Stephen Webb: Out of This World (Notes)

Black Holes and entropy
An organism can be thought of in crude diagrammatic form as an open system with an input and output and a membrane. The net entropy as measured by the growth of the organism (membrane) and the waste (or directed action) output always increases relative to the input.
If this same diagram is used to represent a Black hole, then it looks very much like a perfectly efficient organism - one where all of the input entropy goes into growth in the membrane.

(It was later suggested that black holes are maximum-entropy objects, meaning that the maximum possible entropy of a region of space is the entropy of the largest black hole that can fit into it. This led to the holographic principle.)

The entropy of a black hole can be thought of as a grid of information bits written on its event horizon, where each bit corresponds to 4 Planck unit areas. Obviously, at the Planck length it would seem reasonable to assume that you need 1 bit to describe each Planck unit. In other words you need information to describe the memory registers themselves, as well as the data written to them - a problem.

If you threw bits into a pile - library books full of information, for example, the information would grow based on the size of the pile, the volume, or the cube of its radius. Forget the textual (Shannon) information, and consider the statistical physical entropy (Boltzmann) i.e. the bits of atomic level stuff that make the paper.

Since the volume increases relative to the area, the information (Boltzmann) contained within a large enough pile of books will be equal to the entropy of the area. At exactly this threshold, the pile would collapse under its own weight into a black hole and the 2nd Law would hold.

(Interestingly, Webb’s book, despite the ‘pile of books’ metaphor, it doesn’t talk about the idea that there might actually be something about Shannon entropy which would cause an observer to see a different limit, determined by the diff between the observed Shannon ‘bit size’ and the theoretic Boltzmann limit. Books aren’t perfectly efficient, the number of information bits per unit volume is less than the maximum, i.e the Boltzmann entropy.)

There are plenty of problems with black hole entropy, it puts a maximum limit to the amount of information contained in a volume, but sets that limit as being dependent on its area (a quarter of its area, in natural units). It means that the enclosed volume of a black hole cannot be ’solid’ or more accurately cannot even exists in its entirety, or it entropy as measured by the bits the constitute the volume itself, would contradict the 2nd Law.

There is one saving grace about the surface entropy problem. It makes sense in terms of a relative approach to information (where a bit is a measure of difference between two systems, rather than something which exists when not being measured). It implies that information may only exist at the surface of something, where it can be measured.

(Leonard Susskind and Nobel prizewinner Gerard ‘t Hooft have suggested that a black hole is a two dimensional object extant in three dimensional space. In addition, they believe these results may indicate a solution to the black hole information-loss paradox and that we live in a holographic world.)

****************

Maldacena’s conjecture, 1997: AdS/CFT correspondence. A mathematical validation of holographic universe concept. Using type IIB string theory in 10D with 5 dimensional AdS and a 5D compacted sphere at each point, satisfying Einstein equations and therefore creating a theory of gravity based on strings.

AdS = Anti-de Sitter spacetime. A solution to Einstein field equations where there is no matter in universe (modeled with two negligible mass particles), negative cosmological constant, causing exponential increase in contraction and a boundary at infinity, and a negatively curved space like an inverse sphere.

CFT = Conformal Field Theory, no preferred energy scale.

Boundary of 5D AdS is a 4D surface (just as boundary of a 3D volume is a 2D surface). Maldacena showed that the 5D AdS space was dual to an ordinary quantum theory of point particles without gravity (specifically SU(N) gauge theory with N=4 supersymmetry for maximally large N) residing on the 4D spacetime boundary (like a hologram of the full gravitational theory) and that this looked the same at all energy levels i.e. was a CFT.

The problem with AdS/CFT: we see a messy 4D universe with preferred energy levels (as suggested by non-conformal gauge theories such as QCD), it suggests an empty smooth conformal one.