers approaches infinity?

I, of course, could not even come close to being able to accomplish such a thing. Multivariate calculus is taxing my abilities; I am much better at chemistry than I am at physics. But I'm still very curious!