From: tjoberts137@sbcglobal.net   
      
   On 9/2/23 3:32 AM, Pat Dolan wrote:   
   > Consider a distant observer traveling at .867 c ( gamma=2 ) relative   
   > to the solar system along the line that is collinear with the sun's   
   > axis of rotation. As the clockwork solar system spins beneath him,   
   > the distant observer peers through his powerful telescope at Big Ben   
   > in London. In accordance with special relativity, and after taking   
   > relativistic doppler into account, the distant observer measures Big   
   > Ben's little hand to make one revolution for every two revolutions of   
   > his own wristwatch's little hand.   
      
   This is wrong -- "after taking relativistic doppler into account" the   
   observer would measure Big Ben's little hand to rotate at the same rate   
   as his own wristwatch's little hand. That is, the relativistic   
   Doppler-shift includes the effect due to relative motion and also the   
   effect due to "time dilation".   
      
        [I ignore the mistake that Big Ben is the nickname for   
         the Great Bell of the Great Clock of Westminster, and   
         not the clock itself.]   
      
   > He also observes that Big Ben's little hand still makes 730.5   
   > revolutions for every revolution that the earth makes around the   
   > sun.   
      
   This is correct.   
      
   > From these two observations the distant observer concludes that in   
   > his inertial frame of reference the earth's orbital velocity is only   
   > half the velocity necessary to keep the earth in stable orbit around   
   > the sun.   
      
   Nope, even if we correct the initial claim to leave "time dilation" in   
   the observation. SR does not properly handle the gravitation that keeps   
   earth in its orbit. For that one must use GR, and the components of the   
   metric in the distant observer's inertial frame are quite different from   
   those in the solar system rest frame. A correct calculation using the   
   distant observer's coordinates would model the sun continuing in its   
   orbit as usual.   
      
   IOW: The earth's path through spacetime is independent of the   
   coordinates used to describe it (this should be obvious). GR includes   
   this coordinate independence.   
      
   > Will the earth spiral into the sun? If not, why not?   
      
   It won't. For the simple reason that observations by a distant observer   
   cannot possibly affect the behavior of the solar system. This OUGHT to   
   be obvious.   
      
   Using GR to model the solar system, one immediately knows that the   
   predicted orbit of the earth is independent of coordinates, and the   
   distant observer obtains the same trajectory as an earthbound observer.   
      
        [I ignore the difficulties the distant observer will face   
         in obtaining accurate measurements.]   
      
   > Note: Newtonian gravity is not assumed in this paradox. Invariant   
   > spacetime curvature is assumed to be the cause of the earth's orbit   
   > around the sun.   
      
   Hmmmm. This disclaimer does not accurately describe how this is modeled   
   in GR. Your conclusion about the distant observer calculating earth's   
   orbit is wrong, and did not properly take into account the coordinate   
   independence of paths in GR.   
      
   Tom Roberts   
      
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