Calculations

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.

The top part shows the situation for the moving observer:
- At t0 there is a light signal at the back of the train.
- This light signal reaches observer O2 at the train at t1.
- For the left train at t0 there is also a light signal from the front of the train
- For the right train there is a light signal at t2 from the front of the train
- The light signal from the front reaches observer O2 at the train at t3
The bottom part shows the situation for the Observer O1 at the centre of the track.
- At t0 there is a light signal from the back of the train.
- This light signal reaches observer O1 at the track at t4.
- For the left train at t0 there is also a light signal from the front of the train
- For the right train there is a light signal at t2 from the front of the train
- The light signal from the front reaches observer O1 at the track at t5

Calculator

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

The first line shows the value v/c in red box.
This value can be modified:
Enter a value and select any other point on this diplay in order to excute the calculations.

Enter 0.8, 0.99, 0.999 and 0.

The other values are resp: the speed v, the speed of light c and gamma

Reflection part 1

When there is no Length contraction involved the observer at the track sees both lights simultanous at T4,T5
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.
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.
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

One important remark is that in reality it is not possible to perform this experiment. It is not possible to demonstrate length Contraction.
A second more important remark is that based on what Observer O1 at the track sees is that a garantee that the events that caused those light signals are or are not simultaneous events. IMO that is not for sure. This becomes even more tricky when Length Contraction is involved
- With Length Contraction you have to increase the length of the train in order that O1 sees the signals simultaneous, but IMO that is no garantee that they are simulataneous.
- Even in the case without Length Contraction I have doubt if Observer O1 actual sees the two light signals simultaneous. No experiment can either approve or disapprove this doubt.
And third are we sure that only Length Contraction is involved and no Length Expansion in order for O1 to see the signals simultaneous ?

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Created: 17 January 2006

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l0	gamma	l=l0/gamma

t1	t2	t3-t2	t3

t4	t2	t5