From: PointedEars@web.de   
      
   Maciej Woźniak wrote:   
   > and that means that it's not a matter   
   > of some delusional Laws od Nature   
   > imagined by some idiot mysticians,   
   > it's a matter of how we decide to   
   > adjust clocks. Yes, it's that simple.   
      
   A manipulated clock is not a clock.   
      
   When we say that time is what is measured with a clock we mean a device that   
   is not manipulated, that merely follows Nature.   
      
   Such a clock, when previously synchronized with another clock (or when a   
   note has been taken of the time that it shows), in general shows a different   
   time (offset) than the other clock after it has been at rest in a different   
   reference frame (when it has been in motion relative to the first clock   
   and/or when a different gravitational acceleration would have been measured   
   for it).   
      
   It's that simple.   
      
   But the mathematics of general relativity that describes this correctly and   
   predicts it very precisely is not so simple.  Which is probably why it is   
   probably way over your head, and you keep insisting on your misconceptions.   
      
   Read the following to check yourself: whether you have actually understood   
   what you are talking about.  (If you do not understand it, you are not in a   
   position to talk about it, let alone criticize it.)   
      
   In units where c = 1, the Schwarzschild metric, for the spacetime produced   
   by a spherically-symmetric mass/energy distribution, is in Schwarzschild   
   coordinates:   
      
     ds^2 = −dτ² = −(1 − 2m/r) dt^2 + (1 − 2m/r)^-1 dr^2 + r^2 dΩ²,   
      
        m = G M/c^2,   
        dΩ² = dθ^2 + sin^2(θ) dφ²,   
      
   where M is the mass of the central object, and r, θ, φ are the standard   
   spherical coordinates.   
      
   Then the ratio of proper time (measured with a co-moving clock) and   
   coordinate time (that would be measured at rest at infinite distance from   
   the center of mass) is   
      
     dτ/dt = √[1 − 2m/r − (1 − 2m/r)^-1 dr/dt − r^2 dΩ/dt].   
      
   For an observer at relative rest (dr = dΩ = 0):   
      
     dτ = dt √(1 − 2m/r).   
      
   One can see that, *at relative rest*, more proper time elapses farther away   
   from the center of mass (at larger values of r) than closer to it (at   
   smaller values of r).   
      
   For a(n) observer/clock in a circular orbit (dr = 0) with radius r, though:   
      
     dτ = dt √[1 − 2m/r − r^2 sin^2(θ) dφ/dt]   
        = dt √[1 − 2m/r − v_{orb}²].   
      
   One can see that less proper time elapses the greater the orbital speed   
   v_{orb} is.  But the orbital speed is smaller if the radius of the orbit is   
   greater (as _described_ by Kepler's Second and Third Law).  These effects   
   combine and can be calculated by substituting the Newtonian orbital speed   
   v_{orb} = √(G M/r):   
      
     dτ = dt √[1 − 3G M/(r c^2)].   
      
   So that e.g. aboard GPS satellites, which are in circular orbits with a   
   radius r ≈ 6371 km + 20000 km, more proper time elapses than at rest on the   
   ground (r ≈ 6371 km): approximately 38.3 μs/d more (substitute and   
   calculate it to check; 1 d = 86400 s).   
      
   Thus the GPS sending frequency is nowadays slightly lowered according to GR   
   so that it *appears* on the ground *as if* the same proper time would elapse   
   in orbit as on the ground at rest relative to the ground.  The GPS message   
   payload also includes information that is regularly updated from the GPS   
   ground station for use by a GPS receiver to account for the varying orbital   
   motion of a satellite.  Which makes precise global geopositioning and   
   timekeeping possible in the first place:   
      
   https://link.springer.com/article/10.12942/lrr-2003-1#Sec5   
      
   [In fact, it actually /is/ simple.  You just need to (be willing to) wrap   
   your head around it.]   
      
   --   
   PointedEars   
      
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