XPost: sci.math   
   From: PointedEars@web.de   
      
   Garth Dzhanaev wrote:   
   > Paul B. Andersen wrote:   
   >> The SV clock will typically be microseconds off sync, but a number of   
   >> monitoring stations will track the SV, and measure how the clock   
   >> behaves, and calculate the correction polynomial parameters which are   
   >> uploaded to the SV at least once every three days, typically once a day.   
   >>   
   >> See:   
   >> https://paulba.no/div/GPS_clock_correction.pdf   
   >   
   > thanks, we are great, microseconds accumulates, actually more on earth,   
   > than in space;   
      
   No, one would have to move in outer space sufficiently fast relative to the   
   surface of Earth (or any other celestial body) for less proper time to   
   elapse there than on the surface of Earth.   
      
   Assuming a spherically-symmetric planet, we may use the Schwarzschild metric   
   in Schwarzschild coordinates (with sign convention -+++, and in natural   
   units for simplicity) to find that out:   
      
     ds^2 = -(d tau)^2   
          = -(1 - 2m/r) dt^2 + (1 - 2m/r)^-1 dr^2 + r^2 (d Omega)^2   
          = -dt^2 [(1 - 2m/r) - (1 - 2m/r)^-1 (dr/dt)^2 - r^2 (d Omega/dt)^2],   
      
        m = G M/c^2   
      
   ==> d tau   
         = dt sqrt[(1 - 2m/r) - (1 - 2m/r)^-1 (dr/dt)^2 - r^2 (d Omega/dt)^2]   
         = dt sqrt[(1 - 2m/r) - (1 - 2m/r)^-1 (dr/dt)^2   
                   - r^2 {(d theta/dt)^2 + sin^2(theta) (d phi/dt)^2}]   
      
   As you can see, the elapsed proper time _increases_ with increasing radius   
   (r), i.e. at relative rest, _more_ of it elapses in outer space than on the   
   ground, and decreases with increasing relative speed (dr/dt, d theta/dt,   
   d phi/dt).   
      
   But since GPS satellites are in an *orbit* (with an altitude of ca. 20'000   
   km), and the orbital speed *decreases* with increasing radius, this is not   
   the case for the elapsed proper time aboard the satellites.   
      
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   PointedEars   
      
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