TITLE: Geodesics of a "diagonal" metric
QUESTION [5 upvotes]: Are there any relations that exist to simplify Christoffel symbols/connection coefficients for a diagonal metric which has the same function of the coordinates at each entry? In other words, I have a metric
$g = f(x_1,x_2,\cdots) \; \begin{pmatrix} 1 & 0 & \cdots \\ 0 & 1 &  & \\ \vdots &  & \ddots \end{pmatrix}$
And I want to calculate geodesics. I think they'll be straight lines, i.e. they will take the same shape as they would if the space was Euclidean, but they will be traversed with some varying speed. (That was originally my motivation for looking at this metric, as I have some curves that are 'straight' but not traveled at uniform speed, and so I introduced this metric hoping they would become geodesics).
The geodesic equation is
\begin{equation}
\frac{d^2x^{\lambda}}{ds^2} + \Gamma^{\lambda}_{\mu \nu} \frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}=0
\end{equation}
I'm hoping that the second term will factor into a constant vector $y^{\lambda}$ multiplied by a scalar function of the coordinates $Y(x_1,x_2,\cdots)$. But it isn't obvious to me if this happens. 
On a related note, has anyone ever looked at extending this machinery to spaces of infinite dimension?

REPLY [8 votes]: The previous answers are perfectly good, but let me explain briefly why you should not expect geodesics to be straight lines in the first place.
The conformal factor $f$ can be interpreted as an optic index; for example if you take $f$ to be $1$ on a half space and $2$ on its complement, geodesics through the separating hyperplane will correspond to the travel of light through an interface between two materials of different indexes. Then you see refraction (which for example explains why a straw in a glass of water looks broken). There is a well known formula with sines relating the angles of the geodesic before and after the interface, and you probably know a few other examples of this phenomenon.
Of course, the above example is not smooth, but the same kind of thing happen with smooth $f$ (you get continuous refraction in a material with varying optial index). A way to see this easily is to consider the case where $f$ is equal to $1$ except near a given non straight curve $\gamma$, and $f$ is very small at $\gamma$. Then the length (according to $g$) of $\gamma$ can be much smaller than the length of the line between its endpoints, which therefore cannot be a (minimizing) geodesic.