- #1
JDoolin
Gold Member
- 723
- 9
I've worked through a common-sense argumenthttp://www.mathpages.com/rr/s8-09/8-09.htm" 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 the grr component of the Schwarzschild metric:
[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" 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.
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 gtt is the negative reciprocal of grr?
[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 the grr component of the Schwarzschild metric:
[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" 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 g00 and g11 had to be functions of r, only,
- Setting R00=R11=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 gtt is the negative reciprocal of grr?
Last edited by a moderator: