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Well, I think I finally figured out how to get good values for the local values of the Christoffel symbols (aka local gravitational accelerations) in the Schwarzschild metric. Some of the results are moderately interesting, though there is one point that still makes me wonder a bit.
If we let unit vectors in the r, theta, and phi directions be \hat r \hat \theta \hat \phi, I get the results (using geometric units) that
<br /> \Gamma_{\hat r \hat r \hat r} = \Gamma_{\hat r \hat t \hat t} = -\frac {M} {r^{\frac{3}{2}}\sqrt{r-2M}}<br />
<br /> \Gamma_{\hat r \hat \theta \hat \theta} = \Gamma_{\hat r \hat \phi \hat <br /> \phi} = <br /> \frac{\sqrt{r-2M}}{r^{\frac{3}{2}}}<br />
This isn't a complete set, it's only the set where the first subscript of the symbol is in the r direction.
This was done by first computing the Christoffel symbols in the Schwarzschild coordinates, and then doing a transformation to a locally diagonal metric g'ab via the transformations [EDIT] \Lambda^a{}_{a'} which is given by \Lambda^i{}_i = 1/\sqrt{|g_{ii}|}
Applying the geodesic deviation equation to these Chrsitoffel symbols gives some reasonable looking equations for a "flyby" with a velocity v and time dilation factor \gamma = 1/\sqrt{1-v^2} directed in the \hat \theta or \hat \phi directions
\frac{d^{2}r}{d\tau^2} = \gamma^2(\frac{v^2\sqrt{r-2M}}{r^{\frac{3}{2}}}-\frac{M}{r^{\frac{3}{2}}\sqrt{r-2M}})<br />
which compares favorably to the Newtonian result below
\frac{d^2r}{dt^2} = \frac{v^2}{r} - \frac{GM}{r^2}
(when allowances are made for the geometric units used in the first result).
I'm a little less sure about the result for movement in the r direction - I'm getting a factor of (1+v^2)/(1-v^2) that would be unity instead if one of the signs were different. However, I don't see any sign error.
Also \Gamma_{\hat r \hat t \hat t} gives the correct Newtonian value for the acceleration of gravity when r>>2M, and in addition agrees with the formula in Wald for the required proper acceleration of an observer near the event horizon of a black hole of 1/\sqrt{r-2M}
If we let unit vectors in the r, theta, and phi directions be \hat r \hat \theta \hat \phi, I get the results (using geometric units) that
<br /> \Gamma_{\hat r \hat r \hat r} = \Gamma_{\hat r \hat t \hat t} = -\frac {M} {r^{\frac{3}{2}}\sqrt{r-2M}}<br />
<br /> \Gamma_{\hat r \hat \theta \hat \theta} = \Gamma_{\hat r \hat \phi \hat <br /> \phi} = <br /> \frac{\sqrt{r-2M}}{r^{\frac{3}{2}}}<br />
This isn't a complete set, it's only the set where the first subscript of the symbol is in the r direction.
This was done by first computing the Christoffel symbols in the Schwarzschild coordinates, and then doing a transformation to a locally diagonal metric g'ab via the transformations [EDIT] \Lambda^a{}_{a'} which is given by \Lambda^i{}_i = 1/\sqrt{|g_{ii}|}
Applying the geodesic deviation equation to these Chrsitoffel symbols gives some reasonable looking equations for a "flyby" with a velocity v and time dilation factor \gamma = 1/\sqrt{1-v^2} directed in the \hat \theta or \hat \phi directions
\frac{d^{2}r}{d\tau^2} = \gamma^2(\frac{v^2\sqrt{r-2M}}{r^{\frac{3}{2}}}-\frac{M}{r^{\frac{3}{2}}\sqrt{r-2M}})<br />
which compares favorably to the Newtonian result below
\frac{d^2r}{dt^2} = \frac{v^2}{r} - \frac{GM}{r^2}
(when allowances are made for the geometric units used in the first result).
I'm a little less sure about the result for movement in the r direction - I'm getting a factor of (1+v^2)/(1-v^2) that would be unity instead if one of the signs were different. However, I don't see any sign error.
Also \Gamma_{\hat r \hat t \hat t} gives the correct Newtonian value for the acceleration of gravity when r>>2M, and in addition agrees with the formula in Wald for the required proper acceleration of an observer near the event horizon of a black hole of 1/\sqrt{r-2M}
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