On Friday, May 22, 2020 at 3:57:23 PM UTC-5, Phillip Helbig (undress to reply)   
   wrote:   
   ...   
   > (Why there is such a thing as absolute acceleration is, as far as I   
   > know, an unsolved problem.  Some believe that Mach's Principle explains   
   > it, i.e. acceleration is relative to the bulk of the matter in the   
   > universe.  That implies that if there were no matter in the universe   
   > other than the two twins, then the resolution of the "paradox" might be   
   > different.)   
   ...   
      
   If you combine some simple ideas from QM and SR you can show that   
   acceleration necessarily involves an addition of energy in some   
   frame, and energy-momentum in all other frames:   
      
   -consider the QM wave function of a particle of mass M at rest.   
   Per SR it has an energy Mc^2.  The wave function has zero momentum,   
   therefore k=0, and therefore the particle is not localized but has   
   a frequency w=2pi Mc^2 /hbar.   
      
   -Now consider an observer moving at speed v past this location (i.e.   
   in a differeent inertial frame).  The wave function will be transformed   
   by the Lorentz transform.  Some simple math shows that this observer   
   will see a frequency w' = 2 pi gamma Mc^2/hbar and wave number (i.e.   
   momentum) k'= 2 pi gamma beta Mc^2/hbar.   
      
   -i.e., the particle in the "moving frame" has a higher energy as   
   well as higher momentum.  You much add energy to a particle to get   
   it boosted to a moving frame (which is not a surprise...).   
      
   -The moving particle has non-zero k.  The way to get that is an   
   acceleration.  Per GR, an accelerating frame will show "gravitational"   
   red shift.  A particle initially stationary in such a frame will   
   have different frequencies at different locations.  As a result the   
   wave function at  different locations will get out of phase, which   
   results in the wavenumber changing (i.e. change in momentum).  This   
   is an easy conceptual way to understand how GR metrics, such as   
   Schwartzschild metric, result in gravitational acceleration.   
      
   -This argument can be carried further to show how the classical   
   action is related to the change in phase of the wave function over   
   some path.  The path of least action is the path with the maximum   
   change in wavefunction phase.   
      
   Rich L.   
      
   --- SoupGate-Win32 v1.05   
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