Holographic principle where it stops and why

Question

I just read "Black hole wars" by L. Susskind. Essentially he claims that a boundary contains all the information about the volume it encompasses. He proclaims this the holographic principle (HP).

My questions in regard to the holographic principle are:

1) Does the volume (3D) still exists or everything is 2D which for us (humans) appear as 3D (spatial dimensions only) ? Or is it just clever reductionism that 3D is real but we can described it with information on the boundary (like the entropy of a black hole).

2) If the holographic principle is a general law of nature, how about scaling. If I have a disk (2D) can I describe it with a line (1D) ? Where it stops ? Singularity ?

3) How the HP fits with 9 spatial dimensions in the M-theory (the author proudly claims that the string theory nicely proofs the HP) ?

4) Why the time is untouched by the holographic principle ?

Thanks !

Answer 1

1. The 3d volume (+ time) exists by its own. What holography tells you is that the fundamental description of what is going on with gravity in the volume is encoded in a 2d surface (+ time) where there is no gravity. In other words, there is a mathematical dictionary that allows us to translate what is going on in the gravity side from a theory living in an imaginary 2d surface (+ time). Nevertheless, the 3d volume is just the low energy behaviour of gravity, so how the volume would look at higher energies (at lower distances) may not be even a well posed question, because there might be no volume at all. Nevertheless, holography still would hold in that situation so we could use the 2d surface to try to see what is going on in the complete theory of gravity.

2. The most famous example of holography actually holds when the volume is 4d, and the surface is 3d. There are various other examples in other dimensions, such as when the volume is 2d and the surface is 1d (named AdS$_2/$CFT$_1$). I think there are not known examples when the volume is 1d, so that the surface had to be a point (zero dimensional).

3. To obtain explicit examples of holography (such as the original one that Maldacena published in 1997) it is useful to begin from string theory or M-theory and then go to lower dimensions (what is called a compactification). String and M-theory are thought to be theories that incorporate the complete theory of gravity, so they should naturally give rise to the holographic principle, and they actually do.

4. Time is untouched in the sense that if the theory in the surface had 2 times (or zero times) it would not make sense at all (it would allow closed timelike curves, which violates causality).

All this holographic ideas are only proven in a very particular kind of spacetime called AdS, which is like you put everything inside a box. How to extend this to flat space or even to other spacetimes is not clear, but we think it should be possible somehow.