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).(adsbygoogle = window.adsbygoogle || []).push({});

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 [tex]t=\rho[/tex] 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 [tex]\sqrt{1-\frac{1}{k^2}}[/tex] where [tex]k[/tex] is the integral of motion associated with the timelike Killing vector [tex]\partial_t[/tex]. In other words, [tex]k[/tex] 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 [tex]\sin{t}=\sin{\rho} \Rightarrow t=-\rho[/tex] 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 [tex]k=\infty[/tex]) to have amplitude zero i.e. to remain at [tex]\rho=0[/tex] and be able to resist the acceleration caused by having a negative cosmological constant.

What's going on?

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# Anti de Sitter Geodesics

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