Transatlantic Black Hole Entropy
On a transatlantic flight once more, I get started on the book "An introduction to black holes, information and the string theory revolution" by Leonard Susskind and James Lindesay, one of several cool books World Scientific put out recently.
The text is based on lectures given by the first author. It falls into a much needed category of books, which are above the level of popular science for the large public,
but are still accessible to a large audience of scientifically trained readers. In this book one can finally read a beautifully presented account of black hole entropy.
The main points stressed in the book are "black hole complementarity" that accounts for an extreme red shift between freely falling observers and observers stationed at a distance from the horizon, "infrared/ultraviolet connection" by which one cannot probe arbitrarily small lengths by arbitrarily large energies as beyond the Planck energies the events one wants to observe would be hidden by a black hole horizon, and the "holography principle" for counting microstates of systems involving gravity.
The first half of the book is a must read: as a warm up one gets a brief account of coordinate systems describing near horizon geometry of black holes, first in the simplest case of spherically symmetric, static, uncharged black holes (i.e. those described by a Schwarzschild geometry) and later in the case of electrically charged black holes (Reissner-Nordstrom geometry). This part includes an introduction to the formalism of Penrose diagrams of spacetime. The first chapter also contains a very nice discussion of the equivalence principle and the complementary viewpoints of
static fiducial observers at a distance from the horizon and freely falling observers that cross the black hole horizon. This is followed by a detailed discussion of quantum fields in a background Rindler metric describing the near horizon geometry.
This part contains the clearest discussion of entanglement, von Neumann entropy (entropy of entanglement), density matrix and the thermodynamic properties of black holes I have come across so far. One sees in particular how quantum field theory
seems to overcount the number of states of the system and generate an ultraviolet divergence, which predicts an infinite entropy at the horizon, in contrast with the
Bekenstein-Hawking entropy, which is finite and proportional to the area of the horizon. The latter can be derived from the fact that, to a distant observer, the black hole appears to have a temperature (the vacuum appears as a thermal ensemble
with a density matrix of Maxwell-Boltzmann type, see the discussion in sections 3.4, 4.1, and chapter 5). Thus, in this book one gets (finally!) a very readable but mathematically precise account of all the usual things one reads about in popular
books: evaporating black holes, extremal black holes, loss of quantum coherence, naked singularities, etc. The book is centered on the apparent paradox caused by
comparing the viewpoints of the two different class of observers near black hole horizons. Three main physical principles are involved: information conservation, the
equivalence principle in the theory of gravitation, and the "no clone principle" in quantum physics. The first descends from Liouville's theorem on the volume conservation in phase space and can be phrased in terms of the entanglement
and thermodynamic entropies of the system. At this point the book gives a very nice discussion of the difference between the "fine grain" or entanglement entropy and the "coarse grain" or thermodynamic entropy. The second principle is the usual
local equivalence between gravitational fields and accelerated frames (a freely falling observer only experiences the effect of gravity through the curvature, not through the components of the metric tensor). The third principle is a very simple
consequence of the fact that quantum mechanics is a linear theory and this is not compatible with the existence of an apparatus that can duplicate a given system (just because the square of (a+b) is not the sum of the squares of a and b).
Hawking's paradox of information loss in black holes formation and evaporation is explained in terms of the unitary S matrix that governs the evolution of an initial state and the resulting density matrix that describes the system as it will be seen by an observer outside the black hole. This will not be in a pure state, due to the entanglement with the part of the system that falls into the singularity. The paradox arises from the fact that the purity of the state will not be restored even when the black hole has evaporated, creating a loss of information. This part of the
book also contains a discussion of baryon number violations and a careful analysis of apparent contradictions about exchange of information across the horizon (sections 9.2 and 9.3).
The second part of the book concentrates on the holography principle.
This is a counting principle that produces an estimate for the number of microstates of a system in a region of a certain volume in terms of the area of the boundary of the region. It originates from the fact that, while the number of states of a system described by a quantum field theory typically produce an entropy proportional to the volume, when gravity is involved, the typical for of the entropy estimate is
that of the Bekenstein-Hawking case, which is proportional to the area of the horizon. One produces a holographic image on the boundary by considering light rays on fixed light-like surfaces. This yields an entropy density on the holographic image which can be estimated using the focusing theorem of general relativity (light rays bend around regions of higher gravity). This estimate has cosmological implications. In the Friedman-Robertson-Walker geometry one can see that the requirement that this entropy estimate is not violated implies a lower bound on the expansion rate. The holography principle can be formulated in different geometries. The most interesting cases are de Sitter space, which according to observational data appears to be the geometry most closely related to the observed universe, and anti de Sitter space, which is the geometry in which the holography principle has the nicest mathematical structure. The book gives a review of de Sitter geometry in section 11.5, including how to encode in it the initial phase of inflation. Anti de Sitter geometry is
described at length, especially in the form in which the bulk space is the 10-dimensional product AdS(5) X S(5) where the 5-dimensional sphere S(5) is shrunk to zero at the boundary (hence the boundary can be treated as a 3+1 dimensional geometry rather than in 8+1 dimensions). In this case the holography principle becomes part of the AdS/CFT correspondence, between gravity (in stringy formulation) in
AdS(5)XS(5) and SU(N) super-symmetric Yang-Mills theory on the boundary. Back to black hole geometry, the authors explain that a Schwarzschild black hole in a thermal bath is an unstable object, due to the negative specific heat, which provokes a runaway reaction upon exchanging (either absorbing or emitting) energy with the thermal bath, it is possible to treat the black hole as a stable object if it is immersed not in a thermal bath but in a box that has only a finite environmental heat bath. The AdS space can serve the role of such "box", due to the holography principle: the holographic description of an AdS black hole is a black body thermal gas on the boundary hologram.
The third part of the book presents the string theory approach to the thermodynamics of black holes. In particular, this produces a revision of the principle of locality (which is the source of trouble in Hawking's loss of information paradox).
Black hole complementarity and the extreme red shift between free falling and fiducial observers seems to require modifying the contributions of high energy states beyond what usual cutoffs and other regularization schemes do in quantum field theory. An observer outside the black hole will see the string diffusing over the horizon, while to a free falling observer the string would appear as remaning
of a fixed size. Black holes are non-perturbative effects in a theory of quantum gravity. This requires considering the effects of string interactions in an essential way. The authors discuss longitudinal string motion near the horizon, in D=d+1 dimensional Schwarzschild geometries and entropy estimates.
Believe me, the book is a page turner (and has great pictures too).
It was just the right length to provide full time in-flight entertainment across the Atlantic. If anything, I find the bibliography a bit too short. It could be
expanded to the benefit of those who would like to continue with further readings on the subject.