From: richalivingston@gmail.com   
      
   On Tuesday, April 5, 2022 at 2:54:06 PM UTC-5, Mike Fontenot wrote:   
   >   
   ... {long derivation related to Ehrenfest Paradox and Rendler Coordinates}   
   >   
   > So, for the idealization of an essentially instantaneous velocity   
   > change, the change of the reading on the leading clock is INFINITE   
   > during the infinitesimal change of the rear clock. That means that,   
   > when the traveling twin instantaneously changes his speed from zero to   
   > 0.866 (toward the home twin), the exponential version of the R equation   
   > says that the home twin's age becomes infinite. But we know that's not   
   > true, because the home twin is entitled to use the time dilation   
   > equation for a perpetually-inertial observer, and that equation tells   
   > her that for a speed of 0.866 ls/s, the traveler's age is always   
   > increasing half as fast as her age is increasing. So when they are   
   > reunited, she is twice as old he is, NOT infinitely older than he is, as   
   > the exponential form of the gravitational time dilation equation claims.   
   > The time dilation equation for a perpetually-inertial observer is the   
   > gold standard in special relativity. Therefore the exponential form of   
   > the gravitational time dilation equation is incorrect.   
   >   
      
   Your calculations are all related to the Ehrenfest Paradox and Rindler   
   Coordinates.  What myself and others have been trying to get you   
   to understand is that the acceleration of the observer does not actually   
   change anything about the distant twin.  By your own calculations   
   the age of the distant twin can be anything +/- c*d depending on the   
   relative state of motion and accelerations.  Furthermore, with sufficiently   
   high acceleration the observer cannot "see" the distant twin at all.   
      
   Please note that an inertial observer at the same location as your   
   traveling twin, watching everything as the traveling twin accelerates,   
   does not see any change in the rest of the universe.  All these   
   coordinate transformations are entirely observer dependent and   
   do not represent anything real for the physics.  They only affect the   
   APPEARANCE of the world to that observer.  Any attempt to place   
   greater significance to these appearances is misguided.   
      
   > The correct gravitational time dilation equation turns out to   
   > approximately agree with what Einstein used in his "small acceleration"   
   > analysis, for very small accelerations, but differs substantially for   
   > larger accelerations. And the correct gravitational time dilation   
   > equation agrees with the ages of the twins when they are reunited. It   
   > also exactly agrees with the CMIF simultaneity method for the traveler's   
   > conclusions about the sudden increase in the home twin's age when the   
   > traveler suddenly changes his velocity. The CMIF method provides a   
   > practical way to compute the change in the home twin's age when the   
   > traveler instantaneously changes his velocity. But it is the new   
   > gravitational time dilation equation, and its array of clocks with a   
   > common "NOW" moment, that guarantees that the CMIF result is fully   
   > meaningful to the traveling twin, and that the CMIF method is the ONLY   
   > correct simultaneity method for the traveling twin.   
      
   Again, you should study the Ehrenfest Paradox and Rindler Coordinates.   
   You still don't seem to appreciate the true significance (or lack thereof)   
   of the calculated "now".   
      
   Rich L.   
      
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