Why Do Infinitely Heavy Particles Travel Like Massless Ones in AdS Space?

[adsquestion]

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?

 

[Ambitwistor]

Thank you for your interest in calculating the geodesics of AdS3 and for sharing your query regarding the physics behind it. I have looked into the analysis in the link you provided and I can offer some insights on the behavior of infinitely heavy particles in AdS space.

Firstly, it is important to note that in AdS space, the presence of a negative cosmological constant results in a repulsive force that acts on particles. This means that even infinitely heavy particles will experience acceleration towards the conformal boundary, albeit at a slower rate compared to lighter particles.

In the limit of infinite mass, the acceleration experienced by the particle becomes negligible and it appears as if the particle is traveling along a straight line. This is because the force acting on the particle is directly proportional to its mass, and as the mass increases to infinity, the force becomes infinitely small.

Furthermore, the equation t=-\rho i.e. traveling along the same straight line as a massless particle, only applies in the limit of infinite mass. For particles with finite mass, the equation t=\rho is still valid and they will experience a finite acceleration towards the conformal boundary.

In summary, the behavior of infinitely heavy particles in AdS space is a result of the interplay between the negative cosmological constant, the mass of the particle, and the resulting acceleration. I hope this helps to clarify your query.