From: mlfasf@comcast.net   
      
   I made a careless mistake in my previous post.  Here is that post, down   
   to where my mistake occurred:   
      
   On 3/15/22 2:36 AM, Mike Fontenot wrote:   
   >   
   > Suppose the two twins have just been born when the traveling twin starts   
   > his trip.  So they are each zero years old then.  If the traveling twin   
   > (he) travels for 4 years of his time, he will be 4 yours old when he   
   > stops.  (His age when he stops is an EVENT that all observers must agree   
   > about, and so she also agrees that he is 4 years old when he stops. He   
   > says he has traveled (0.866)(4) = 3.464 lightyears away from his twin   
   > (her) then.  She says he traveled 8 years of her time ... i.e., she says   
   > she is 8 years old when he stops.  She says he is (0.866)(8) = 6.928   
   > lightyears away when he stops.   
   >   
   > After he stops, she says that he remains 6.928 lightyears away from her   
   > after that.  And she says they age at the same rate after he stops.  He   
   > says that, during his essentially instantaneous stopping time, his age   
   > essentially doesn't change during his stopping.  She agrees with that.   
      
   The above is all correct.  But here is the sentence where I made the   
   careless error:   
      
   > But he says that during his essentially instantaneous stopping, SHE   
   > essentially instantaneously gets older by 4 years ... i.e., he says that   
   > she essentially instantaneously goes from being 4 years old to being 8   
   > years old during the essentially instantaneous time in his life it takes   
   > him to stop.   
      
   According to him, her age right before he stops is 2 years old, not 4   
   years old as I said above.  Immediately after he stops, they both agree   
   about their respective ages.  He says she is now 8 years old, so he says   
   her age increases by 6 years during his essentially instantaneous   
   stopping, not by 4 years as I stated in my previous post.   
      
   I should also have pointed out that each of the twins, during his   
   outbound trip, are entitled to use the famous time dilation equation for   
   inertial observers:  That equation says that any inertial observer will   
   conclude that a person moving at speed "v" with respect to them is   
   ageing at a rate gamma times slower than they are, where   
      
      gamma = 1 / sqrt { 1 / (1 - v * v) },   
      
   where the asterisk indicates multiplication.  For v = 0.866, gamma =   
   2.0.  So on the outbound trip, each twin says the other twin is ageing   
   half as fast as their own rate of ageing.  So right before he stops, she   
   says he is 4 and she is 8, but he says he is 4 and she is 2.  And   
   immediately after he stops, they both agree that he is 4 and she is 8.   
   So he says she essentially instantaneously gets 6 years older during his   
   essentially instantaneous stopping.   
      
   There is an equation that I derived long ago that makes it easy to   
   calculate how her age changes (according to him) whenever he essentially   
   instantaneously changes his velocity.  I call it the "delta_CADO" equation:   
      
      delta_CADO = -L * delta_v,   
      
   where L is their distance apart, according to her, and   
      
      delta_v = v2 - v1,   
      
   where v1 is his velocity before the change, and v2 is his velocity after   
   the change (with positive v being taken as the velocity when they are   
   moving apart).   
      
   So in this example,   
      
      L = 6.928 lightyears   
      
      v1 = 0.866 ly/y   
      
      v2 = 0.0 ly/y,   
      
   and so   
      
      delta_v = -0.866   
      
   and   
      
      delta_CADO = -6.928 * (-0.866) = 6.0 years.   
      
   We didn't need this equation to get the answer in the above particular   
   scenario, because his new velocity was zero, which makes things   
   especially easy.  But when the two velocities are completely general,   
   it's necessary to either use the delta-CADO equation, or else do it   
   graphically with a Minkowski diagram, drawing lines of simultaneity.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)