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Understanding proper distance in Schwarzschild solution.

  1. Dec 21, 2011 #1
    I'm trying to understand the Schwarzschild solution concept of proper distance. Given the proper distance equation
    [tex]
    d\sigma=\frac{dr}{\left(1-\frac{R_{S}}{r}\right)^{1/2}}
    [/tex]
    how would I calculate the coordinate distance. For example - assuming the distance from the Earth to the Sun is 150,000,000km, is it a valid question to ask what the coordinate distance is, and how would I calculate it?

    I know [itex]R_{S}[/itex] is about 3km.

    Many thanks
     
  2. jcsd
  3. Dec 21, 2011 #2

    A.T.

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    Science Advisor
    Gold Member

    If you use Schwarzschild coordinates, then the radial coordinate distance is simply the difference in the radial coordinate "r" between two points: |rA-rB|

    To get the radial proper distance you have to integrate your formula between rA and rB.
     
  4. Dec 21, 2011 #3
    Thanks. For me, integrating that looks hard. Can I make any simplifying approximations?
     
  5. Dec 21, 2011 #4

    PeterDonis

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    2016 Award

    Staff: Mentor

    That general form of integral actually appears in standard tables of integrals, so you can find an exact antiderivative for it (you may have to modify the form of the integrand somewhat to fit it into a standard form). Try here for one such table:

    http://integral-table.com/

    Or, particularly if r is very large compared to R_s (which it certainly is in your case), you can expand the binomial in the denominator in a power series, as here:

    http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html

    You should only need the first couple of terms to see how things will go for the case of R_s / r very small.
     
  6. Dec 21, 2011 #5
    It is very easy to solve it with a math program such as Mable, Mathematica or Matlab. But effectively R = rho as the Sun's mass is not large enough to be considered a strong field. You have to go many numbers behind the decimal point to find a discrepancy.
     
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