[continued from previous message]   
      
   I made two mistakes in converting this to Unicode by copying it from your   
   PDF document.  In the original you actually write "c²dt²" and "dφ²" as it   
   is   
   correct.   
      
   >>      [where I have put rectangular brackets for the corresponding rendering   
   >>       in Unicode of LaTeX-generated formulae].   
   >>   
   >>      That is not quite correct.  First of all, a (spacetime) metric is   
   >>      defined via its line element ds², not proper time.  That the line   
   >>      element is equal to ±c²dτ² where we call τ the proper time is a   
   >>      consequence of the (spacetime) metric (coming from the Minkowski   
   >>      metric) and the *definition* of proper time -- namely when all spatial   
   >>      differentials are equal to 0 --, and the chosen sign convention   
   >>      (see below).   
   >   
   > Schwarzschild metric with SI-units and signature -+++ is.   
   >    c²ds² = -(1 − 2GM/[c²r])c²dt² + 1/(1 − 2GM/[c²r])dr²   
   >                     + r² (dθ² + sin² θ dφ)   
      
   Sorry, no, that is *definitely* wrong.  Instead, it is what I wrote before.   
      
   > Since we are only interested time like intervals where   
   > s² is negative it is better to set ds² = - dτ² and we get   
   > the equation numbered (2) in my article.   
      
   That is NOT how it works.  Unless you divide the metric by a constant value,   
   it is *always* _ds²_ LHS (or RHS) *regardless* what kind of interval it is   
   [possibilities are ds² < 0, ds² = 0, and ds² > 0. In the sign convention   
   (+−−−), ds² > 0 means timelike; in (−+++), spacelike].  So you can   
   write, if   
   you want   
      
      c²ds² = -(1 − 2GM/[c²r]) c⁴dt² + 1/(1 − 2GM/[c²r]) c²dr²   
              + c²r² [dθ² + sin²(θ) dφ²),   
      
   i.e. you can multiply the whole equation by c².  It does not change   
   anything, but it is also not particularly helpful.   
      
   But you can NOT write what you have written above.  That you can set c ≔ 1   
   for convenience if you want (as I indicated) does NOT mean that you can put   
   c where you want: For example, the equation still has to be dimensionally   
   consistent, and c²ds² is NOT the square of a length (remember that we put   
   the c² in c²dt² merely because initially we chose the temporal coordinate to   
   be x⁰ ≔ c t so that we would have a quantity with dimensions of length like   
   x¹ ≔ x (the Euclidean one), x² ≔ y, and x³ ≔ z, and the metric would   
   be   
   dimensionally consistent).   
      
   > It is the exact same metric as the one above, with the same signature.   
      
   No.   
      
   >>      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:   
      
   Nonsense.  In your paper you clearly use the sign convention (+−−−)   
   [which   
   is OK, just less common in GR], and that is what I was referring to.   
      
   >     c²ds² = - c²dτ² = -(1 − 2GM/[c²r])c²dt² + 1/(1 −   
   2GM/[c²r])dr²   
   >                     + r² (dθ² + sin² θ dφ)   
      
   That is nonsense (see above), and not how you wrote it in your paper (in   
   multiple respects).   
      
   > 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.   
      
   You appear to be missing the points of my comment in this regard.   
      
   In particular, while you *can* use the Schwarzschild metric (in whatever   
   coordinates you want) to describe the "gravitational field" of Earth, the   
   metric is NOT *defined* based on the gravitational field of Earth, but   
   describes a spacetime corresponding to *any* spherically-symmetric   
   mass/energy distribution.  That distinction is missing from your paper, and   
   that is misleading.   
      
   >>      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?   
      
   Square roots are complicated?   
      
   >>      [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.   
      
   Prove it, then.   
      
   >>        "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.   
      
   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.   
      
   Please trim your quotes to the relevant minimum next time.   
      
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   PointedEars   
      
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