From: nicolaas.vroom@pandora.be   
      
   On Friday, 22 May 2020 21:22:21 UTC+2, Hendrik van Hees  wrote:   
   > On 22/05/2020 21:15, Tom Roberts wrote:   
      
   > > If twin A remains at rest in an inertial frame, it is easy to calculate   
   > > the age (elapsed proper time) of each twin: simply integrate   
   > >   
   > > 	T = \integral sqrt(1-v^2/c^2) dt   
   > >   
   > > where T is the elapsed proper time, the integral is taken over the path   
   > > of the twin relative to A's rest frame, v is the speed of the twin   
   > > relative to that frame (as a function of t), and t is the time   
   > > coordinate of the frame.   
      
   > One should note that the proper time is an invariant and as such   
   > completely independent of the coordinates and the parametrization you   
   > choose to calculate it.  You can introduce arbitrary generalized   
   > coordinates (most of which then correspond to the description of a   
   > non-inertial observer) and an arbitrary parameter to parametrize the   
   > worldline.  As in Euclidean space the length of a curve also proper   
   > time, which is in a somewhat generalized sense nothing else than the   
   > length of a time-like worldline, is independent of these choices.   
      
   Is the above text not much too complicated?   
      t is the time of the clock at rest.   
      T is the time of the moving clock.   
   I personally would prefer to write:   
   t is the time calculated based on the observed clock counts of a clock.   
   Why calling T an invariant?   
      
   The whole issue is if the function or equation is correct.   
   That means is the prediction using this equation in accordance   
   with an actually performed experiment.   
   See also my comments on Tom's posting.   
   What makes this equation tricky is that it uses two parameters v and c.   
   Both of these parameters have to be calculated based on observations   
   as an integral part of this experiment.   
   Declaring c a constant makes this problem slightly simpler but does   
   not solve the physical issue.   
      
   Nicolaas Vroom   
      
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