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I'm trying to understand why timelike geodesics in Anti de-Sitter space are plotted as sinusoidal waves on a Penrose diagram (a nice example of the Penrose diagram for AdS is given in Figure 2.3 of this thesis: http://www.nbi.dk/~obers/MSc_PhD_files/MortenHolm_Christensen_MSc.pdf).
Bearing in mind that the Penrose diagram shows conformally compactified AdS in order to draw points at infinity, we need to be clear whether we are drawing the timelike geodesics of a) the UNCOMPACTIFIED AdS metric or b) the COMPACTIFIED AdS metric.
Considering each in turn:
a) UNCOMPACTIFIED AdS3 metric can be written in global coordinates as [tex]ds^2=L^2(-\cosh^2{\mu} dt^2 + d \mu^2 + \sinh^2{\mu} d \theta^2)[/tex]
with [tex]t \in [-\pi,\pi), \theta \in [0,2 \pi), \mu \geq 0[/tex].
We can then substitute ##\sinh{\mu}=\tan{\rho}## to bring this to the form
[tex]ds^2=L^2(-\sec^2{\rho} dt^2 + \sec^2{\rho} d \rho^2 + \tan^2{\rho} d \theta^2)[/tex] with [tex]\rho \in [0,\pi/2][/tex]
Now let's suppose that we take a timelike observer (##ds^2=-1##) who is moving radially (##d \theta=0##). Their equation of motion will take the form
[tex]-1=L^2(-\sec^2{\rho} \dot{t}^2 +\sec^2{\rho} \dot{\rho}^2)[/tex].
We know that time translation invariance (stationary metric) gives rise to a conserved energy given by ##E=-g_{tt} \dot{t} = L^2 \sec^2{\rho} \dot{t} \Rightarrow \dot{t} = \frac{E}{L^2 \sec^2{\rho}} = \frac{E \cos^2{\rho}}{L^2}##
And if we substitute this into the above equation of motion, we can rearrange to get [tex]-1=-\frac{E^2 \cos^2{\rho}}{L^2} + \frac{\dot{\rho}^2 L^2}{\cos^2{\rho}} \Rightarrow \frac{\dot{\rho}^2 L^2}{\cos^2{\rho}} = \frac{E^2 \cos^2{\rho}}{L^2}-1[/tex].
At this point we note that since we want to plot ##\rho## against coordinate time not proper time on the Penrose diagram, we use ##\dot{\rho} = \frac{d \rho}{d t} \dot{t} = \frac{d \rho}{dt} \frac{E \cos^2{\rho}}{L^2}## to find ##\frac{d \rho}{dt} = \pm \sqrt{1- \frac{L^2}{E^2 \cos^2{\rho}}}## and this can be integrated up (by substitution) to obtain:
[tex]\sin{\rho(t)} = \sqrt{1 - \frac{L^2}{E^2}} \sin{(t + t_0)}[/tex]
This gives the sinusoidal behaviour that we see in the standard AdS Penrose diagram and it's also interesting to note that the greater the energy of the particle, the greater the amplitude of the trajectory and the closer the particle gets to infinity - this looks great and has all the properties we want BUT the problem is that we calculated it using the UNCOMPACTIFIED metric.
b) So let's see what happens when we use the geodesics of the COMPACTIFIED metric (recall the Penrose diagram requires using the COMPACTIFIED metric in order to be able to plot points at infinity at a finite point).
If we start with the UNCOMPACTIFIED metric in global coordinates [tex]ds^2=L^2(-\sec^2{\rho} dt^2 + \sec^2{\rho} d \rho^2 + \tan^2{\rho} d \theta^2)[/tex], we can multiply by the conformal factor ##\Omega^2=\cos^2{\rho}## to define the COMPACTIFIED metric [tex]\tilde{ds}^2 = \Omega^2 ds^2 = L^2(-dt^2 + d \rho^2 + \sin^2{\rho} d \theta^2)[/tex]
Note that points at infinity of the original metric (##\mu=\infty##) now appear at finite position ##\rho=\frac{\pi}{2}## in the COMPACTIFIED metric which is what we need in order to plot the Penrose diagram). But what do the radial (##d \theta=0##), timelike (##ds^2=-1##) geodesics look like. Well their equation of motion looks like
[tex] -1=L^2(-\dot{t}^2 + \dot{\rho}^2)[/tex]
As before this is a stationary metric so energy is conserved and given by ##E=-g_{tt} \dot{t} = L^2 \dot{t} \Rightarrow \dot{t} = \frac{E}{L^2}##
If we substitute this in we get [tex] -1=-\frac{E^2}{L^2} + L^2 \dot{\rho}^2 \Rightarrow \dot{\rho}^2 = \frac{1}{L^4} ( E^2 - L^2) \Rightarrow \dot{\rho} = \pm \frac{1}{L^2} \sqrt{E^2-L^2}[/tex]
Again, in order to plot ##\rho## against coordinate time rather than proper time, we need to make use of ##\dot{\rho} = \frac{d \rho}{dt} \dot{t} = \frac{d \rho}{dt} \frac{E}{L^2}## which allows us to write
##\frac{d \rho}{d t} = \pm \sqrt{1-\frac{L^2}{E^2}}## which integrates up directly to give
[tex] \rho(t) = \sqrt{1-\frac{L^2}{E^2}} t + t_0[/tex]
Disappointingly, whilst this does have the property that increasing energy increases the amplitude of the geodesics, it does not have the sinusoidal behaviour that we see drawn on AdS Penrose diagrams and, furthermore, it appears that as time increases, the radial coordinate increases indefinitely i.e. ##\rho \rightarrow \infty## but this is not allowed since we know ##\rho## is bounded by ##\frac{\pi}{2}## which represents spatial infinity for AdS space.
QUESTION: Clearly what we really want to do is plot the geodesics of UNCOMPACIFIED AdS space on the Penrose diagram of COMPACTIFIED AdS space but this makes absolutely no sense to me and I can see absolutely no justification for doing this and yet authors continually do this in books an papers etc without giving any reason. Is there any explanation for this? Or is there perhaps a more general reason that I am missing to do with plotting of timelike geodesics in general spacetimes (does something happen to them under conformal compactification that alters the shape of their trajectory and thus means we need to use the information from the uncompactified metric)?
The best "reason" I've managed to come up with is that everything COULD be ok because I calculated the geodesics of the uncompactified metric using the ##\rho## coordinate which had finite range (same range as Penrose diagram) rather than the ##\mu## coordinate which has infinite range? BUT I didn't explicitly compactify in calculation a) and if I actually do the compactification I end up with straight line geodesics as in calculation b) and I'm really unsatisfied by that!Thanks very much for any help!
Bearing in mind that the Penrose diagram shows conformally compactified AdS in order to draw points at infinity, we need to be clear whether we are drawing the timelike geodesics of a) the UNCOMPACTIFIED AdS metric or b) the COMPACTIFIED AdS metric.
Considering each in turn:
a) UNCOMPACTIFIED AdS3 metric can be written in global coordinates as [tex]ds^2=L^2(-\cosh^2{\mu} dt^2 + d \mu^2 + \sinh^2{\mu} d \theta^2)[/tex]
with [tex]t \in [-\pi,\pi), \theta \in [0,2 \pi), \mu \geq 0[/tex].
We can then substitute ##\sinh{\mu}=\tan{\rho}## to bring this to the form
[tex]ds^2=L^2(-\sec^2{\rho} dt^2 + \sec^2{\rho} d \rho^2 + \tan^2{\rho} d \theta^2)[/tex] with [tex]\rho \in [0,\pi/2][/tex]
Now let's suppose that we take a timelike observer (##ds^2=-1##) who is moving radially (##d \theta=0##). Their equation of motion will take the form
[tex]-1=L^2(-\sec^2{\rho} \dot{t}^2 +\sec^2{\rho} \dot{\rho}^2)[/tex].
We know that time translation invariance (stationary metric) gives rise to a conserved energy given by ##E=-g_{tt} \dot{t} = L^2 \sec^2{\rho} \dot{t} \Rightarrow \dot{t} = \frac{E}{L^2 \sec^2{\rho}} = \frac{E \cos^2{\rho}}{L^2}##
And if we substitute this into the above equation of motion, we can rearrange to get [tex]-1=-\frac{E^2 \cos^2{\rho}}{L^2} + \frac{\dot{\rho}^2 L^2}{\cos^2{\rho}} \Rightarrow \frac{\dot{\rho}^2 L^2}{\cos^2{\rho}} = \frac{E^2 \cos^2{\rho}}{L^2}-1[/tex].
At this point we note that since we want to plot ##\rho## against coordinate time not proper time on the Penrose diagram, we use ##\dot{\rho} = \frac{d \rho}{d t} \dot{t} = \frac{d \rho}{dt} \frac{E \cos^2{\rho}}{L^2}## to find ##\frac{d \rho}{dt} = \pm \sqrt{1- \frac{L^2}{E^2 \cos^2{\rho}}}## and this can be integrated up (by substitution) to obtain:
[tex]\sin{\rho(t)} = \sqrt{1 - \frac{L^2}{E^2}} \sin{(t + t_0)}[/tex]
This gives the sinusoidal behaviour that we see in the standard AdS Penrose diagram and it's also interesting to note that the greater the energy of the particle, the greater the amplitude of the trajectory and the closer the particle gets to infinity - this looks great and has all the properties we want BUT the problem is that we calculated it using the UNCOMPACTIFIED metric.
b) So let's see what happens when we use the geodesics of the COMPACTIFIED metric (recall the Penrose diagram requires using the COMPACTIFIED metric in order to be able to plot points at infinity at a finite point).
If we start with the UNCOMPACTIFIED metric in global coordinates [tex]ds^2=L^2(-\sec^2{\rho} dt^2 + \sec^2{\rho} d \rho^2 + \tan^2{\rho} d \theta^2)[/tex], we can multiply by the conformal factor ##\Omega^2=\cos^2{\rho}## to define the COMPACTIFIED metric [tex]\tilde{ds}^2 = \Omega^2 ds^2 = L^2(-dt^2 + d \rho^2 + \sin^2{\rho} d \theta^2)[/tex]
Note that points at infinity of the original metric (##\mu=\infty##) now appear at finite position ##\rho=\frac{\pi}{2}## in the COMPACTIFIED metric which is what we need in order to plot the Penrose diagram). But what do the radial (##d \theta=0##), timelike (##ds^2=-1##) geodesics look like. Well their equation of motion looks like
[tex] -1=L^2(-\dot{t}^2 + \dot{\rho}^2)[/tex]
As before this is a stationary metric so energy is conserved and given by ##E=-g_{tt} \dot{t} = L^2 \dot{t} \Rightarrow \dot{t} = \frac{E}{L^2}##
If we substitute this in we get [tex] -1=-\frac{E^2}{L^2} + L^2 \dot{\rho}^2 \Rightarrow \dot{\rho}^2 = \frac{1}{L^4} ( E^2 - L^2) \Rightarrow \dot{\rho} = \pm \frac{1}{L^2} \sqrt{E^2-L^2}[/tex]
Again, in order to plot ##\rho## against coordinate time rather than proper time, we need to make use of ##\dot{\rho} = \frac{d \rho}{dt} \dot{t} = \frac{d \rho}{dt} \frac{E}{L^2}## which allows us to write
##\frac{d \rho}{d t} = \pm \sqrt{1-\frac{L^2}{E^2}}## which integrates up directly to give
[tex] \rho(t) = \sqrt{1-\frac{L^2}{E^2}} t + t_0[/tex]
Disappointingly, whilst this does have the property that increasing energy increases the amplitude of the geodesics, it does not have the sinusoidal behaviour that we see drawn on AdS Penrose diagrams and, furthermore, it appears that as time increases, the radial coordinate increases indefinitely i.e. ##\rho \rightarrow \infty## but this is not allowed since we know ##\rho## is bounded by ##\frac{\pi}{2}## which represents spatial infinity for AdS space.
QUESTION: Clearly what we really want to do is plot the geodesics of UNCOMPACIFIED AdS space on the Penrose diagram of COMPACTIFIED AdS space but this makes absolutely no sense to me and I can see absolutely no justification for doing this and yet authors continually do this in books an papers etc without giving any reason. Is there any explanation for this? Or is there perhaps a more general reason that I am missing to do with plotting of timelike geodesics in general spacetimes (does something happen to them under conformal compactification that alters the shape of their trajectory and thus means we need to use the information from the uncompactified metric)?
The best "reason" I've managed to come up with is that everything COULD be ok because I calculated the geodesics of the uncompactified metric using the ##\rho## coordinate which had finite range (same range as Penrose diagram) rather than the ##\mu## coordinate which has infinite range? BUT I didn't explicitly compactify in calculation a) and if I actually do the compactification I end up with straight line geodesics as in calculation b) and I'm really unsatisfied by that!Thanks very much for any help!
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