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About Schwarzschild Metric

  1. Dec 8, 2006 #1
    I need to write Schwarzschild Metric, that is in spherical coordinates, into the form that has the metric tensor.

    Now, if the first the term of the metric is:
    [tex]\Large (ds)^2=f(r)c^2dt^2-....[/tex] and x0=ct,
    then the first component gij of the metric tensor g is supposed to be:
    [tex]\Large <\frac{\partial}{\partial x^i} \ | \ \frac{\partial}{\partial x^j}> \ ,i=j=0 \Rightarrow
    (\frac{d}{dx^0}f(r)c^2dt^2 | \frac{d}{dx^0}f(r)c^2dt^2)[/tex]

    But I do not actually understand that last statement. I guess dx0=cdt, but I do not know how to proceed from that.

    So I know this: the component g00 of g is supposed to be f(r), and I know that f(r) should come from the inner product, but I do not understand how. Basically, what does [tex]\Large \frac{d}{dx^0}f(r)c^2dt^2[/tex] mean?

    I apologize if this should have been in the introductory section, or in the calculus section.
    Last edited: Dec 8, 2006
  2. jcsd
  3. Dec 8, 2006 #2


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    Staff Emeritus
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    I don't see why you need to use the inner products. If you've already written down the metric in the form [tex]\Large (ds)^2=g_{ij}dx^idx^j[/tex] then you can simply read off the components of the metric tensor.
    i.e. [tex] (ds)^2=f(r)(dx^0)^2 \Rightarrow g_{00}=f(r) [/tex]
  4. Dec 8, 2006 #3
    I guess I was trying to do it the hard way, for some unclear reason. I didn't understand that the solution you suggested would do.

    Thanks for the answer!
  5. Dec 11, 2006 #4
    I think that f(r) in the schwarzschild metric is:


    You don't need to do any calculus.
  6. Dec 11, 2006 #5


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    Staff Emeritus
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    I'm not sure that the actual form of the function was required in the question.
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