From: tjroberts137@sbcglobal.net   
      
   On 6/26/19 2:21 AM, Phillip Helbig (undress to reply) wrote:   
   > There is a simple argument to show that the acceleration is not the   
   > cause of the slower aging of the turnaround twin: the magnitude of   
   > the effect, i.e. the difference in ages, depends only on the speed   
   > travelled and the length of the journey: the longer the journey, and   
   > the faster the journey, the greater the difference, even if the   
   > acceleration is the same.   
      
   Right. And here is a simple and direct way to show that. This   
   counterexample is in the context of SR:   
      
   Arrange two turntables to have their centers at rest in some inertial   
   frame, and arrange them so their rims are tangent. On each rim place a   
   clock, and arrange their rotation rates to be a rational number, such   
   that the clocks periodically meet. By carefully choosing the rotation   
   rates and the turntables' radii, you can arrange for the clock with the   
   SMALLER acceleration to have a larger speed relative to the inertial   
   frame, and thus accumulate less proper time between meetings.   
      
   That is contrary to the notion that "acceleration is all that matters".   
      
   For the twin paradox, what matters is the paths of the clocks/twins   
   through spacetime. Acceleration is incidental, and affects the result   
   only insofar as it affects the paths.   
      
        This OUGHT to be obvious, as acceleration simply does not   
        appear in the calculation of the elapsed proper time of   
        a twin; the twin's path does. Indeed, in SR the only   
        quantity that can vary is the path:   
            ElapsedProperTime = sqrt( \integral g_ij dx^i dx^j )   
        Where the integral is taken over the path through spacetime   
        and the {g_ij} are the metric components relative to the   
        coordinates {x^i}; here the metric signature is +---.   
      
   Tom Roberts   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)