Scientifica/Black Hole Thermodynamics

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Black Hole Thermodynamics is a potentially very important subject for developing a theory of quantum gravity. In short, is a set of principles which extend thermodynamic concepts and laws, which appear to fail when applied to black holes at first glace, into the realm of general relativity.

Logical Foundations[edit]

Recall basic thermo logic[edit]

(1) We observe conservation of energy in non-dissipative systems after we define (i) potential/internal energy and (ii) work/kinetic energy: dU = dW

(2) We observe that work can be converted completely into "hotness" (something proportional to heat) at a constant exchange rate in dissipative systems. This suggest we define heat (in units of energy) such that for single material,

dU = dQ + dW.

(3) For a single material which is not a heat engine, dU=dQ+dW is almost a tautology. However, by allowing multiple materials with different heat capacities interact, dU=dQ+dW is non-trivial.

(4) Thermodynamically, we get the second law by starting with Lord Kelvin's postulate (can't turn heat purely into energy) or Clausius' postulate (heat flows from hot to cold) and showing that the quantity dS =dQ/T cannot decrease.

(4') Statistical-mechanically, we get the second law by counting states, assuming the fundamental assumption, and showing that S *almost* always increases.

Surface Gravity-Temperature Analogy[edit]

It can be derived based on the laws of classical GR that the surface gravity, k, is constant everywhere along the horizon of a *stationary* BH (Schwarzchild, Kerr, whatever). At first, this is not very interesting because many quantities (e.g. curvature, radius) are constant along the horizon of a BH due to the high symmetry of stationary BHs in 4 dimensions. However, in higher dimensions BHs can be much more complicated, so that things like curvature and radius are not constant along the horizon. Nevertheless, the surface gravity is *always* constant in arbitrary dimensions once the BH has settled down.

This weakly suggests that the surface gravity is a monotonic function of the temperature T ( :=dQ/dS ) in the true theory of quantum gravity.

Area-Entropy Analogy[edit]

It can be derived based on the laws of classical GR that, in a system with one or more BHs, the sum of the horizon areas never decreases. Note that the total mass is *not* conserved. In general, some energy will be radiated away in the form of gravity waves, so the final mass of a BH created by merging other BHs together can be less than the initial total mass.

This moderately suggests that the BH area A is proportional to the entropy S ( := ln g) in the true theory of quantum gravity.1 (Applied to a single BH, we would only have monotonicity. But since we can add BH's together, we get proportionality.)

Hawking Radiation[edit]

If you drive a motorized wheelchair, you can derive based on QFT in curved space that stationary BH's radiate like black bodies with temperature T = k/2*pi . This holds true even though the relationship between T and other BH parameters (q, M, J) is different for different types of BHs. Apparently, it's a pretty convincing derivation, although see [1] and references therein for objections.

This strongly suggests that the surface gravity and the temperature of a BH are related by

T = k/2*pi

in the true theory of quantum gravity.

Conservation of Energy Analogy[edit]

It can be derived based on the laws of classical GR that

dM = k dA / 8pi + w dJ + Vdq,

where

J=ang momentum added to BH,
w=ang velocity of BH,
V=external electric potential,
q=charge added to BH.

This is analogous to dU=dQ+dW. The identifications are

dU --> dM ,
dW --> w dJ + V dq ,
dQ --> k dA / 2pi .

The first two are not just analogies; in relativity, the mass of a system at rest is its energy, and wdJ+Vdq is precisely the work done on system, BH or no BH.

This moderately suggests that the

k dA / 2pi = dQ = T dS

which supports our suspicion that k and A are functions of T and S, respectively. Given the Hawking radiation relation, T = k/2*pi, we get

S = A/4

in the true theory of quantum gravity.

A bit more evidence[edit]

(1) Work cannot be extracted from isolated BH's, but work can be extracted by bringing two BH's together. (= Work cannot be extracted from isolated systems in thermo equilibrium, but but work can be extracted by bringing two systems in equilibrium together.) The analogy is imperfect because work can be extracted by combining two BH's at same temp, but usually one can't extract work from two systems at same temp. This imperfection may just be due to the fact that there is potential energy between separated BHs, but not between textbook thermo systems.

(2) The proportionality constant between A and S was estimated by Bekenstein before the discovery of Hawking radiation fixed it exactly. Basically, he considered assigning a minimum of one bit of entropy to an isolated particle and seeing how much area was added to a BH if you dropped the particle in from a distance of one Compton wavelength above the horizon. He argued that you couldn't have dropped it from any closer to the horizon (thereby reducing arbitrarily the energy--i.e. area--added to the BH for the given amount of entropy) because of the finite-width of the wavepacket. This is independent support (within an order of magnitude) for S=A/4.


Summary of the Historial Literature[edit]

J. D. Bekenstein, "Black Holes and Entropy", Phys. Rev. D 7, 2333 (1972) [2]

Bekenstein points out analogies between BH's and thermo:
(1) BH surface area (which is proportional to the "irreducible mass") never decreases. [ = Entropy never decreases]
(2) No energy may be extracted from an isolated Schwarzchild BH, but two SBH's may be brought together to release energy (through grav. radiation), but the final composite BH must have an irreducible mass at least as large as the orginal system (M_irr^2 = M_1^2 + M_2^2). [ = No energy can be extracted from an isolated system in equilibrium at some temperature, but energy can be extracted by bringing two systems of different temperature together]
(3) there is a BH analog of dE = TdS - PdV
(4) more, but I haven't finished reading

S.W. Hawking, "Particle creation by black holes", Communications in Mathematical Physics 43, 199 (1975) [3]

Hawking discovers Hawing radiation and derives a temperature for the BH of a given mass. This gives the exact proportionality constant between surface area and entropy. (I haven't read this yet.)

J. D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems", Phys. Rev. D 23, 287 (1980) [4]

Bekenstein uses BH's to establish upper bound on entropy.

R. D. Sorkin, "Toward a Proof of Entropy Increase in the Presence of Quantum Black Holes", Phys. Rev. Lett. 56, 1885 (1986) [5]

Sorkin makes stronger argument that second law of thermo is preserved with quantum BH's. (I haven't read this yet; it was mentioned in Bekenstein's popular article in Scientific American).

G. 't Hooft, "Dimensional Reduction in Quantum Gravity", arXiv:gr-qc/9310026 (1993) [6]

't Hooft argues that the correct theory of QG should have degrees of freedom that scale with the area, not the volume. I think this scaling feature is the precise way to state the holographic principle, not bs like "the universe really exists as a hologram on a far away surface" (as if that surface existed somewhere in a 3D space). Bekenstein himself states the holographic principle as "[the correct theory of QC] defined only on the 2-D boundary of [a] region completely describes the 3-D physics". I assume he means that the fundamental 2-surface (which exista a priori to the 3-surface of our experience) has the same topology as the boundary of the 3-surface of our experience.

L. Susskind, "The World as a Hologram", J. Math. Phys. 36, 6377 (1995) [7] Susskind connects this to string theory. Although I don't think you or I care much about the implications for string theory, this is considered an important paper, and I think it had some good discussion at the beginning independent of strings.

Review Articles[edit]

R. Busso, "The holographic principle", Rev. Mod. Phys. 74, 825 (2002) [8]