# Anti de Sitter Geodesics

1. Nov 2, 2015

I'm interested in calculating the geodesics of AdS3. I've been following the analysis in this link (http://www.ncp.edu.pk/docs/snwm/Pervez_hoodbhoy_002_AdS_Space_Holog_Thesis.pdf).

I actually agree with all of the mathematics in the calculation and just have a query regarding the physics behind it.

In Section 2.5, null geodesics i.e. massless particles are shown to travel along straight lines with equation $$t=\rho$$ i.e. they travel to the conformal boundary of AdS and back in finite coordinate time.

In Section 2.4, timelike geodesics are shown to travel along sin curves with amplitude $$\sqrt{1-\frac{1}{k^2}}$$ where $$k$$ is the integral of motion associated with the timelike Killing vector $$\partial_t$$. In other words, $$k$$ is the Energy or mass of the particle.
This means that more massive particles travel along the more zig-zag geodesics, or, to put it another way, they get closer to the conformal boundary before being turned around'' by the infinite potential well. In fact, an infinitely heavy particle would obey $$\sin{t}=\sin{\rho} \Rightarrow t=-\rho$$ i.e. travel along the same straight line curve as a massless particle.

Whilst I accept the periodic motion of freely-falling timelike observers in AdS space, I don't understand why the infinitely heavy particles travel along the same straight line curves as massless particles. Naively, I would expect infinitely heavy particles (with $$k=\infty$$) to have amplitude zero i.e. to remain at $$\rho=0$$ and be able to resist the acceleration caused by having a negative cosmological constant.

What's going on?

2. Nov 7, 2015