THE TWINS PARADOX (2)
=====================

There is another way of describing the situation of the twins. It consists in placing clocks at regular intervals in all the frames of reference. Of course, these clocks will have to be synchronized. We will therefore use the method recommended by Einstein's theory. Each observer will send signals to each clock placed in his frame, and admit that the time indicated by the clock when the signal reaches it equals half of the round trip time as mesured by the observer. We saw that, doing so, each observer considers his clocks to be synchronized, but those in the other frames unsynchronized.
We could use Lorentz's formulas to calculate the time indicated by the clocks, but it seems preferable, in order to better understand the situation, to do the calculation in detail.

These are the (very simple) formulas to be used:
If in a frame of reference, moving at v, a clock is placed at a distance d "after" the observer, the signal will take the time t1 = d / (c + v) to reach the clock, and t2 = d / (c - v) to come back. The observater supposes that the signal hit the clock at the moment tm = (t1 + t2) / 2.

Let's say B travels at 0,8660254c, speed to which his clocks slow down half.
If B places his clocks at 0,866.. light-minute of interval (distance as mesured by A), this is the result of B's synchronization, seen by A:
t1 = 0.8660254 / 1.8660254 = 0,4641016
t2 = 0.8660254 / 0.1339746 = 6.4641014
tm = (0.4641016 + 6.4641014) / 2 = 3.4641015
Each clock in B's reference frame advances tm - t1 = 3 minutes compared to the "preceding" clock.

However, as the clocks in B's reference frame slow down half, due to the speed, we must divide all figures by 2.
B's watch thus indicates 3.4641015 minutes when the signal returns, tm(B) = 1.7320508, t1(B) = 0.2320508, and according to the indications of B's clocks, the advance is 1.5 minute per light-minute.

To siplify the drawings, we will place the clocks at the double distance, in both frames of reference. So, one minute passes, on B's watch, between each passage by A's clocks. To allow B to make half-turn while remaining in the framework of special relativity, we add a third reference frame which moves in the direction opposite to B, at the same speed.
Let's compare the points of vue of A and B.
                  VERSION A                                        VERSION B

                                              |                                                           
  --+--+--+--+--+--+--B->                     |   --+--+--+--+--+--+--B-
   18 15 12  9  6  3  0                       |     0  0  0  0  0  0  0
                      |                       |                       |
         -------------A--+--+--+--            |                     <-A--+--+--+--
                      0  0  0  0              |                       0 1.5 3 4.5
                      |                       |                       |
                    <-+--+--+--+--+--+--+--   |                    <<-+-----+-----+-----+--
                      0  3  6  9 12 15 18     |                       0
                                              |                                                          
                                              |                                             
     --+--+--+--+--+--+--B->                  |   --+--+--+--+--+--+--B-
      19 16 13 10  7  4  1                    |     1  1  1  1  1  1  1
                         |                    |                       |
         -------------A--+--+--+--            |                  <-A--+--+--+-
                      2  2  2  2              |                   0.5 2 3.5 5
                         |                    |                       |
                 <-+--+--+--+--+--+--+--      |              <<-+-----+-----+-----+--
                   1  4  7 10 13 16 19        |                       7
                                              |                                             
                                              |                                                          
        --+--+--+--+--+--+--B->               |   --+--+--+--+--+--+--B-
         20 17 14 11  8  5  2                 |     2  2  2  2  2  2  2
                            |                 |                       |
         -------------A--+--+--+--            |               <-A--+--+--+--
                      4  4  4  4              |                 1 2.5 4 5.5
                            |                 |                       |
              <-+--+--+--+--+--+--+--         |        <<-+-----+-----+-----+--
                2  5  8 11 14 17 20           |                      14
                                              |                                             
                                              |                                                          
                                              |                                                          
           --+--+--+--+--+--+--B->            |   --+--+--+--+--+--+--B-
            21 18 15 12  9  6  3              |     3  3  3  3  3  3  3
                               |              |                       |
         -------------A--+--+--+--            |            <-A--+--+--+--
                      6  6  6  6              |             1.5 3 4.5 6
                               |              |            --A--+--+--+->
           <-+--+--+--+--+--+--+--            |            10.5 9 7.5 6
             3  6  9 12 15 18 21              |                       |
                             B=3              |   --+--+--+--+--+--+--+--
                                              |    21 21 21 21 21 21 21
                                              |                                                          
                                              |                                                          
                                              |                                                          
              --+--+--+--+--+--+--+->         |   --+--+--+--+--+--+--B-
               22 19 16 13 10  7  4           |     4  4  4  4  4  4  4
                            |                 |                       |
         -------------A--+--+--+--            |               --A--+--+--+->
                      8  8  8  8              |                11 9.5 8 6.5
                            |                 |                       |
        <-+--+--+--+--+--+--+--               |   --+--+--+--+--+--+--+--
          4  7 10 13 16 19 22                 |    22 22 22 22 22 22 22
                          B=4                 |                                                         
                                              |                                                          
                                              |                                                          
                                              |                                                          
                 --+--+--+--+--+--+--+->      |   --+--+--+--+--+--+--B-
                  23 20 17 14 11  8  5        |     5  5  5  5  5  5  5
                         |                    |                       |
         -------------A--+--+--+--            |                  --A--+--+--+->
                     10 10 10 10              |                 11.5 10 8.5 7
                         |                    |                       |
     <-+--+--+--+--+--+--+--                  |   --+--+--+--+--+--+--+--
       5  8 11 14 17 20 23                    |    23 23 23 23 23 23 23
                       B=5                    |                                                           
                                              |                                                          
                                              |                                                          
                                              |                                                          --+--+--+--+--+--+--+->   |   --+--+--+--+--+--+--B-
                     24 21 18 15 12  9  6     |     6  6  6  6  6  6  6
                      |                       |                       |
         -------------A--+--+--+--            |                     --A--+--+--+->
                     12 12 12 12              |                      12 10.5 9 7.5
                      |                       |                       |
  <-+--+--+--+--+--+--+--                     |   --+--+--+--+--+--+--+--
    6  9 12 15 18 21 24                       |    24 24 24 24 24 24 24
                    B=6                       |                                                         
                                              |                                                          
                                              |
The drawings above show well how much the explanation by special relativity is incomplete.
In A's version, at the time of the turnaround, B switches to another reference frame, in which the clock that was synchronized at the beginning of the trip marks 21 minutes. But that doesn't have any influence: B, whoses watch marks 3 minutes, holds its own time.
In B's version, it's A's frame that changes, but in this case, whe ought to suppose that A's time "jumps" 9 minutes ahead so that at the end of the journey, A, who according to B should have aged only 3 minutes, would in fact have aged 12 minutes.

It is clear that, even if the formulas make it possible to calculate the two versions, this is a rather absurd situation.

Can general relativity, which handles noninertial reference frames, bring a solution?
At first sight, yes.
We learn from RG that time slows down in a gravitational field.
According to the equivalence principle, the acceleration forces can be assimilated to a (pseudo-) gravitational force.
During the half-turn (which, in reality, is obviously not instantaneous), B will ondergo such a force that he could be allowed to consider that the time of A runs so rapidly, compared to his time, (- in fact, his time would slow down -), that, after the turnaround, A will have "advanced" the 9 minutes necessary to bring the situation in agreement with the observation.
But, not only the calculations are likely to be extremely difficult, (if there is an explanation in general relativity, it is surely much more complex and abstract that the one we are looking for), moreover, this explanation does not make B's version coherent!
Suppose B sends a signal to A just before the turnaround and another juste after.
If the half-turn lasts 1 minute (time of B), these signals will reach A at 2 minutes of interval (time of A). How will B explain that?

It seems to me that the only acceptable way to explain the problem of the twins is to consider that, from the departure on, the symmetry of the situation is an illusion.




Bruno Van Rossum


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