From: PointedEars@web.de   
      
   Paul B. Andersen wrote:   
   > Den 26.11.2025 12:52, skrev J. J. Lodder:   
   >> Summary: The GPS sat clocks are steered such that they are synchronous   
   >> with the GPS master clocks at USNOP.   
   >> (applying the appropriate corrections, Doppler and relativity)   
   >   
   > There is no correction for the Doppler shift due to the velocity   
   > of the SVs relative to the receiver.   
   > The gravitational Doppler shift is included in the (1-4.4647e-10)   
   > rate correction of the SV clock.   
      
   Gravitational redshift is NOT a Doppler shift (it arises from spacetime   
   curvature, not relative motion¹), but my professor in General Relativity has   
   confirmed to me that the interpretation of gravitational redshift as   
   successive relativistic Doppler shifts within a infinite family of observers   
   moving relative to each other due to different gravitational acceleration   
   (as explained by Veritasium) is valid.  (Still I think that it is an   
   unnecessarily complicated interpretation of the phenomenon.)   
      
   [By contrast, *cosmological* redshift cannot be properly understood as a   
   relativistic Doppler shift; cf. Davis & Lineweaver, "Expanding Confusion",   
   2003.]   
      
   ___   
   ¹ For the Schwarzschild metric in Schwarzschild coordinates, with sign   
     convention (−+++) and in natural units,   
      
       ds² = −dτ²   
           = −(1 − 2m/r) dt² + (1 − 2m/r)⁻¹ dr² + r² dΩ²   
           ≡ −A(r) dt² + A(r)⁻¹ dr² + r² dΩ²,   
      
         m ≡ G M/c²,   
       dΩ² ≡ dθ² + sin²(θ) dφ²,   
      
     for a stationary observer (dr² = dΩ² = 0) we have   
      
       dτ = dt √[A(r)].   
      
     For two observers at the radial coordinates r₁ and r₂, r₂ > r₁,   
     we have (trivially)   
      
               dτ₁ = dt √[A(r₁)]   
               dτ₂ = dt √[A(r₂)],   
      
     therefore   
      
           dτ₂/dτ₁ ~ f₁/f₂ = c λ₂/(c λ₁) = λ₂/λ₁,   
      
     where f₁ and λ₁ are the frequency and wavelength of the EM radiation   
     emitted by the observer at r₁, and f₂ and λ₂ are the frequency and   
     wavelength of the same radiation as received and observed by   
     the observer at r₂.   
      
       ==>      λ₂ ~ λ₁ dτ₂/dτ₁   
                   ~ λ₁ √[A(r₂)/A(r₁)]   
                   ~ λ₁ √[(1 − 2m/r₂)/(1 − 2m/r₁)]   
                   > λ₁. ∎   
      
       [r₂ > r₁ ⇔ 2m/r₂ < 2m/r₁   
                ⇔ 1 − 2m/r₂ > 1 − 2m/r₁   
                ⇔ (1 − 2m/r₂)/(1 − 2m/r₁) > 1.]   
      
   --   
   PointedEars   
      
   Twitter: @PointedEars2   
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