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I'm having some trouble doing an easy computation with the 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?

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you should check if this parametrizaition $\lambda$ is a linear function of the length of along the geodesic. otherwise, the geodesic equation will be modified –  mastrok Jul 11 at 8:56

1 Answer 1

up vote 4 down vote accepted

The geodesic equation (GE)

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

depends on the parametrization: The 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 reparametrization contains an extra term proportional to the velocity.

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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. –  David Pravos Jul 11 at 9:34

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