Christoffel symbols in the Schwarzschild metric

[pervect]

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

$$\Gamma_{\hat r \hat r \hat r} = \Gamma_{\hat r \hat t \hat t} = -\frac {M} {r^{\frac{3}{2}}\sqrt{r-2M}}$$
$$\Gamma_{\hat r \hat \theta \hat \theta} = \Gamma_{\hat r \hat \phi \hat <br /> \phi} = <br /> \frac{\sqrt{r-2M}}{r^{\frac{3}{2}}}$$

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

$$\frac{d^{2}r}{d\tau^2} = \gamma^2(\frac{v^2\sqrt{r-2M}}{r^{\frac{3}{2}}}-\frac{M}{r^{\frac{3}{2}}\sqrt{r-2M}})$$

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 $$1/\sqrt{r-2M}$$

 

[R0man]

for a stationary observer, so that's reassuring as well.

Overall, I'm quite happy with these results and am excited to see how they can be applied in various scenarios. The Christoffel symbols in the Schwarzschild metric provide a useful tool for understanding the effects of gravity on objects moving through spacetime. From your results, it seems that the local gravitational accelerations in the r, theta, and phi directions are affected by both the mass of the object and the distance from the center of the gravitational field (r-2M). It's interesting to see how these values compare to the Newtonian equations and how they can be used to calculate the proper acceleration of an observer near a black hole.

The discrepancy in the result for movement in the r direction may require further investigation, but overall your results seem to align with known theories and formulas. It's exciting to see how these calculations can be applied to different scenarios, and I'm sure they will continue to provide valuable insights into the nature of gravity and spacetime. Keep up the great work!