On Thursday, September 21, 2017 at 1:30:37 AM UTC-5, Luigi Fortunati wrote:   
   > Newton claims that gravity is a force, Einstein denies it: for him it   
   > is not a force (it is something else).   
      
   Let's throw a wrench into the gears here!   
      
   The metric (and 0 constraint)   
      
   dx^2 + dy^2 + dz^2 - 2aU/(1 + 2aU) (x dx + y dy + z dz)^2/(x^2 + y^2 + z^2)   
    + 2 dt du + a du^2 - 2U dt^2 = 0   
   with the proper time ds given by   
   ds = dt + a du   
      
   gives you the Schwarzschild solution to Einstein's equations when a > 0 (with   
   light speed c = root(1/a)) and Newton's law of gravity when a = 0! I   
   n that case, the metric (and 0 constraint) becomes   
   dx^2 + dy^2 + dz^2 + 2 dt du - 2U dt^2 = 0,   
   with proper time ds = dt.   
      
   Apply the geodesic equation (and use the above 0-constraint) to a test body   
   and you will, indeed, find that the body conforms to what's expected for   
   a > 0 AND a = 0. The coordinate u -- in all cases -- will yield the negative   
   of the action per unit mass.   
      
   To obtain the mass shell condition, invert the metric and apply the   
   correspondence:   
   del <-> p, -(d/dt)_u = H = kinetic energy, (d/du)_t = m   
   and note that corresponding to the invariant ds is the invariant m.   
      
   In flat space (where U = 0), this produces the result   
      p^2 - 2mH - a H^2 = 0   
   which for a > 0 is just the familiar mass shell constraint, provided you take   
   E = H + m/a. For a = 0 it is also the mass shell constraint, only here   
   the mass-energy-momentum vector is a 5 vector (p_x, p_y, p_z, H, m) (and   
   transforms as such). The transform for a > 0 is similar, except it can be   
   factored out by replacing H with E.   
      
   > It is obvious that both can not be right: if gravity is a force, Newton   
   > is right and Einstein is wrong, and if it is the reverse Newton is   
   > wrong and Einstein is right.   
      
   I think we've just blown the entire question out of the water (and, you're   
   welcome). Even Cartan would envy this direct simplicity. :)   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)