Timelike Boundary
I was reading in a paper (see 1st paragraph of introduction section in http://arxiv.org/pdf/1510.00709.pdf) that in AdS space, waves can reach the boundary in finite time and, since said boundary is timelike, they can be reflected back into the bulk.
I want to try to understand this statement. First of all, it is shown in the final answer of the following thread (http://physics.stackexchange.com/questions/116813/ads-space-boundary-and-geodesics) that null rays take finite COORDINATE TIME to reach the AdS boundary but that this corresponds to infinite affine parameter. Q1: Does this mean an observer staying inside the bulk and measuring coordinate time could see the light ray go to infinity and back in finite time? But, if the light ray itself needs infinite affine parameter to reach r=infinity, how could it ever get back into the bulk? How are these two ideas consistent? I think I'm getting confused about coordinate time - I know there's no proper time for null rays but I still don't understand what's happening here.
Q2: Secondly, assuming our wave is travelling on a null geodesic, what does the boundary being timelike have to do with the ability to reflect the wave back into the bulk? What would have happened if it was a null boundary or a spacelike boundary? I have thought about this from the perspective of <PERSON> diagrams and, if I drew the boundary as null (at 45 degrees), then any null wave that hit it (coming in at 45 degrees), would be reflected and start travelling along the r=infinity surface, right? Similarly, if the boundary was spacelike, it would be drawn at >45 degrees to the vertical and I'm not sure any incoming 45 degree null wave would be able to end up back in the bulk. On the other hand, if it were timelike boundary and <45 degrees to vertical then I can see how null waves could continuously reflect off it. However, this "picture argument" needs tightening up - can anyone justify the physics? Why does the boundary being timelike allow for reflection?
Q3: How does one show the boundary is timelike? For this, I took AdS in <PERSON> coordinates and constructed the normal to surfaces of r=const. These are $n=g^{rr} \partial_r$ and so the components of this normal vector are $n^r=g^{rr}$. Then I compute the norm of this normal vector as $n^2=g_{\mu \nu} n^\mu n^\nu = g_{rr} g^{rr} g^{rr} = g^{rr} = \frac{r^2}{L^2} \rightarrow \infty$. 275
We see that the norm is always positive for surfaces r=const and in fact, it actually becomes infinite on the boundary. This means the normal to the boundary is spacelike and I assume this means the tangent to the boundary is timelike i.e. the boundary is a timelike hypersurface. Is this correct?
Q4: If the above argument is correct, then why do I get a weird result when I apply it to <PERSON> space? Here I write <PERSON> in spherical polar coordinates and again the normal to surfaces r=const is $n=g^{rr} \partial_r$ and so $n^2=g_{rr}g^{rr}g^{rr}=g^{rr}=1$ and this is always equal to 1 regardless of what the value of r is on the hypersurface. This means the boundary to Minkowski has a spacelike normal and, by the above reasoning, this boundary would again be timelike. But we know from the <PERSON> diagram that the boundary of Minkowski is null with a null normal vector so what has gone wrong? And indeed, assuming my reasoning is wrong, how does one correctly show AdS has a timelike boundary and Minkowski has a null boundary?
Q5: In <PERSON>, null waves take an infinite coordinate time to arrive at the boundary. Is this the reason there is no reflection back into the bulk or is it because the boundary itself is null? My argument using <PERSON> diagrams above would suggest that it's not possible for null boundaries even if the wave was able to arrive there in finite coordinate time, right?
Thanks very much for your help :) 474