Geodesics in $\text{AdS}_3$

Question

I'm having some trouble doing an easy computation with the $\text{AdS}$ space. I'm considering $\text{AdS}_3$ space with the Poincaré coordinates, so the metric reads

$$ds^2 = \frac{R^2}{z^2}(dz^2 - dt^2 + dx^2).$$

I want to compute the geodesics for a $t=\text{const.}$ slice, in order to obtein the holographic entanglement entropy for the region $x\in[-l/2,+l/2]$, as described in this paper (eq. 12 to 14).

So, I set $t = \text{const.}$ and I compute the geodesics equations:

$$\ddot{z} + \frac{1}{z}(-\dot{z}^2 + \dot{x}^2) = 0.$$

$$\ddot{x} - \frac{2}{z}\dot{z}\dot{x}=0.$$

As the paper says, the solution should be the semicircunference $x = \sqrt{(\frac{l}{2})^2-z^2}$, or written in parametric form:

$$x = - \frac{l}{2}\cos \pi\lambda$$

$$z = \frac{l}{2}\sin \pi\lambda$$

with $\lambda\in[0,1]$.

But if I substitute this solution into the geodesics equations I don't get they are satisfied. So, what do you suggest is my problem?

Answer 1

The affine geodesic equation (GE)

$$ {d^2 x^{\mu} \over d\lambda^2} + \Gamma^{\mu}_{\alpha\beta} {dx^{\alpha} \over d\lambda} {dx^{\beta} \over d\lambda} ~=~ 0\tag{1}$$

depends on the parametrization: The affine GE (1) holds when the parameter $\lambda$ is affinely related to the arc length $s=a\lambda+b$ of the geodesic.

This can e.g. be deduced from the fact that eq. (1) is not invariant under world-line reparametrizations $\lambda\to\tilde{\lambda}$. The GE for a generic parametrization contains an extra term proportional to the velocity: $$ {d^2 x^{\mu} \over d\lambda^2} + \Gamma^{\mu}_{\alpha\beta} {dx^\alpha \over d\lambda} {dx^\beta \over d\lambda} ~\propto~ {d x^{\mu} \over d\lambda}.\tag{2}$$