Christoffel symbol in polar coordinates

Question

I'm working through this problem in Problem Book in Relativity and Gravitation. The book shows the full solution. I'm trying to understand it all.

"For a 2-dimensional flat, Euclidean space described by polar coordinates $r$, $\theta$, assume that geodesics are the usual straight lines. Find the connection coefficients $\Gamma^\alpha _{\beta \gamma}$ using your knowledge of these geodesics, and the geodesic equation"

$$ \frac{d^2x^\mu}{ds^2} + \frac{dx^\alpha}{ds}\frac{dx^\beta}{ds}\Gamma^\mu _{\alpha \beta} = 0 $$

Using the equation of a line in polar coordinates, this can be parameterized in terms of $\theta$.

$$ R_{0} = r\cos(\theta - a) $$

$$ \frac{d^2x^\mu}{d\theta^2} - 2\ tan(\theta - a)\ \frac{dx^\mu}{d\theta} + \frac{dx^\alpha}{d\theta}\frac{dx^\beta}{d\theta}\Gamma^\mu _{\alpha \beta} = 0 $$

Then the text states to "consider the point $\theta = a$" where we get:

$$ \Gamma^\theta _{\theta \theta} = 0 $$

$$ \Gamma^r_{\theta \theta} = -r $$

But then the text states "Since $a$ and $R_0$ are arbitrary (this) is true in general."

This is what I don't understand. The point $\theta$ = $a$ refers to a very specific point on the line: the point closest to the origin. So we have found that $\Gamma^\theta _{\theta \theta} = 0$ and $\Gamma^r_{\theta \theta} = -r$ at the point closest to the origin. How does that generalize to all other points? For any point on a line one could to a change of coordinates so that that point was closest to the origin, but then the derived Christoffel symbol would be be for a different coordinate system.

Answer 1

I have no idea what they're talking about. You compute the Christoffel symbols from the parametrization. Indeed, $\Gamma^\theta_{\theta\theta}=0$. This has nothing to do with any curve in the surface.

EDIT: Now that I correctly conjectured what must have been in the text, there's no problem with what is written. You are told to assume that every line in the plane is a geodesic. Given any value $(r_0,\theta_0)$, choose $R_0 = r_0$ and $a=\theta_0$, and you've deduced what the author claims. But this tells you the values of the Christoffel symbols at an arbitrary point of the plane.