On Thursday, June 27, 2019 at 3:43:15 PM UTC-5, Jonathan Thornburg [remove   
   -animal to reply] wrote:   
   > To assess Rock Brentwood's contention that the acceleration is the   
   > *only* thing which matters, let's consider the standard spacetime   
   > diagram of the twin paradox:   
      
   More precisely, consider the trajectory (depicted in the original) given by:   
   x(t) = X - 2X/T |t - T/2| for 0 <= t <= T   
   with   
   v(t) = x'(t) = -2R/T sgn(t - T/2)   
   and   
   a(t) = v'(t) = -4R/T delta(t - T/2) != 0   
      
   This "non-accelerating" example of accelerating motion is probably   
   the worst of the culprits that have contributed to the folklore   
   myth that that we've actually seen propagated in text books(!) ...   
      
   > I think this shows that the acceleration isn't the *only* thing   
   > which matters.   
      
   ... when in fact it is an illustration that acceleration IS the only   
   thing of relevance here!   
      
   This can be made precise in the following way -- as already seen   
   and alluded to before.   
      
   From the following data r(0), r'(0) and (s in [0,S] |-> A(s) =   
   acceleration in the rest frame), one can derive a unique trajectory   
   (t in [0,T] |-> r(t)) such that the accelerating r''(t) is A(s)   
   transformed to the coordinate frame by instantaneous velocity r'(t)   
   and s = s(t) = integral_0^t root(1 - r'(t)^2) dt is the proper time   
   as a function of t such that S = s(T). The function A is only assumed   
   to be integrable and possibly singular.   
      
   The clock time for the inertial motion is given by root(T^2 - alpha   
   R^2), using alpha = 1/c^2 (and alpha = 0 for non-relativistic   
   theory).   
      
   Define U by S = T + alpha U.   
      
   The difference root(T^2 - alpha R^2) - S can be expressed PURELY   
   as a function of r(0), r'(0) and A = (s in [0,S] |-> A(s)), as A   
   ranges over all integrable singular functions (taken modulo equivalence   
   ==).   
      
   It is 0 if and ONLY if A == 0; otherwise it is positive -- thus   
   demonstrating that acceleration is the only thing that matters or   
   counts.   
      
   Moreover, it also a non-relativistic form, upon dividing out by   
   -alpha. In both cases, if R = 0, then difference is just the total   
   action for the trajectory taken from time 0 to time T.   
      
   Being extremal it is 0 if and only if the motion is geodesic (i.e.   
   a straight line). Otherwise it is negative and the above time   
   difference is positive.   
      
   In the case where R = 0, U can be expressed simply as integral du,   
   where du is derived from the 4+1 line element   
   dx^2 + dy^2 + dz^2 + 2 dt du + alpha du^2 by setting it to 0.   
      
   Moreover, as the existence of a non-relativistic version of the   
   time difference shows (i.e. the non-relativistic limit of   
   u = (t - s)/alpha); even the consideration of special relativity   
   is, itself, a red-herring.   
      
   [[Mod. note -- As both Tom Roberts and I have noted, in a curved   
   spacetime you can have a twin paradox even with ZERO acceleration.   
      
   	[Digression:   
   	I suspect, but have not worked out in detail, that for   
   	a flat spacetime it is possible to have twin paradox if   
   	the spacetime is multiply connected.  In particular,   
   	consider paths going on opposite sides of a "cone"   
   	singularity, one where the circumnavigation angle is   
   	different from 2*pi radians.]   
      
   For Minkowski spacetime:   
      
   It is obvious that if you know x(t), you can determine the proper time   
   along the worldline and hence the extent & amount of any twin paradox.   
      
   It is also obvious that if you know v(t), you can integrate it once   
   to determine x(t) (up to an integration constant), and from this you   
   can determine the proper time along the worldline and hence the extent   
   & amount of any twin paradox.   
      
   And it is also obvious that if you know a(t), you can integrate it twice   
   to determine v(t) and x(t) (up to integration constants), and from these   
   you can determine the proper time along the worldline and hence the extent   
   & amount of any twin paradox.   
      
   So I do not see how one can logically say that a(t) is the *only*   
   quantity that matters, as opposed to (say) x(t) or v(t).   
   -- jt]]   
      
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