From: rockbrentwood@gmail.com   
      
   On Wednesday, September 7, 2022 at 1:22:46 AM UTC-5, Luigi Fortunati wrote:   
   > A geodesic passes from A and also from B.   
   >   
   > Is the direction from A to B fully equivalent to the direction from B   
   > to A?   
   >   
   > Or can it happen that one of the two directions prevails over the   
   > other?   
      
   In any metric geometry, around each point is a neighborhood U,   
   such that between any two points A and B in U   
   lies a unique geodesic between A and B.   
   Uniqueness means it's therefore the same regardless of direction.   
      
   This even includes non-Riemannian geometries,   
   like the chronogeometry of Newtonian gravity,   
   which can be regarded as residing on a light cone   
   of the 4+1 dimensional pseudo-Riemannian chronogeometry   
   given by the metric whose line element is   
      
   dx^2 + dy^2 + dz^2 + 2 dt du - 2V dt^2   
      
   where V = -GM/r (with r^2 = x^2 + y^2 + z^2)   
   is the gravitational potential per unit mass.   
      
   This also works perfectly well for multi-body potentials,   
   V = -sum_a (-GM_a/|r - r_a|).   
      
   Its geodesics comprise the orbital motions of Newtonian gravity   
   AND the geometrical instantaneous geodesics of space at any instant.   
      
   You may be hard-pressed, in this case, to account for   
   what the corresponding geodesic *distances* are,   
   since everything is zero on a light cone!   
   All the geodesics for Newtonian gravity   
   are null curves in 4+1 dimensions.   
   But it's the same in both directions.   
      
   This may be compared to the Schwarzschild metric   
   which is equivalently given as light cones   
   in the 4+1 dimensional chronogeometry   
   whose metric has the following line element   
      
   dx^2 + dy^2 + dz^2 + 2 dt du - 2V dt^2   
   + alpha du^2 - 2 alpha V dr^2/(1 + 2 alpha V)   
      
   where dr^2 = (x dx + y dy + z dz)^2/r^2 and alpha = (1/c)^2.   
      
   This is a deformation of the Newtonian chronogeometry,   
   from alpha = 0 to alpha > 0.   
   Here, however, V = -GM/r is only for one body, not many.   
   There may exist similar deformations for the many-body cases.   
      
   The geodesic distances on the light cones   
   in the (4+1)-dimensional chronogeometry are all 0,   
   since all the curves on the light cones are null.   
   The light cones, however (unlike the Newtonian case)   
   comprise metric geometries in their own right -   
   that is: the Schwarzschild chronogeometries,   
   whose the metric yields - for time-like curves -   
   a proper time equal to s = t + alpha u.   
   It's the same in both directions.   
      
   In the alpha = 0 case, which is Newtonian gravity,   
   the (3+1)-dimensional chronogeometry continues to have   
   s = t + alpha u as an invariant ... which is now just s = t.   
   So, it can still be taken as a metric of sorts for time-like curves.   
   Ordinary time is proper time for Newtonian gravity.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)