About Schwarzschild Metric

  • Thread starter parsifal
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  • #1
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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.
 
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Answers and Replies

  • #2
cristo
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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]
 
  • #3
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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!
 
  • #4
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I think that f(r) in the schwarzschild metric is:

[tex]f(r)=1-\frac{2m}{r}[/tex]

You don't need to do any calculus.
 
  • #5
cristo
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I think that f(r) in the schwarzschild metric is:

[tex]f(r)=1-\frac{2m}{r}[/tex]

You don't need to do any calculus.

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

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