- #1
Widdekind
- 132
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ROUGH DRAFT
I have a beginner's basic question:
1. Schwarzschild Metric components
Let [tex]\epsilon[/tex] = rs / r, where rs is the Schwarzschild Radius. Then, as is is well-known:
B/c this Schwarzschild Metric Tensor gij is Diagonal, its Inverse gij is also Diagonal, w/ components equal to "one over" those above.2. Christoffel Symbol components
As is well-known:
But, since the Schwarzschild Metric Tensor is diagonal, [tex]g^{im} = \delta^{im} \; g^{ii}[/tex]. So:
Thus, in this Schwarzschild Polar Coordinate System, w.h.t.:
Or, noting that [tex]\partial \epsilon / \partial r = - \epsilon / r[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
(1 - \epsilon)^{-1} {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
{1 \over 2} \left( 1 - \epsilon \right) \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - \left( 1 - \epsilon \right)^{-2}{\epsilon \over r}{\partial r \over \partial s}^{2} - 2 r {\partial \theta \over \partial s}^{2} - 2 r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \right) \\
{2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} - {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]4. Zero-Gravity limit (??)
If [tex]\epsilon = 0[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
0 \\
- r {\partial (\theta) \over \partial s}^{2} - r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
0 \\
- r {\partial (\phi) \over \partial s}^{2} \\
0 \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Is this the equation of a straight line in Polar Coordinates ?5. Weak-Gravity limit (??)
If [tex]\epsilon << 1[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
{\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
{1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\theta) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \right) \\
{2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
{\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
{1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\phi) \over \partial s}^{2} \right) \\
0 \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Does this reduce to Newton's equations ?
I have a beginner's basic question:
1. Schwarzschild Metric components
Let [tex]\epsilon[/tex] = rs / r, where rs is the Schwarzschild Radius. Then, as is is well-known:
[tex]g_{00} = 1 - \epsilon[/tex]
[tex]g_{11} = - \left( 1 - \epsilon \right)^{-1}[/tex]
[tex]g_{22} = - r^{2}[/tex]
[tex]g_{33} = - r^{2} \; sin^{2}(\theta)[/tex]
[tex]g_{11} = - \left( 1 - \epsilon \right)^{-1}[/tex]
[tex]g_{22} = - r^{2}[/tex]
[tex]g_{33} = - r^{2} \; sin^{2}(\theta)[/tex]
B/c this Schwarzschild Metric Tensor gij is Diagonal, its Inverse gij is also Diagonal, w/ components equal to "one over" those above.2. Christoffel Symbol components
As is well-known:
[tex]\Gamma^{i}_{k\ell} = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m})[/tex]
But, since the Schwarzschild Metric Tensor is diagonal, [tex]g^{im} = \delta^{im} \; g^{ii}[/tex]. So:
[tex]\Gamma^{i}_{k\ell} = {1 \over 2} g^{ii} (g_{ik,\ell} + g_{i\ell,k} - g_{k\ell,i}) + 0[/tex]
Thus, in this Schwarzschild Polar Coordinate System, w.h.t.:
[tex]\Gamma^{0}_{k\ell} = {1 \over 2} g^{00} \[ \left( \begin{array}{cccc}
0 & g_{00,1} & 0 & 0 \\
g_{00,1} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right)\][/tex]
[tex]\Gamma^{1}_{k\ell} = {1 \over 2} g^{11} \[ \left( \begin{array}{cccc}
-g_{00,1} & 0 & 0 & 0 \\
0 & g_{11,1} & 0 & 0 \\
0 & 0 & -g_{22,1} & 0 \\
0 & 0 & 0 & -g_{33,1} \end{array} \right)\][/tex]
[tex]\Gamma^{2}_{k\ell} = {1 \over 2} g^{22} \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & g_{22,1} & 0 \\
0 & g_{22,1} & 0 & 0 \\
0 & 0 & 0 & -g_{33,2} \end{array} \right)\][/tex]
[tex]\Gamma^{3}_{k\ell} = {1 \over 2} g^{33} \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & g_{33,1} \\
0 & 0 & 0 & g_{33,2} \\
0 & g_{33,1} & g_{33,2} & 0 \end{array} \right)\][/tex]
0 & g_{00,1} & 0 & 0 \\
g_{00,1} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right)\][/tex]
[tex]\Gamma^{1}_{k\ell} = {1 \over 2} g^{11} \[ \left( \begin{array}{cccc}
-g_{00,1} & 0 & 0 & 0 \\
0 & g_{11,1} & 0 & 0 \\
0 & 0 & -g_{22,1} & 0 \\
0 & 0 & 0 & -g_{33,1} \end{array} \right)\][/tex]
[tex]\Gamma^{2}_{k\ell} = {1 \over 2} g^{22} \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & g_{22,1} & 0 \\
0 & g_{22,1} & 0 & 0 \\
0 & 0 & 0 & -g_{33,2} \end{array} \right)\][/tex]
[tex]\Gamma^{3}_{k\ell} = {1 \over 2} g^{33} \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & g_{33,1} \\
0 & 0 & 0 & g_{33,2} \\
0 & g_{33,1} & g_{33,2} & 0 \end{array} \right)\][/tex]
Or, noting that [tex]\partial \epsilon / \partial r = - \epsilon / r[/tex], w.h.t.:
[tex]\Gamma^{0}_{k\ell} = {1 \over 2} (1 - \epsilon)^{-1} \[ \left( \begin{array}{cccc}
0 & \epsilon / r & 0 & 0 \\
\epsilon / r & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right)\][/tex]
[tex]\Gamma^{1}_{k\ell} = -{1 \over 2} \left( 1 - \epsilon \right) \[ \left( \begin{array}{cccc}
-\epsilon / r & 0 & 0 & 0 \\
0 & \left( 1 - \epsilon \right)^{-2}(\epsilon / r) & 0 & 0 \\
0 & 0 & 2 r & 0 \\
0 & 0 & 0 & 2 r \; sin^{2}(\theta) \end{array} \right)\][/tex]
[tex]\Gamma^{2}_{k\ell} = -{1 \over 2} r^{-2} \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & -2 r & 0 \\
0 & -2 r & 0 & 0 \\
0 & 0 & 0 & r^{2} \; sin(2 \theta) \end{array} \right)\][/tex]
[tex]\Gamma^{3}_{k\ell} = -{1 \over 2} r^{-2} \; sin^{-2}(\theta) \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -2 r \; sin^{2}(\theta) \\
0 & 0 & 0 & -r^{2} \; sin(2 \theta) \\
0 & -2 r \; sin^{2}(\theta) & -r^{2} \; sin(2 \theta) & 0 \end{array} \right)\][/tex]
3. Geodesic Equation (?!)0 & \epsilon / r & 0 & 0 \\
\epsilon / r & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right)\][/tex]
[tex]\Gamma^{1}_{k\ell} = -{1 \over 2} \left( 1 - \epsilon \right) \[ \left( \begin{array}{cccc}
-\epsilon / r & 0 & 0 & 0 \\
0 & \left( 1 - \epsilon \right)^{-2}(\epsilon / r) & 0 & 0 \\
0 & 0 & 2 r & 0 \\
0 & 0 & 0 & 2 r \; sin^{2}(\theta) \end{array} \right)\][/tex]
[tex]\Gamma^{2}_{k\ell} = -{1 \over 2} r^{-2} \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & -2 r & 0 \\
0 & -2 r & 0 & 0 \\
0 & 0 & 0 & r^{2} \; sin(2 \theta) \end{array} \right)\][/tex]
[tex]\Gamma^{3}_{k\ell} = -{1 \over 2} r^{-2} \; sin^{-2}(\theta) \[ \left( \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -2 r \; sin^{2}(\theta) \\
0 & 0 & 0 & -r^{2} \; sin(2 \theta) \\
0 & -2 r \; sin^{2}(\theta) & -r^{2} \; sin(2 \theta) & 0 \end{array} \right)\][/tex]
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
(1 - \epsilon)^{-1} {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
{1 \over 2} \left( 1 - \epsilon \right) \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - \left( 1 - \epsilon \right)^{-2}{\epsilon \over r}{\partial r \over \partial s}^{2} - 2 r {\partial \theta \over \partial s}^{2} - 2 r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \right) \\
{2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} - {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]4. Zero-Gravity limit (??)
If [tex]\epsilon = 0[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
0 \\
- r {\partial (\theta) \over \partial s}^{2} - r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
0 \\
- r {\partial (\phi) \over \partial s}^{2} \\
0 \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Is this the equation of a straight line in Polar Coordinates ?5. Weak-Gravity limit (??)
If [tex]\epsilon << 1[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
{\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
{1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\theta) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \right) \\
{2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Further restricting [tex]\theta = {\pi \over 2}[/tex], w.h.t.:
[tex]{\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}
c t \\
r \\
\theta \\
\phi \end{array} \right) + \left( \begin{array}{c}
{\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\
{1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\phi) \over \partial s}^{2} \right) \\
0 \\
{2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0[/tex]
Does this reduce to Newton's equations ?
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