Calculation 2

Example 2

For Example 1 See: Calculation 1
This example is based on page 23 of the book "Introducing Einstein's Relativity" by Ray d'Inverno. In that book the same example is discussed but IMO the full picture is much more complex as described on that page. There is no mention that length contraction is involved in the experiment; IMO there is.
On that page they highlight the concept of Simultaneity of Relativity. Of course it is difficult to establish the correct timing and order of a sequence of events, but that is a general problem inherent in all our observations and has nothing to do with Special Relativity.

Consider a train on a track with a length 2 * l0 = 2 * 300000 km
The train has a variable speed v

                          t1
             /          ./           /        /         /         /
            /         . /           /        /         t1        /
           /        .  /           /        /        ./         /
          /       .   /           /        /       . /         /
         /      .    /           /        /      .  t3        /
        /     .     /           /        /     .   /   .     /
       /    .      /           /        /    .    /      .  /
      /   .       t3          /        /   .     /         t2
     /  .        /  .        /        /  .      /         /|
    / .         /     .     /        / .       /         / |
t0 /.        O2/        .  /      t0/.     O2 /         /  |
  B-----------------------F        B---------/---------F-------->
 M1 .         |O1       . M2      M1 .         |O1      \  M2
  |   .       |       .   |        |   .       |         \ |
  |     .     |     .     |        |     .     |          \|
  |       .   |   .       |        |       .   |           |t2
  |         . | .         |        |         . |         . |
            t4,t5                  |           t4      .   |
                                   |           |     .     | 
                                   |           |   .       | 
                                   |           | .         | 
                                               t5

    No Lengt Contraction             Length Contraction
         
           B = Back of the train
           F = Front of the train
           . = Light signal
The above left figure shows the train when there is no length contraction involved
The above right figure shows the train when there is Length Contraction involved

Each figure is subdivided into two parts.

Calculator

In order to do the example for a particular value of v/c you can use the following calculator:
v/c v c

Calculations without Length Contraction.
l=l0 t1 t3 t4 t5

Calculations with Length Contraction.
l0 gamma l=l0/gamma
t1 t2 t3-t2 t3
t4 t2 t5


Reflection part 1

  1. When there is no Length contraction involved the observer at the track sees both lights simultanous at T4,T5
  2. When there is no Length contraction involved the observer at the train sees the light at T3 from the front before the light at T1 from the back.
  3. When there is Length contraction involved the observer at the track sees the light at T4 from the back of the train before the light at T5 from the front. This is opposite as compared with the moving observer.
  4. When there is Length contraction involved the observer at the train sees the light at T3 from the front before the light at T1 from the back. The time difference is much smaller compared with the No Length Contraction case i.e. case 3.
The main reason for the difference between what observer O1 sees, that in the case with Length Contraction, the train has to move a certain extra distance between the moment that the back hits M1 and for the front of the train to reach M2. As a result O1 will see light from the front later. In the case without Length Contraction there is no extra distance involved.
For O2 this extra distance results that O2 at the train will see the two signals closer together.


Reflection part 2


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Created: 17 January 2006

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