From: fortunati.luigi@gmail.com   
      
   Luigi Fortunati il 17/12/2023 09:29:50 ha scritto:   
   > Gravity is a force and, therefore, is a vector.   
   >   
   > Is spacetime curvature a vector? Does it have a direction and a verse?   
   >   
   > Luigi Fortunati   
   >   
   > [[Mod. note --   
   > In order to answer questions like this, we need to be precise in   
   > our terms.  You write that "Gravity is a force".  But what do you mean   
   > by "gravity"?   
      
   By gravity I mean the Newtonian force F directly proportional to the product   
   of the masses m1 and m2 and inversely proportional to the square of their   
   distance.   
      
   > Starting with Newtonian mechanics for simplicity,   
   > "gravity" could plausibly mean any of several things:   
   > * gravitational potential energy (which is a scalar in Newtonian mechanics).   
   > * the Newtonian "little g" (which is a 3-vector at any given position   
   >   and time   
   > * the *difference* in the Newtonian "little g" between nearby objects   
   >   at a given time; this difference is what you can measure about the   
   >   gravitational field if you're in a freely falling elevator.  This   
   >   difference is a 3-vector which depends on the separation between the   
   >   nearby objects,   
   >     difference = M * separation   
   >   where M is a 3x3 matrix and "*" denotes matrix multiplication.  This   
   >   3x3 matrix M (which is really a rank 2 tensor) provides a complete   
   >   description of the local gravitational field at a given position and   
   >   time.   
      
   Does all this exclude that, in classical mechanics, gravity is a fundamental   
   force and, therefore, is necessarily a vector?   
      
   > In general relativity (GR) things are (not surprisingly) more complicated.   
   > To fully describe spacetime curvature at an event (a point in space, at   
   > a particular time) requires generalizing the 3x3 matrix (rank 2 tensor) M   
   > to the Riemann curvature tensor, which is a 4x4x4x4 4-dimensional matrix   
   > (really a rank 4 tensor), i.e., it's a set of 4x4x4x4 = 256 numbers.   
   > The Riemann curvature tensor has a bunch of symmetries, so it actually   
   > has only 20 independent components.   
   >   
   > So in GR, the best answer to your question is that spacetime curvature "is"   
   > the Riemann curvature tensor.  This doesn't have a single "direction" any   
   > more than the 3x3 matrix M has a "direction" in Newtonian mechanics.   
      
   In the Wikipedia entry "Spacetime" there is the figure of the space-time sheet   
   that curves *downward* when we place a mass on it.   
      
   If there are many directions in the GR, why does the elastic sheet sink   
   *always and only* downwards, i.e. towards a single direction?   
      
   Luigi Fortunati   
      
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