I've worked through a common-sense argument(adsbygoogle = window.adsbygoogle || []).push({}); ^{http://www.mathpages.com/rr/s8-09/8-09.htm" [Broken]}showing the time-time component of the Schwarzschild metric

[tex]g_{tt} = \left (\frac{\partial \tau}{\partial r} \right )^2\approx 1-\frac{2 G M}{r c^2} [/tex]

On the other hand, I've not worked through any common-sense argument for thegcomponent of the Schwarzschild metric:_{rr}

[tex]\left (\frac{\partial s}{\partial r} \right )^2\approx \frac{1}{1-\frac{2 G M}{r c^2}}[/tex]

I can see there is a derivation in the http://www.blatword.co.uk/space-time/Carrol_GR_lectures.pdf" [Broken] on pages 168-172. That remains a goal for me, to work through that derivation as well, but I'm not comfortable with most of the concepts involved here.

I am interested in the reasons behind these steps.

- assuming [tex]g_{11}=-1/g_{00}[/tex]
- finding the connection coefficients
- finding the nonvanishing components of the Reimann tensor
- taking the contraction (as usual?) to find the Ricci tensor
- setting all the components of the Ricci tensor to zero
- discovering that the g
_{00}and g_{11}had to be functions ofr, only,- Setting R
_{00}=R_{11}=0- Doing some differential equations with boundary conditions, and deriving the metric

My trouble is that I don't have a common-sense understanding of the Reimann tensor or the Ricci tensor. I'm also having trouble distinguishing the relevant equations, like definitions of these tensors, as I find myself, as I read through the Carroll Lectures, filling up page after page of undefined components, but never really getting to the heart of the matter.

I'm only beginning to have some common-sense understanding of the connection (Christoffel) coefficient, based on the Cartesian to polar connection coefficients, diagrammed on page 6, http://mysite.verizon.net/vze11jx21/GR2c-Derivatives.pdf".

Where should I begin? Perhaps with step 1. Why do we start with the assumption that g_{tt}is the negative reciprocal of g_{rr}?

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# Deriving the Schwarzschild Metric

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