[continued from previous message]   
      
       The Schwarzschild metric in Schwarzschild coordinates _can_ thus   
       _be written_ (this is the wording that I prefer to use)   
      
         ds² = −(1 − 2m/r) dt² + (1 − 2m/r)⁻¹ dr² + r²dΩ²,   
           m = G M/c²,   
         dΩ² = dθ² + sin²(θ) dφ.   
      
       Then it follows that   
      
         −dτ² = −(1 − 2m/r) dt² + (1 − 2m/r)⁻¹ dr² + r²dΩ².   
      
       One might even want to define a function of r,   
      
          f(r) = 1 − 2m/r   
      
       to reduce further repetition as g_rr = −(g_tt)⁻¹.  [My professor in   
       GR prefers to do that.]   
      
       For disambiguation, I also write and would recommend to write the   
       argument of a function in parentheses if a factor follows that does   
       not belong to the argument; here "sin²(θ) dφ²" instead of just   
       "sin²θ dφ²" or (your notation which is slightly more ambiguous than   
       the preceding one) "sin² θ dφ".   
      
         "G is the gravitational constant"   
      
       My professor in GR is fond of emphasizing that G is _Newton's_   
       (or "the Newtonian") gravitational constant and even writes "G_N"   
       all the time.  While I think that the subscript N is superfluous,   
       it is a good idea to emphasize that one means the _Newtonian_   
       constant, not the Gaussian one, in astronomical discussions   
       (astronomers are fond of using the Gaussian gravitational constant).   
      
          "M is the mass of the Earth"   
      
       As written, this is not correct.  The Schwarzschild metric is not   
       limited to Earth (the IAU recommends to drop the "the"), and   
       M is a mass only by convenience.  (The textbook Schwarzschild   
       black hole is *eternal*; it always was there and will ever be there,   
       and there is no explanation what M physically means -- in essence it   
       is just a convenient measure for energy to have an energy density.)   
       It is somewhat clear that you are discussing a special case, but then   
       you should say so explicitly.   
      
          "A very good approximation is:"   
      
       You do not give a reason why that is "a very good approximation"   
       in general or, more likely, *in this case*.  Maybe you want to borrow   
       the argument that I gave in my previous follow-up.   
      
       However, I do not see a good reason for an approximation.  Instead, I   
       find it a rather remarkable fact that an experimentally confirmed result   
       can be obtained from the Schwarzschild metric directly *without* a   
       previous approximation.   
      
       [A few months before I registered for the GR course, motivated by a need   
       for a better explanation of "time dilation", I did a similar, first more   
       general and then in its application more detailed calculation, published   
       on Quora (currently unavailable; they blocked me without good reason);   
       when previously for the Schwarzschild spacetime I had considered   
       "kinetic time dilation" and "gravitational time dilation" separately,   
       simply arguing conceptually that one offsets the other.]   
      
         "The difference between (4) and (5) is less than 10^-25 for all r"   
      
       That is a questionable claim (for *all* r?); you should substantiate it.   
      
         "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.   
      
     * You write:   
      
         "If we set GM = µ, the geocentric gravitational constant, equation (2)   
          can simplifies to:"   
      
       This should be "... can be simplified to" or "... equation (2)   
       simplifies to" (but not both).   
      
   - References:   
      
       * When you are referring to other scientific papers, you should not   
         refer to their copies in your Web space unless they were written by   
         you.  For example, Neil Ashby's paper is freely available, and you   
         should refer that instead:   
      
           https://link.springer.com/article/10.12942/lrr-2003-1   
      
       * When you are referring to sources whose content could change, such   
         as Web sources, you should indicate the date of retrieval; in   
         case of Wikipedia articles you can and should even refer to the   
         particular version that you used, e.g.   
      
      
      
      
         It can be obtained via the Wikipedia menu "Tools", and   
         "Permanent link" there.   
      
       * You should use a standard citation style that includes at least   
         the year and location of publication.  I recommend to use   
         BibLaTeX/biber to achieve that.  Let me know in private if you need   
         help regarding this.   
      
      
   \\//,   
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   PointedEars   
      
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