From: tjroberts137@sbcglobal.net   
      
   On 3/17/17 3/17/17   8:59 PM, Richard D. Saam wrote:   
   > It would appear that gravity travels instantaneously   
   > among all "now" time objects (galaxies etc).   
      
   Such sound bites are rarely correct in GR. It will take me a page to   
   explain....   
      
   This depends on what model you use, and how you apply it, and whether you can   
   use an approximation. As an illustration of the difficulties, remember that in   
   GR "now" is not well defined except at a single point.   
      
   First Model:   
   In Newtonian mechanics (NM), gravity does not propagate at all. It is in   
   general   
   a function of the universal time, but anywhere you measure it the value depends   
   only on the physical situation at the time of the measurement. Gravity is a   
   function of each mass's position at the time of the measurement (Poisson's   
   equation). Speaking loosely, gravity "propagates with infinite speed".   
      
   In GR this is not so, and the situation is considerably more complicated. First   
   a digression to SR.   
      
   Interlude: classical electrodynamics (CE):   
   In CE one can compute the E and B fields at a a point (x,y,z,t) in inertial   
   frame S [#] via the Lienard-Wiechert potential. This evaluates the location and   
   velocity of each charge at a retarded point (x',y',z',t') [@] such that the   
   interval between these two points is null -- i.e. a vacuum light ray would   
   propagate from the primed point to the unprimed point. E and B are functions of   
   the charge's position and velocity at the primed point. If you examine the   
   math,   
   you find that this potential in effect makes a linear extrapolation of each   
   charge's position to t ("now") in S. As a result, E from a given charge points   
   very close to where the charge is "now" (in S), and exactly there if the charge   
   is moving inertially; even though the EM field propagates at c.   
      
   	[#] Remember that in CE the EM field is really a 2-form which   
   	must be projected onto an inertial frame to obtain E and B.   
   	This requirement foreshadows much greater complexities in GR.   
      
   	[@] Note the primes denote a different point in S, NOT a   
   	different frame.   
      
   Second model:   
   In the linearized approximation to GR, the equivalent construction does a   
   similar thing, in that it uses the retarded position of each mass,   
   extrapolating   
   its position, velocity, and acceleration to "now". So in this approximation the   
   "gravitational force" at each point points very close to where the mass is   
   "now", with magnitude 1/distance^2; exactly so if the mass is following a   
   second-order curve in the equivalent to S. In this approximation the   
   "gravitational force" from a mass points at its location "now"; even though   
   gravity propagates at c.   
      
   Third model:   
   In GR itself (i.e. NOT that approximation), there is no "gravitational force",   
   the above construction does not apply, one cannot add "gravitational fields"   
   from multiple masses (equations are nonlinear), energy other than mass   
   contributes, and things are VASTLY more complicated. But still, one can say   
   that   
   CHANGES in gravity propagate at c, which yields gravitational waves.   
      
   Conclusion:   
   GR goes smoothly into NM as long as one looks at a small enough region of   
   spacetime in which all masses and velocities are "small" (so the above   
   approximation applies). This is so even though speaking loosely one might say   
   "gravity propagates at c in GR but with infinite speed in NM" -- as discussed   
   above the actual situation is more complicated than such a sound bite can   
   capture.   
      
   Tom Roberts   
      
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