[continued from previous message]   
      
   Is it really necessary?   
      
   sqrt(1 - x) ~ (1 - x/2) when x << 1   
      
   The approximation gets worse the higher x is.   
   In (4) and (5) the highest value x can have is when   
   r is smallest, which is the radius of the Earth.   
      
   So just compare (4) and (5) with r = Earth's radius   
      
   >   
   >>>         "It is the Schwarzschild coordinate time that is fast, and its   
   seconds   
   >>>          are shorter than the SI-second."   
   >>>   
   >>>       I do not subscribe to your interpretation, and I find it misleading.   
   >>>       It is better to state that more or less proper time, measured in   
   >>>       the same (SI) seconds, elapse, than to say that seconds differ, that   
   >>>       time would be running fast or slow.   
   >>   
   >> I am sure you know that the rate of Schwarzschild coordinate time   
   >> is equal to the rate of a clock at infinity, and is faster closer   
   >> to the gravitating mass.   
      
   Same with UTC. The rate of UTC is faster further down.   
   The rate is equal to a proper clock at the ground,   
   and slower than a proper clock higher up.   
      
      
   >   
   > AISB, I do not subscribe to the idea and interpretation of the "rate of a   
   > clock".  IMHO, the effect should be argued in terms of potentially different   
   > elapsed proper times along different worldlines instead.   
      
   Exactly.   
   Consider the circular word line of a GPS satellite.   
   What would a clock in the satellite measure the proper time   
   of an orbit to be?   
   And what would the same orbit be measured by UTC ?   
   We know the answers.   
   The orbital time of the GPS satellite is half a sidereal day   
   measured in UTC, that's  86164.0905/2 UTC seconds   
   measured along the world line of the satellite.   
   Measured by a clock in the satellite the proper time   
   is (86164.0905/2)(1 + 4.4647e-10) proper seconds.   
      
   So measured along the world line, the UTC second is longer   
   than a proper second, and the rate o UTC is slower than   
   the rate of a proper clock.   
   >   
   > Please trim your quotes to the relevant minimum next time.   
   >   
      
      
   --   
   Paul   
      
   https://paulba.no/   
      
   --- SoupGate-Win32 v1.05   
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