[continued from previous message]   
      
   >>>        exposure :)]   
   >>>   
   >>>     * You write   
   >>>   
   >>>         "The Schwarzschild metric is:   
   >>>   
   >>>            c² dτ² = (1 − 2GM/[c²r]) c²dtc − 1/(1 − 2GM/[c²r])   
   >>>                     − r² (dθ² + sin² θ dφ)."              (2)   
   >   
   > 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.   
      
   OK.   
      
   >   
   >>>       [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.   
      
   It is a typo! (copy an paste error) I meant to write:   
      ds² = -(1 − 2GM/[c²r])c²dt² + 1/(1 − 2GM/[c²r])dr²   
                          + r² (dθ² + sin² θ dφ)   
   >   
   >> 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.   
      
   This is wrong. It should be:  ds² = - c²dτ²  (in SI units)   
      
   >   
   > 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),   
      
   Which would be pointless.   
      
   >   
   > 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)   
      
   Typo again.   
   I meant to write:   
      
   ds² = - c²dτ² = -(1 − 2GM/[c²r])c²dt² + 1/(1 − 2GM/[c²r])dr²   
                          + r² (dθ² + sin² θ dφ)   
   Which is correct.   
      
   When s² < 0 and thus time like, then  (-c²τ²) < 0 and thus time like.   
      
   So : ds² = -(1 − 2GM/[c²r])c²dt² + 1/(1 − 2GM/[c²r])dr²   
                          + r² (dθ² + sin² θ dφ)   
   and: -c²dτ² = -(1 − 2GM/[c²r])c²dt² + 1/(1 − 2GM/[c²r])dr²   
                         + r² (dθ² + sin² θ dφ)   
                          - r² (dθ² + sin² θ dφ)   
   >   
   > That is nonsense (see above), and not how you wrote it in your paper (in   
   > multiple respects).   
      
      
   In the paper I wrote:   
   c²dτ² = (1 − 2GM/[c²r])c²dt² - 1/(1 − 2GM/[c²r])dr²   
                          - r² (dθ² + sin² θ dφ)   
      
   You can argue that the signature in this case is (+---)   
   But then the same metric as above would be:   
   -ds² = (1 − 2GM/[c²r])c²dt² - 1/(1 − 2GM/[c²r])dr²   
      
   >   
   >> 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.   
      
   This is nonsense.   
   Of course I can use the Schwarzschild metric to find the proper   
   times of satellites orbiting the Earth without explaining that   
   it also could be used on othe massive bodies than the Earth.   
      
   >   
   >>>       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.   
      
      
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