From: ram@zedat.fu-berlin.de   
      
   Luigi Fortunati  writes:   
   >What I want to point out is that in (1) there is the   
   >*dilation* of terrestrial time (which runs slower in the   
   >spaceship frame) and in (3) there is the *contraction* of   
   >terrestrial time (which runs faster in the spaceship frame).   
      
     This is true from the point of view of the space traveler if   
     he considers himself to be at rest. (But which traveler would   
     seriously consider himself to be at rest?)   
      
     However, if we do not look at this from a special coordinate system,   
     we can say that each twin travels a different world line (travel   
     route) in spacetime between the start of the spacecraft and its   
     landing. The length of each of these world lines, measured in the   
     metric of spacetime, gives the the time that has passed for each   
     of them. And these lengths are independent of any particular   
     reference system, so it is actually easier to consider this invariant   
     point of view than "simultaneities" that depend on reference systems.   
      
     Quantitatively, let dS and dT denote the length (elapsed proper   
     time) of the world line of the traveller and the other twin,   
     respectively, between the launch of the rocket and its return.   
     According to the metric, ( c dS )^2 = ( c dt )^2 - dx^2, where   
     dt is the difference of the times of launch and landing in   
     coordinate time of the resting twin, and dx is the total length   
     traveled by the rocket in the frame of the resting twin, so   
     dS = sqrt( 1 -( v/c )^2 )dt, where v = dx/dt, the speed of the   
     rocket. (For simplification, I assumed that v^2 is constant and   
     I did not distinguish between v and -v, as it does not matter   
     for terms where v is squared. Actually, the calculation should   
     be split into one summand for the outward and one summand for   
     the return trip, but the result would be the same.) For the   
     twin resting on Earth dT is simply dt, i.e., his proper time is   
     the coordinate time as he is not moving. So, when the twin on   
     Earth at the time of the reunion has aged by 1 unit since the   
     launch, dT=1, dt=1, and dS for the twin travelling with v is   
     sqrt( 1 -( v/c )^2 ), i.e., 1/2 given v=0.866c.   
      
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