# Geodesics in $\text{AdS}_3$

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?

$${d^2 x^{\mu} \over d\lambda^2} + \Gamma^{\mu}_{\alpha\beta} {dx^{\alpha} \over d\lambda} {dx^{\beta} \over d\lambda} ~=~ 0\tag{1}$$
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: $${d^2 x^{\mu} \over d\lambda^2} + \Gamma^{\mu}_{\alpha\beta} {dx^\alpha \over d\lambda} {dx^\beta \over d\lambda} ~\propto~ {d x^{\mu} \over d\lambda}.\tag{2}$$
• Thank you. I have it. The geodesics equation for a non-affine parametrization is $$\ddot{x}^\mu + \Gamma^{\mu}_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = f(\lambda)\dot{x}^\mu$$ where, $f(\lambda)$ is some function to be determined. The solution satisfies the GE with $f(\lambda) = -\pi cotan \pi\lambda$ for my parametrization. Jul 11, 2014 at 9:34