From: fortunati.luigi@gmail.com   
      
   Luigi Fortunati domenica 11/09/2022 alle ore 11:38:28 ha scritto:   
   > The time coordinate is part of the geodesic.   
   >   
   > If the time coordinate has a preferred direction, the geodesic also has   
   > a preferred direction.   
   >   
   > [[Mod. note --   
   > A useful mental model for a geodesic at a point is motion on the surface   
   > of the Earth.  That is, starting at some specified point, we can move   
   > in a specified compass direction (e.g., you might start out moving due   
   > west).  If, once moving, we don't turn left and you don't turn right,   
   > our motion will be along a geodesic on the Earth's surface.   
   >   
   > Thinking of our starting point again, you could have started moving   
   > in any direction   
      
   Exactly, I could have started moving in any direction and, among all, I   
   would have been forced, finally, to choose only one.   
      
   > in any direction (e.g., instead of bearing 090 degrees = due west, we   
   > could have chosen any other compass bearing).  So there are a whole   
   > (infinite) family of possible geodesics passing through that starting   
   > point (one for each possible compass bearing).   
      
   That's right, the choice is one of an infinite number of possible   
   geodesics.   
      
   > If I understand you correctly, you're asking "once a particle is moving,   
   > how does it know to continue moving in that direction?".   
      
   No! I am asking how does it know how to "start" to move (along   
   4-geodesic) and not how to "continue" to move.   
      
   Once the particle has moved, its choice (among the infinite possible)   
   has already been made!   
      
   > The answer is basically conservation of momentum   
      
   There is no conservation of 4-momentum when the elevator cables break   
   and the elevator goes from constrained condition to free fall.   
      
   > unless there is some external force pushing on the particle, it's going to   
   > continue moving in the *same* direction it was already moving in.   
      
   The elevator where the cables break, does not keep moving in the *same*   
   4-direction it was moving before.   
      
   > In terms of geodesics in relativity (the original context of your question),   
   > it's essential to realise that (as others have noted) the trajectoris of   
   > free particles are geodesics in *spacetime*, not geodesics in *space*.   
   > That means the most useful particle velocity to think about is the   
   > 4-velocity, which is *never* zero (you're always moving forward in time),   
   > and corresponding momentum is the 4-momentum, which is also never zero.   
      
   Ok, let's talk about 4-momentum.   
      
   When the cables break, the elevator does not retain the 4-momentum it   
   had before but switches from a certain 4-momentum (the one it had when   
   standing at the floor) to another completely different 4-momentum (the   
   one it assumes during free fall).   
      
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