Length Contraction

Just as in the previous post where we found out that time is dependent on the relative speed of an observer it is also true for length.

Consider the experiment in post 2 where you are on a space station and a space ship flies past. Now have the experiment on the ship carried out in a way so that the flash of light sent out is in the direction of motion of the passing space ship. So now we have a flash of light sent out and reflected back to the source by a mirror.

Let’s define some events

The light travels from the source to the mirror a total distance of (L) which we can write as ( C ) the speed of light times the time taken. (CT)

On the space station we measure the light travelled a distance of L plus the distance the ship has travelled which is the time taken for the light to reach the mirror times the speed of the spaceship so the distance is (L)+(U)(T), where (U) is the speed of the ship and (T) is the time taken.

So we now have (L)+(U)(T)= (CT) with a little bit of maths this can be expressed as (T)= (L)/( C )-(U) (just re-arraging equation)

Working out for the return trip of the flash of light in the same way we get (T2) = (L)/( C )+(U)

Now the total time taken would be (T) time of light to the mirror plus (T2) the return journey, which is (L)/( C )-(U) plus (L)/( C )+(U) = total time = (2L)/( C )(1-(U²)/( C² ))

If you look closely this looks familiar to 1/square root (1-(U²)/( C² ) , what we defined to be gamma. So if we multiply both sides we get the change in time divided by gamma = (2L)/( C )

Some more maths of combining previous equations we get an expression for length contraction

Where length measured from the reference frame of an observer on the space station is the proper length divided by gamma.

Example

Superman is a handsome 2 meters tall and towers over me by only 1 inch (that 1 inch makes all the difference). As I am setting my telescope up for a good night viewing I see superman fly past at 0.99( C ). Using the equation from above for length contraction

Gamma = 1/square root (1-(0.99C)/( C )) = 7.1

Height of superman 2 meters divided by gamma 7.1 we get 0.282 meters (Hmmm not so big now are we)

Summary

We have found that time and length are both dependent on the relative speed of the inertial reference frame but are there any quantities that are independent of the reference frame. The answer is yes the speed of light is independent it is constant in every reference frame.

After thought

Time dilation and length contraction both use the fact that the speed of light is 299 792 458 m / s, that nothing can travel faster than light and that the speed of light is the same in all inertial reference frames. I have only being told these facts so I was wondering if anybody knows of any experiments that have been carried out or observations to validate these claims.