From: tjroberts137@sbcglobal.net   
      
   On 4/15/22 6:06 PM, Mike Fontenot wrote:   
   > You (Richard Livingston and moderator JT) are both confusing the   
   > conclusions of the people who are undergoing acceleration with the   
   > conclusions of people who are perpetually-inertial and who are   
   > observing the people who are accelerating.  The perpetually-inertial   
   > people DO conclude that the distance between the accelerating people   
   > decreases when they accelerate, but the people who are accelerating   
   > don't agree.   
      
   This is wrong in two quite different ways:   
    1. You presume that the principle of equivalence (POE) applies,   
       and base your conclusion on it: you are discussing Case 2   
       (below), and incorrectly believe the POE permits you to apply   
       the constancy of Case 1 (also below); it doesn't.   
    2. You are claiming a coordinate dependence for measurements that   
       are invariant [@] -- all observers ("people") will agree on   
       the results of a measurement (or series of measurements),   
       regardless of whether the observers are inertial or accelerated.   
      
   Bottom line: the POE applies ONLY in regions of spacetime that are small   
   enough that any curvature is negligible (compared to measurement   
   accuracies). For the case you have in mind, with helper friends   
   ("people") near a distant friend while the accelerated observer ("AO")   
   roams the universe, those people span an enormous spatial region, over a   
   very long time. (Parentheticals relate your earlier and later notations.)   
      
   Case 1: all people are at rest in a uniform and static gravitational   
   field [#]. They will indeed measure their pairwise distances to all be   
   constant over time [@]. As the field is uniform, they all have identical   
   proper accelerations.   
      
       [#] I.e. their 4-velocities are parallel to the timelike   
       Killing vector.   
      
       [@] Measuring pairwise distances in the AO's instantaneously   
       co-moving inertial frame (ICIF) at a given time.   
      
   Case 2: all people are in in flat spacetime, initially at rest in some   
   inertial frame, each with an identical proper acceleration. They will   
   NOT measure their pairwise distances to be constant [@], because this is   
   not Born rigid motion (for which lower people must have larger proper   
   accelerations than higher people).   
      
       [In Case 2, if one measured those distances in the initial   
        inertial frame, they would be constant. But the changing   
        inertial frames of successive measurements, and the   
        structure of Lorentz transforms between frames, makes them   
        be not constant in successive ICIFs of the AO. See Bell's   
        spaceship paradox.]   
      
   This has nothing to do with different measurement methods, or the use of   
   inertial or non-inertial coordinates, or any supposed coordinate   
   dependence of invariant measurements; it is due the different ways the   
   two collections of people evolve in the two cases.   
      
   Tom Roberts   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)