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I'm having trouble understanding something about Susskind's Holographic principle. Susskind speaks about the surface of the universe? In what sense does the universe have an outer surface? I'm a bit of an amateur when it comes to cosmology, so be gentle.

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There is a surface more or less at 13.7 billion light years from us which is where we see back to the big-bang (looking back in time, if you define "now" in a global way, although there is no reason that we should't define "now" by what we are seeing "now", i.e. along a past light cone). This surface is analogous to a black hole horizon, except it surrounds us instead of being localized in a region.

This thing is called the "cosmological horizon", and the general idea of the holographic principle suggests that everything inside the cosmological horizon is described by oscillations of this horizon. This is hard to make precise because the horizon has a finite area and growing, and so has a finite maximum entropy associated with it (which is growing), and this is paradoxical seeming, because it suggests that the Hilbert space for our universe is growing.

The number of states in a quantum mechanical system can't increase, so this leads many people to renounce the idea of string theory in our kind of universe, choosing instead to describe the dynamics in terms of the asymptotic future, where presumably the universe will vacuum decay to a supersymmetric state. This is one approach, another is to try and formulate a real theory with a finite Hilbert space. I think a possible third approach is to consider finite area horizons as somehow density-matrix like, so that they, unlike black holes, have fundamental decoherence. Nobody knows the answer, and this is the major unsolved problem of string theory today.

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  • $\begingroup$ Could you elucidate on "the number of states in a quantum mechanical system can't increase"? I thought that Hilbert space is infinite anyway, even for the hydrogen atom. infinite + or - or times anything is infinite. Not to forget bosons which can always appear out of energy. $\endgroup$ – anna v May 2 '12 at 4:25
  • $\begingroup$ Just thought I'd point out that the comoving radius of the observable universe is closer to 47 billion light years, instead of 13.7 billion light years. $\endgroup$ – kleingordon May 2 '12 at 4:34
  • $\begingroup$ @kleingordon: This is not useful for the size of the cosmological horizon, which is defined in the correct way using the "now" slice which goes back in time, in which the radius is 13.7 billion light years (a little less because of cosmological constant making the expansion of the cosmological horizon stop). The comoving radius is nonsense people use to feel superior to people who "mistakenly" use the age of the universe as a distance (which is correct). $\endgroup$ – Ron Maimon May 2 '12 at 6:43
  • $\begingroup$ @annav: Hilbert space is infinite when you have no upper bound on radius or energy, so particle in a box has an infinite hilbert space, as does H-atom (the infinity at n large is just the large radius infinity, there is a second infinity in the continuum states, which you can regulate in a box). The point is that a deSitter space is like a particle in a box with an upper bound on the energy (sort of, it's not like that because it is thermal and lossy, but it's like that in terms of state counting), and this gives a finite number of states, the exponential of the horizon area. $\endgroup$ – Ron Maimon May 2 '12 at 6:46
  • $\begingroup$ @RonMaimon: Don't people usually define the cosmological event horizon as the boundary of the events which the observer will ever be able to see? In fig1 here for example, it looks like it's at 16GLy radius. $\endgroup$ – twistor59 May 2 '12 at 7:11
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In AdS/CFT, which is the closest thing to a concrete realization of gravitational holography, space has a "boundary at infinity". This is hard to visualize, but it's similar to what happens in the Poincaré and Klein disc models of hyperbolic geometry, which were used by M.C. Escher in his "circle limit" woodcuts. These models cram an infinite amount of hyperbolic area into a finite disc, with the density increasing as you get near the circular boundary. The circular boundary can be seen as points at infinity which extend the hyperbolic plane in a natural way. Similarly, hyperbolic 3-space has a sphere at infinity.

Anti de Sitter space has a hypercylinder at infinity: a hypersphere times $\mathbb R$ (time). It's the field theory on that hypersphere that corresponds to gravity in the AdS "bulk". Because the hypersphere effectively has infinite radius, distances (with units of length) are meaningless, but you can still identify points by angles (latitude and longitude). Smaller wavelengths (in the angular sense) are associated with bulk features that are closer to the boundary in the disc model (hence smaller in the disc).

In de Sitter space (empty spacetime with a positive cosmological constant), there is no spatial infinity (only a past and future infinity), and nothing similar to AdS/CFT can work. But for any particular observer (worldline), there is a spherical cosmological horizon which behaves something like a black hole turned inside out. (For the real-world value of the cosmological constant, the radius of the horizon is about 16 billion light years.) This has led people to suggest that de Sitter quantum gravity has finitely many degrees of freedom (the Bekenstein-Hawking entropy of the horizon, around $10^{122}$ bits in the real world), and that there's a dual description of the world on the horizon. See hep-th/0010252.

Ron Maimon's answer seems to have some serious errors. The basic idea of a cosmological horizon as the "outer surface of the universe" is right, but there is no surface 13.7 billion light years from us. He says that this is the size of our past light cone back to the big bang, but that's a flat-spacetime value that's meaningless in the actual curved spacetime. It's like saying that if two people head off at right angles from the north pole of a sphere, and meet at the south pole after travelling $20000$ km, that they are $20000\sqrt2$ km away from each other. He also says that this horizon is expanding, but past light cones don't expand. It's possible that what he said makes sense in some way that I'm missing, but I think he was just confused about cosmology.

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