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I have a simple question regarding basic thought experiments to do with special relativity (which we just started in school). The thought experiment our class went through to explain time dilation was of a train travelling at relativistic speeds with a pulse of light moving up and down in a straight line within the train (reflecting off a mirror on the roof). Relative to an outside stationary observer, the light pulse is moving over a greater distance than just up and down (it is travelling the hypotenuse of a right angled triangle because of the train's motion on the x-direction), but because light travels at 'c' in every reference frame, the pulse must still travel at the same speed 'c' relative to the outside observer. Hence, because it travels a greater distance with the same speed, it must take longer to do so and hence time will appear to be running slower within the train - relative to the man outside. I understand the logic behind this however it seems to me that the thought experiment could be turned on its head by changing the direction the pulse is fired at. Say for example that instead of straight up and down, the light pulse is shot in a direction opposite to that of the trains motion, with an x-component equal to the motion of the train (i.e. if the train is travelling right at 0.5 'c', then the light pulse is fired left with an x-component of 0.5 'c'). If this were the case, the stationary man outside would now see the pulse travelling straight up and down relative to him and therefore see the pulse travelling a smaller distance. because the pulse still travels at 'c' relative to the outside observer he will now see the pulse travelling a smaller distance with the same speed, i.e. in a quicker time. To the outside observer, time now looks as if it has sped up inside the train (but of course this must be wrong because 'time dilation' says that time will always slow down for a stationary observer looking into a fast moving reference frame) I am wondering what exactly is wrong with this thought experiment. it uses the same logic as the original thought experiment (which was taught in my syllabus) but achieves the opposite result, it seems to prove time 'contraction' rather than time dilation. Help would be greatly appreciated.
Question Date: 2011-01-22
Answer 1:

My mind is busy with other things now, so I'll just ask you a dumb question: Would 'length contraction' help explain your thought experiment?

You explain how light can travel at its proper speed if it travels a longer distance in a longer time - basically, the light is traveling in a WWWWWWW pattern to the outside observer, while the observer on the train just sees it going up and down in a single vertical line, 'I'.

So if you are worried about time contraction, you could get back to the proper speed of light by invoking 'length contraction' - the light travels a shorter distance in a shorter time.

I came up with this possibility by googling : 'special relativity train light' and 'time contraction relativity'.As usual, Wikipedia seemed to be the best source of info. They have an article on 'length contraction' that might [or might not] be useful to you.

Best wishes,

Answer 2:

In the theory of relativity, speed is calculated in the usual way: distance covered over time elapsed. However, the tricky part is to get both, distance and time right. Here is an example:

Say you want to travel to Proxima-Centaury, a star about 4.2 light years away, at a speed of 80% that of light.

From the definition of speed,
v = d / t,
you will take a time
t = d / v = 4.2 ly / (0.80c) = 5.25 y = 5
years and 3 months,
where c = 1 ly / y is the speed of light using ly (light year) for the unit of length and y (short for year) the unit of time. This will be the time as measured on Earth.

One of the features of relativity is time dilation, where the elapsed time according to clocks on board your spaceship, is given by
t' = t (1-(v/c)2)(1/2)
or t' = Sqrt(1-(v/c)2),
where Sqrt is short for 'Square Root of'. Using our numbers we get
t' = 5.25 y Sqrt(1-(0.8)2) = 5.25 y (0.6) = 3.15 y
so that you (along with everything on board your spaceship) will only age 3.15 years instead of the 5.25 years your trip will take according to Earth based clocks.

Since 3.15 years is less than the 4.2 years it takes light to travel from earth to Proxima Centauri, you might be tempted to conclude that you will be traveling faster than light. How do we reconcile this numbers with the fact that you are really traveling at only 80% the speed of light? The answer lies in another feature of relativity where distances contract along the line of motion. To better understand this, imagine that as you begin your journey you accelerate to 0.8c in the neighborhood of Earth (maybe just going round in circles) so that you start your trip already moving at 0.8c. Even though you are still near the Earth, when you measure the distance to Proxima-Centauri you will find that it is only (using the length contraction formula)
d' = d Sqrt(1-(v/c)2) = 4.2 ly Sqrt(1-(0.8)2) = 4.2 ly (0.6) = 2.52 ly
away from you, as if Proxima-Centauri had been magically been brought closer to you as you start the journey at a place nearby the Earth.

So, what you see now is a star that is 2.52 light years away from you and approaching (remember that motion is relative) with a speed of 0.8c. To find the time of arrival of the star at your spaceship we use the same formula we used above to find:
t = d / v = 2.52 ly / (0.8 c) = 3.15 y.

So everything adds up; no strange things happening here other than, of course, relativity's funny features of time dilation and length contraction.

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