[continued from previous message]   
      
   the equation numbered (2) in my article.   
   It is the exact same metric as the one above, with the same signature.   
      
   >   
   >      Second, it is very important to realize that this is just the   
   >      Schwarzschild metric _in Schwarzschild coordinates_, and using   
   >      the "mostly minus" (+−−−) sign convention; see above and below.   
      
   No, it is the metric with signature (-+++) because:   
      
       c²ds² = - c²dτ² = -(1 − 2GM/[c²r])c²dt² + 1/(1 −   
   2GM/[c²r])dr²   
                       + r² (dθ² + sin² θ dφ)   
      
   I am using the Schwarzschild metric to calculate:   
   "The rate of clocks in circular orbit compared to clocks on the geoid"   
      
   I should possibly have called the paper:   
   "The rate of clocks in circular orbit around the Earth compared to   
   clocks on the geoid".   
   but the word "geoid" should make it obvious that  the clocks are   
   orbiting the Earth.   
      
   Since the mass of the Earth, the orbital radii , the orbital   
   periods, the radius of the Earth, the gravitational constant   
   all are given in SI-units, the Schwarzschild metric must   
   also be written in SI-units.   
      
   So I find most of the below a bit off the point.   
      
   >   
   >      Further, as I learned and indicated in my previous recent   
   >      postings, the metric becomes more concise, and thus easier   
   >      to understand and handle mathematically, if   
   >   
   >        - one writes it in natural units where c = 1;   
   >        - one defines a variable for the Schwarzschild radius or G M/c²;   
   >        - one defines a variable for the spherical surface element (dΩ²).   
   >   
   >      Also, it is my impression, and I have been told explicitly by them,   
   >      that theoretical physicists doing research in general relativity   
   >      prefer to use the "mostly plus" (−+++) sign convention.  The reason   
   >      being, IIUC, that purely spatial discussions are more obvious   
   >      (cf. the Pythagorean theorem), and that the oddity here is the temporal   
   >      coordinate, not the spatial ones.  [Of course, theoretical physicists   
   >      doing research in quantum field theory and related topics   
   >      beg to differ :)]   
   >   
   >      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).   
      
   I am using μ = GM, the geocentric gravitational constant as it   
   is defined in the GPS Interface Specification Document.   
      
   >   
   >         "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.   
      
   I consider the first order approximations: √(1−x)≈(1−x/2) when |x|<< 1,   
       and (1-x)⋅(1+y) ≈ 1-x+y when |xy| << 1   
   to be well known by most readers   
   If not, the statements:   
   "The difference between (4) and (5) is less than 10^-25 for all r."   
   and   
   "The difference between (8) and (9) is less than 10^−17 for all r."   
   should make it clear that the approximations are good.   
      
   >   
   >      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.   
      
   You can obviously calculate the exact equation for dτ/dt_utc for   
   a satellite, or you can use equation (10) in my paper:   
   dτ/dt_utc = (1 + δ_utc - 1.5⋅µ/c²r)   
      
   The error in the latter compared to the former is in the order of   
   1e-16. The error in r will be many order order of magnitudes   
   greater than the error in the equation.   
      
   So what's the point with using a very complicated equation in   
   stead of a very simple one?   
      
   This is practical calculation, not research.   
      
   >   
   >      [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 true, tough.   
      
   >   
   >        "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.   
      
   If you use 'second' as time unit, the Schwarzschild coordinate time   
   'second' is shorter than the second of the clock.   
   A clock always run at it's proper rate, so it is the coordinate   
   time that is fast.   
      
   >   
   >    * 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).   
      
   OK. Will be corrected.   
      
   >   
   > - References:   
   >   
   >      * When you are referring to other scientific papers, you should not   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)