From: goldenfieldquaternions@gmail.com   
      
   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).   
   >   
   > 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.   
   >   
   > It can not be otherwise!   
   >   
   > Einstein is newer and Newton is more "ancient", so gravity should not   
   > be a force.   
   >   
   > So how do we "measure" gravity with the dynamometer that measures the   
   > forces?   
   >   
   > If it were not a force we should not be able to measure it with the   
   > dynamometer!   
   >   
   > --   
   > Credere e' piu' facile che pensare   
   > Believing is easier than thinking   
   > Luigi Fortunati   
   >   
   > [[Mod. note -- The simple answer is that (from the perspective of general   
   > relativity) "gravity" is a fictitious force quite similar to "centrifugal   
   > force" in Newtonian mechanics, arising from NOT being in a freely-falling   
   > inertial reference frame.   
   > -- jt]]   
      
   The geodesic equation tells us that the 4-velocity U changes with   
   proper interval s as   
      
   dU/ds = G*U*U   
      
   where G is the Gamma or Christoffel symbols. The system here is not   
   very forgiving with writing equations so I am leaving it at this.   
   These equation is very ambiguous. The connection coefficients are   
   elements of differential one-forms act on a section of a fiber   
   bundle s as ds. we can use the Lorentz group elements U to transform   
   this so   
      
   ds ---> Uds.   
      
   The section s transforms as s' = Us, so we transform this to the   
   section s' as   
      
   Uds = Ud(U^{-1}Us),   
      
   where we have done nothing by insert the unit 1 = U^{-1}U into the   
   equation. Now use product or chain rule of calculus   
      
   Uds = (UdU^{-1})Us + UU^{-1}d(Us) = (UdU^{-1})s' + ds'.   
      
   This transforms badly or not homogeneously.   
      
   We may consider the deviation between two geodesic. For V = U   
   =E2=80=93 U'  this gives an equation   
      
    dV/ds = (G - G')*U*V*U.   
      
   This is pretty cryptically written, but the deviation in two   
   connection or Christoffel terms G - G' gives the Riemann curvature   
   tensor so   
      
   dV/ds = R*U*V*U.   
      
   This is the geodesic deviation equation which is the most covariant   
   way to think of gravitation; it is a measure of how geodesics   
   deviated from each other. However, if one of the connection   
   coefficients is zero, say this is from a distant region with   
   negligible spacetime curvature, then this converges to the plain   
   old geodesic equation. This is why for central gravity fields we   
   often just default to the geodesic equation.   
      
   Where does the force come in. suppose with the geodesic deviation   
   equation you have a spring along the separation vector V between   
   two geodesics. This spring is distended by the geodesic separation,   
   and is then a sort of weight scale. This is where force comes into   
   play; it is not with gravitation so much as with the interaction   
   with matter that counters the action of gravitation. This is the   
   situation just standing on the Earth's surface. We have material   
   of the Earth pushing up, which counteracts the geodesic motion that   
   would otherwise have us falling further down. This geodesic motion   
   for a weak gravity field involves the gravitational acceleration g   
   = 9.8m/s^2 and the Earth pushes up with the normal force N that   
   counters mg = F_g.   
      
   LC   
      
   --- SoupGate-Win32 v1.05   
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