From: mlfasf@comcast.net   
      
   (This is an improved version of a posting that I submitted yesterday   
   [4-3-22], but which hasn't shown up yet)   
      
   In 1907, Einstein published a VERY long paper (in several volumes) on   
   his "relativity principle".  In volume 2, section 18, page 302, titled   
   "Space and time in a uniformly accelerated reference frame", he   
   investigated how the tic rates compare for two clocks separated by the   
   constant distance L, with both clocks undergoing a constant acceleration   
   "A".  He restricted the analysis to very small accelerations (and very   
   small resulting velocities).  His result (on page 305) was that the   
   leading clock tics at a rate   
      
      R  =  1  +  L A   
      
   faster than the rear clock.  Note that that result agrees with my   
   equation, for very small "L" and "A".  But he then said:   
      
      "From the fact that the choice of the coordinate origin must not   
   affect the relation, one must conclude that, strictly speaking, equation   
   (30) should be replaced by the equation R = exp(L A). Nevertheless, we   
   shall maintain formula (30)."   
      
   I've never understood that one sentence argument he gave, for replacing   
   his linear equation with the exponential equation.  But I DID assume he   
   was right (because he was rarely wrong), until I tried applying his   
   exponential equation to the case of essentially instantaneous velocity   
   changes that are useful in twin "paradox" scenarios in special   
   relativity.  Specifically, I worked a series of examples where the   
   separation of the two clocks is always   
      
      L  =  7.52 ls  (lightseconds)   
      
   and where the final speed (with the initial speed being zero) is always   
      
      v  =  0.866 ls/s.   
      
   That speed implies a final "rapidity" of   
      
      theta  =  atanh(0.866)  =  1.317 ls/s.   
      
   ("Rapidity" is a non-linear version of velocity.  They have a one-to-one   
   correspondence.  In special relativity, velocity can never exceed 1.0   
   ls/s in magnitude, but rapidity can have an infinite magnitude.  An   
   acceleration of "A" ls/s/s lasting for "t" seconds changes the rapidity   
   theta by the product of "A" and "t".)   
      
     So in this case, when we are starting from zero velocity and thus zero   
   rapidity, at the end of the acceleration,   
      
      theta  =  A tau,   
      
   where tau is the duration of the acceleration (according to the rear   
   clock).  So, if we know theta and tau, we then know the acceleration:   
      
      A  =  theta / tau.   
      
   I do a sequence of calculations, each starting at t = 0, with the two   
   clocks reading zero, and with zero acceleration for t < 0.   
      
   First, I set the duration tau of the acceleration to 1 second. The   
   acceleration then needs to be   
      
      A  =  theta / tau  =  1.317 / 1.0  =  1.317 ls/s/s    (that's roughly   
   40 g's).   
      
   So for this first case, the tic rate of the leading clock during the one   
   second acceleration is   
      
      R = exp( L A )   =  exp(9.9034),   or about  20000.   
      
   (I picked that weird value of "L" so that this value of "R" for the fist   
   case would be a round number, just to make the calculations easier.)   
      
   Since R is constant during the acceleration, the CURRENT reading on the   
   leading clock (which I'll denote as AC, for "Age Change") is   
      
      AC  =  tau R  =  2x10sup4  =  20000   
      
   (where 10sup4 means "10 raised to the 4th power").   
      
   I then start over and work a second case, with ten times the   
   acceleration (13.17 ls/s/s), but with tau ten times smaller (0.1   
   second).  That keeps the final rapidity the same as in the first case,   
   and the final speed is also 0.866, as before.  For the second case,   
      
      AC  =  1.02x10sup42.   
      
   So when we made the acceleration an order of magnitude larger, and the   
   duration an order of magnitude smaller, the current reading "AC" on the   
   leading clock got about 38 orders of magnitude larger.   
      
   Next, I start over again and work a third case, again increasing the   
   acceleration by a factor of 10, and the decreasing the duration by a   
   factor of 10, so "A" = 131.7 ls/s/s and tau = 0.01 second.  Then, AC  =   
     1.27x10sup428.  So this time, when we increased "A" by a factor of 10,   
   and decreased tau by a factor of ten, AC got about 380 orders of   
   magnitude larger.   
      
   AC is not approaching a finite limit as tau goes to zero and "A" goes to   
   infinity.  In each iteration, the change in AC compared to the previous   
   change gets MUCH larger.  Clearly, the clock reading is NOT converging   
   to a finite limit.  It is going to infinity as tau goes to zero.   
      
   We can see this, even without doing the above detailed calculations.  Since   
      
      AC  =  tau exp(L A),   
      
   the tau factor goes to zero LINEARLY as tau goes to zero, but exp(L A)   
   goes to infinity EXPONENTIALLY as tau goes to zero, so their product is   
   obviously not going to be finite as tau goes to zero.   
      
   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.   
      
   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.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)