Christoffel Symbols for Schwarzschild Metric (?)

[Widdekind]

ROUGH DRAFT

I have a beginner's basic question:

1. Schwarzschild Metric components

Let $$\epsilon$$ = r_s / r, where r_s is the Schwarzschild Radius. Then, as is is well-known:

$$g_{00} = 1 - \epsilon$$
$$g_{11} = - \left( 1 - \epsilon \right)^{-1}$$
$$g_{22} = - r^{2}$$
$$g_{33} = - r^{2} \; sin^{2}(\theta)$$​

B/c this Schwarzschild Metric Tensor g_ij is Diagonal, its Inverse g^ij is also Diagonal, w/ components equal to "one over" those above.2. Christoffel Symbol components

As is well-known:

$$\Gamma^{i}_{k\ell} = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m})$$​

But, since the Schwarzschild Metric Tensor is diagonal, $$g^{im} = \delta^{im} \; g^{ii}$$. So:

$$\Gamma^{i}_{k\ell} = {1 \over 2} g^{ii} (g_{ik,\ell} + g_{i\ell,k} - g_{k\ell,i}) + 0$$​

Thus, in this Schwarzschild Polar Coordinate System, w.h.t.:

$$\Gamma^{0}_{k\ell} = {1 \over 2} g^{00} \[ \left( \begin{array}{cccc}<br /> 0 & g_{00,1} & 0 & 0 \\<br /> g_{00,1} & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & 0 \end{array} \right)\]$$

$$\Gamma^{1}_{k\ell} = {1 \over 2} g^{11} \[ \left( \begin{array}{cccc}<br /> -g_{00,1} & 0 & 0 & 0 \\<br /> 0 & g_{11,1} & 0 & 0 \\<br /> 0 & 0 & -g_{22,1} & 0 \\<br /> 0 & 0 & 0 & -g_{33,1} \end{array} \right)\]$$

$$\Gamma^{2}_{k\ell} = {1 \over 2} g^{22} \[ \left( \begin{array}{cccc}<br /> 0 & 0 & 0 & 0 \\<br /> 0 & 0 & g_{22,1} & 0 \\<br /> 0 & g_{22,1} & 0 & 0 \\<br /> 0 & 0 & 0 & -g_{33,2} \end{array} \right)\]$$

$$\Gamma^{3}_{k\ell} = {1 \over 2} g^{33} \[ \left( \begin{array}{cccc}<br /> 0 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & g_{33,1} \\<br /> 0 & 0 & 0 & g_{33,2} \\<br /> 0 & g_{33,1} & g_{33,2} & 0 \end{array} \right)\]$$​

Or, noting that $$\partial \epsilon / \partial r = - \epsilon / r$$, w.h.t.:

$$\Gamma^{0}_{k\ell} = {1 \over 2} (1 - \epsilon)^{-1} \[ \left( \begin{array}{cccc}<br /> 0 & \epsilon / r & 0 & 0 \\<br /> \epsilon / r & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & 0 \end{array} \right)\]$$

$$\Gamma^{1}_{k\ell} = -{1 \over 2} \left( 1 - \epsilon \right) \[ \left( \begin{array}{cccc}<br /> -\epsilon / r & 0 & 0 & 0 \\<br /> 0 & \left( 1 - \epsilon \right)^{-2}(\epsilon / r) & 0 & 0 \\<br /> 0 & 0 & 2 r & 0 \\<br /> 0 & 0 & 0 & 2 r \; sin^{2}(\theta) \end{array} \right)\]$$

$$\Gamma^{2}_{k\ell} = -{1 \over 2} r^{-2} \[ \left( \begin{array}{cccc}<br /> 0 & 0 & 0 & 0 \\<br /> 0 & 0 & -2 r & 0 \\<br /> 0 & -2 r & 0 & 0 \\<br /> 0 & 0 & 0 & r^{2} \; sin(2 \theta) \end{array} \right)\]$$

$$\Gamma^{3}_{k\ell} = -{1 \over 2} r^{-2} \; sin^{-2}(\theta) \[ \left( \begin{array}{cccc}<br /> 0 & 0 & 0 & 0 \\<br /> 0 & 0 & 0 & -2 r \; sin^{2}(\theta) \\<br /> 0 & 0 & 0 & -r^{2} \; sin(2 \theta) \\<br /> 0 & -2 r \; sin^{2}(\theta) & -r^{2} \; sin(2 \theta) & 0 \end{array} \right)\]$$​

3. Geodesic Equation (?!)

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> (1 - \epsilon)^{-1} {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\<br /> {1 \over 2} \left( 1 - \epsilon \right) \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - \left( 1 - \epsilon \right)^{-2}{\epsilon \over r}{\partial r \over \partial s}^{2} - 2 r {\partial \theta \over \partial s}^{2} - 2 r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \right) \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} - {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$4. Zero-Gravity limit (??)

If $$\epsilon = 0$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> 0 \\<br /> - r {\partial (\theta) \over \partial s}^{2} - r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Further restricting $$\theta = {\pi \over 2}$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> 0 \\<br /> - r {\partial (\phi) \over \partial s}^{2} \\<br /> 0 \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Is this the equation of a straight line in Polar Coordinates ?5. Weak-Gravity limit (??)

If $$\epsilon << 1$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\<br /> {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\theta) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r \; sin^{2}(\theta) {\partial (\phi) \over \partial s}^{2} \right) \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial (\phi) \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Further restricting $$\theta = {\pi \over 2}$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\<br /> {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} + {\epsilon \over r}{\partial (r) \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial (\phi) \over \partial s}^{2} \right) \\<br /> 0 \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Does this reduce to Newton's equations ?

 

[Widdekind]

Possibly pertinent posts:

https://www.physicsforums.com/archive/index.php/t-8259.htmlbr/t-114999.html
https://www.physicsforums.com/showpost.php?p=621802&postcount=32
https://www.physicsforums.com/showthread.php?p=2283415#post2283415
https://www.physicsforums.com/showthread.php?t=270346

L. P. Hughston, K. P. Tod. An introduction to general relativity, pg. 113.

And, LaTex matrices:
http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/Matrices.html

 

[Widdekind]

Verifying Geodesic Equation by applying Euler-Lagrange Equation to Scwarzschild Metric-derived Lagrangian

We apply the Euler-Lagrange Equation to the Scwarzschild Metric-derived Lagrangian:

$$L \equiv g_{\mu \nu} {d x^{\mu} \over ds} {d x^{\nu} \over ds} = 1$$​

Explicitly, w.h.t.:

$$L \equiv (1 - \epsilon) c^{2} {dt \over ds}^{2} - (1 - \epsilon)^{-1} {dr \over ds}^{2} - r^{2} {d \theta \over ds}^{2} - r^{2} sin^{2}(\theta) {d \phi \over ds}^{2}$$​

Applying the Euler-Lagrange Equation:

$${\partial L \over \partial x^{\mu}} - {d \over ds}{\partial L \over \partial ({\partial x^{\mu} \over \partial s})} = 0$$​

w.h.t.:

$$\left( \begin{array}{c}<br /> 0 \\<br /> c^{2} {\epsilon \over r}{\partial t \over \partial s}^{2} + (1 - \epsilon)^{-2} {\epsilon \over r}{\partial r \over \partial s}^{2} - 2 r {\partial \theta \over \partial s}^{2} - 2 r sin^{2}(\theta) {\partial \phi \over \partial s}^{2}\\<br /> - r^{2} sin(2 \theta) {\partial \phi \over \partial s}^{2} \\<br /> 0 \end{array} \right) - \left( \begin{array}{c}<br /> 2 c^{2} {\epsilon \over r} {\partial r \over \partial s}{\partial t \over \partial s} + 2 c^{2} (1 - \epsilon) {\partial^{2} t \over \partial s^{2}} \\<br /> 2 (1 - \epsilon)^{-2} {\epsilon \over r}{\partial r \over \partial s}^{2} - 2 (1 - \epsilon)^{-1}{\partial^{2} r \over \partial s^{2}} \\<br /> -4 r {\partial r \over \partial s}{\partial \theta \over \partial s} - 2 r^{2}{\partial^{2} \theta \over \partial s^{2}} \\<br /> -2 r^{2} sin^{2}(\theta){\partial^{2} \phi \over \partial s^{2}} - 2 r^{2} sin(2 \theta) {\partial \theta \over \partial s}{\partial \phi \over \partial s} - 4 r {\partial r \over \partial s}{\partial \phi \over \partial s} \end{array} \right) = 0$$​

This seems to be substantially similar to the above-derived Geodesic Equation (so far).

 

[Widdekind]

UPDATE

4. Zero-Gravity limit (??)

If $$\epsilon = 0$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> 0\\<br /> - r {\partial \theta \over \partial s}^{2} - r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} - {sin(2 \theta) \over 2} {\partial (\phi) \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Further restricting $$\theta = {\pi \over 2}$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> 0 \\<br /> - r {\partial (\phi) \over \partial s}^{2} \\<br /> 0 \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Is this the equation of a straight line in Polar Coordinates ?


5. Weak-Gravity limit (??)

If $$\epsilon << 1$$, w.h.t.:

$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\<br /> {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - {\epsilon \over r}{\partial r \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial \theta \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r \; sin^{2}(\theta) {\partial \phi \over \partial s}^{2} \right) \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \theta \over \partial s} + {sin(2 \theta) \over 2}{\partial \phi \over \partial s}^{2} \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} + 2 cot(\theta){\partial \theta \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Further restricting $$\theta = {\pi \over 2}$$, w.h.t.:


$${\partial^{2} \over \partial s^{2}} \left( \begin{array}{c}<br /> c t \\<br /> r \\<br /> \theta \\<br /> \phi \end{array} \right) + \left( \begin{array}{c}<br /> {\epsilon \over r} {\partial (ct) \over \partial s} {\partial r \over \partial s} \\<br /> {1 \over 2} \left( {\epsilon \over r}{\partial (ct) \over \partial s}^{2} - {\epsilon \over r}{\partial r \over \partial s}^{2} - \left( 1 - \epsilon \right) 2 r {\partial \phi \over \partial s}^{2} \right) \\<br /> 0 \\<br /> {2 \over r}{\partial r \over \partial s} {\partial \phi \over \partial s} \end{array} \right) = 0$$

Does this reduce to Newton's equations ?

 

[cristo]

There doesn't really appear to be a question here, which is probably why you haven't received any replies. If I were you, I would re-check your first calculation, and compare to well known results. (See, for example, http://arxiv.org/abs/0904.4184 for a useful catalogue of spacetimes).