Comments about Special Relativity in 14 Easy Steps

Introduction

Prof George Collins at University of Illinois has written a document to explain Special Relativity in 14 Easy Steps.
For the index of this document select: http://web.hep.uiuc.edu/home/g-gollin/relativity/index.html

Step 5

In Step 5 the length of Sulaco is measured at two different ways, called Method 1 and Method 2
In method 1 the length is measured with one clock.
The problem is you can not measure the length in this way, because what is the speed v of Sulaco ?
In order to measure speed v you need two clocks, for example one at the nose of Nostromo and one at the tail of Nostromo which have to be synchronized.
synchronized means that you send out a signal from the center of Nostromo. When this signal reaches a clock, the clock is set to zero.
The time interval for a light signal to go from tail to nose is dt1
The time interval for the front of Sulaco to coast past one of the clocks is dt2
The length of Sulaco is l * dt1 /dt2, with l being the length of Nostromo

Comment Step 5

No mention is made if this measured length is a function of v.

Step 6

In Step 6: Time dilation / Lorentz contraction both Time dilation and Lorentz Contraction are explained.
Time dilation is demonstrated by means of 2 clocks: one clock at rest (which measures a time dts) versus one moving clock (which measures a time dtn). That is a correct experiment.
Length contraction is explained by means of a speed v from two different points of view: Nostromo at rest and Sulaco moving versus Sulaco at rest and Nostromo moving.
This is complex. In my opinion the experiment described in step 5 should be used to demonstrate Length Contraction.

Comment Step 6

No mention is made that if also two clock's at Nostromo are used, that the final readings of the two clocks at the tail, when they meet, should be the same.

Step 8

The whole idea behind the experiment in step 8 is to have one event (a collision between the antennas at the center of each ship) and to observe the arrival time of the flashes at the two periscopes (front and tail) at each ship.
What that experiment teaches us is that the arrival time are the same
My suggestion is to make a small modification to this experiment: Replace the two periscopes at each ship by two mirrors and place the two periscopes at the center of the ship, such that you can see the flashes.
The following sketch shows this situation from the point of view from Nostromo at rest 
                                              t3                 
                                              .                 
                                            ..  .               
                                          . .     .            
                                  <--------B-------->t2
                                        .  .        .                
       Sulaco moving --->             .   .       .                
                                    .    .      .           
                                  .     .     .             
                                .      .    .  
                              .       .   . 
                          t1<--------B-------->
                            .        .  . 
                              .     . . 
                                .  .. 
                         <--------B--------> t0
----------------------------------X-----------------------------
                      <-----------------------> t0
                                . . . 
    Nostromo at rest          .   .   . 
                            .     .     .
                          .       .       .
                        .         .         .
                      .           .           .
                      <-----------A----------->t4
                        .         .         .
                          .       .       .
                            .     .     .
                              .   .   .
                                . . .
                                  .
                                  t5
  1. There is a collision at t0. The length of Nostromo is l0.
  2. The flashes are received at the mirrors of Nostromo at t4. t4 =0.5 * l0/c
  3. The flashes are received simultaneous at the center of Nostromo at t5. t5 = l0/c
  4. There is a collision at t0. The length of sulaco is l. l =
  5. The flash towards the tail of Sulaco reaches the mirror at t1. c*t + v*t = 0.5*l. t1 = 0.5*l/(c+v)
  6. The flash towards the front of Sulaco reaches the mirror at t2. c*t = 0.5*l + v*t . t2 = 0.5*l/(c-v)
  7. The time towards the tail reaches the center at t3. t3 = 0.5*l/(c+v) + 0.5*l/(c-v)= l*c/(c^2-v^2)
  8. The time towards the front also reaches the center at t3. That means an Observer at the center sees both flashes simultaneous.
  9. Accordingly to SR there is length contraction involved. That means that l = l0/gamma. gamma = 1/sqrt(1-v*v/c*c)
  10. Accordingly to SR the moving clock runs slower. That means you have to divide the time by gamma.
  11. As such t3 for a moving clock = l0*c/(c*c-v*v)/gamma^2 = l0*c/(c*c-v*v) * (1-v*v/c*c) = l0*c/c*c = l0/c
At the bottom of the page we read:
The crews on Sulaco and Nostromo disagree about the simultaneity of the arrival of light flashes at the various ships' periscopes.
The problem lies in the word simultaneity. The crews disagree about the arrival times etc.
But there is more:
  1. The observers at the center of each ship agree that each wil see the light flashes simultaneous. That does not mean, they agree about when.
  2. There are two events at each ship that each flash hits a mirror.
  3. Both Observers agree that of those two pairs of events, that not both pairs can be simultaneous.
  4. Both Observers agree that of those two pairs of events only one can be simultaneous. In dependent of the fact that each observer actual sees them simultaneous.
  5. Both Observers agree that of those two pairs of events most probably both are not simultaneous. They do not know the answer.
  6. Both observers agree that the problem about when is difficult because you can not perform this experiment in practice.
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Created: 7 January 2008

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