Question

Definitions of Thermodynamics and Holography.

There are many differences between the laws of thermodynamics and the laws of black hole thermodynamics (BHT):

Zeroth Law: In thermodynamics, the Zeroth Law establishes the notion of thermal equilibrium and defines temperature. It states that if objects A and B are in thermal equilibrium, and B and C are in thermal equilibrium, then A and C are also in thermal equilibrium. This allows for the definition of an empirical temperature scale. In BHT, there is no notion of different black holes being in equilibrium with each other. The statement that surface gravity is constant over the event horizon is more analogous to a consequence of the Zeroth Law, not the law itself.

First Law: The thermodynamic First Law defines internal energy and relates it to heat and work. In BHT, the mass of the black hole plays the role of internal energy, but there is no clear notion of heat or work. The First Law relates changes in mass to changes in area and angular momentum, but the physical meaning is unclear.

Second Law: In thermodynamics, the Second Law states that entropy increases in any spontaneous process. In BHT, the area theorem states that the area of the event horizon never decreases, which is analogous but conceptually different than entropy increase. There is also no underlying statistical mechanical explanation for why black hole area should increase.

Third Law: The thermodynamic Third Law says entropy approaches zero as temperature approaches absolute zero. There is no parallel statement in BHT, and surface gravity can approach zero independently of area.

E.T. Jaynes said the word entropy has been very abused and documented six different types:

SE
- Experimental entropy of Clausius, Gibbs, and Lewis, defined based on macroscopic laboratory measurements.
SI
- Information entropy, defined as −∑pilog(pi)
.
S(A)
- Maximum of SI
subject to constraints A1...An
.
SB
- Boltzmann entropy, defined with single-particle distribution function.
SG
- Gibbs entropy, defined in terms of N-particle distribution function.
SBEP
- Entropy of Boltzmann, Einstein and Planck, defined as klogW
.
Is Bekenstein-Hawking entropy another?

Question: What might this imply about holography and the AdS/CFT correspondence?

(Aside: Frankly, I never really understood what was so profound about holography, given that the fundamental field equations are second order. Many ordinary boundary value problems involve data from a dimension lower. The holographic duality between quantum gravity theories on anti-de Sitter space and conformal field theories on the boundary is mathematically elegant but whether our universe exhibits anti-de Sitter geometry is another matter.)

Answer 1

The Zeroth Law of Thermodynamics states that systems in thermal equilibrium have the same temperature. The First Law relates internal energy to heat and work, but in black hole thermodynamics, there is no clear notion of heat or work. The Second Law states that entropy increases in spontaneous processes, whereas in black hole thermodynamics, the area of the event horizon never decreases.

Step-by-step explanation:

The Zeroth Law of Thermodynamics states that when two systems are in thermal equilibrium with each other, and one of them is in thermal equilibrium with a third system, then the two systems are also in thermal equilibrium with each other.

The First Law of Thermodynamics relates changes in internal energy to heat and work. However, in black hole thermodynamics, there is no clear notion of heat or work, and the mass of the black hole plays the role of internal energy.

The Second Law of Thermodynamics states that entropy increases in any spontaneous process. In black hole thermodynamics, the area theorem states that the area of the event horizon never decreases, which is conceptually similar to entropy increase but has no underlying statistical mechanical explanation.