On Tuesday, June 25, 2019 at 2:33:20 AM UTC-5, Tom Roberts wrote:   
   > Moreover [sic], in General Relativity   
      
   Nobody was talking about General Relativity, except you. This has   
   nothing to do with what you're replying to and is therefore irrelevant.   
      
   In any case, the generalization to General Relativity is that the 0-G   
   motion (i.e. the free fall motion) is the one that has the largest time   
   -- also a likewise consequence of the action principle ... up to the   
   limit alluded to below on geodesic (non-)uniqueness and cut points.   
      
   > each with ZERO proper acceleration, have different   
   > elapsed proper times between meetings: put one twin in circular orbit   
   > around a mass, put the other in a highly elliptical orbit around the   
   > same mass, arrange for their orbital periods to have a ratio that is a   
   > rational number, and orient them so the orbits periodically intersect.   
      
   ... and the time difference is 0 (at least to the order alpha^2 = 1/c^4,   
   if not exactly), for orbits with the same mean radii, independent of   
   shape.   
      
   The time difference S0 - S1 between the two motions is given in terms of   
   the non-relativistic action A by S0 - S1 = alpha (A1/m1 - A0/m0) +   
   O(alpha^2) and the non-relativistic action A is a function only of r(0),   
   r(T) and HT, where H is the (kinetic + potential energy) -- which is   
   independent of the orbit's shape.   
      
   Moreover, since the difference is a function of the gravitational   
   coefficient G which is 0 when G = 0, this shows that the time difference   
   (for orbits of different mean radii) therefore has nothing to do with   
   Special Relativity.   
      
   For orbits with different mean radii, it is a function only of winding   
   number N that -> 0 as G -> 0 (A1/m1 - A0/m0) = 3 pi (N1 root(GM a1) - N0   
   root(GM a0)), where M is the mass of the center of force N0,N1 the   
   respective winding numbers and a0,a1 the respective mean radii)   
   ... meaning it is a function of winding number alone, since a0 and a1   
   are determined by N0, N1 and T.   
      
   For orbits of different mean radii, the difference between respective   
   global actions gets into the matter of global geometry; once you go   
   beyond the cut radius where geodesics are no longer uniquely determined   
   in Riemannian geometry by start and end point alone.   
      
   But geodesic non-uniqueness is a property that ONLY applies to curved   
   space-times and therefore has nothing to do with special relativity; and   
   it is a bit misleading on your part to be bringing it up here at all in   
   a discussion about the Twin Paradox in SPECIAL relativity.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)