Entropy

7.3 Entropy

Let us now summarize the ideas behind counting of surface microstates that leads to the expression of entropy. To incorporate dynamics in this canonical approach, we have to first construct physical states by imposing quantum Einstein equations (i.e., quantum constraints). While the procedure is technically quite involved, the result is simple to state: What matters is only the number of punctures and not their locations. To calculate entropy, then, one constructs a micro-canonical ensemble as follows. Fix the number $n$ of punctures and allow only those (non-zero) spin-labels $jI$ and charge labels $α qI$ on the polymer excitations which endow the horizon with a total area in an interval $(a0 − ε,a0 + ε)$ and charges in an interval $(Q α0 − εα, Qα0 + εα)$ . (Here $I = 1,2,...,n$ and $ε$ and $εα$ are suitably small. Their precise values will not affect the leading contribution to entropy.) We denote by $ℋBH n$ the sub-space of $ℋ = ℋ ⊗ ℋ V S$ in which the volume states $Ψ V$ are chosen with the above restrictions on $jI$ and $α qI$ , and the total state $Ψ$ satisfies the quantum horizon boundary condition as well as quantum Einstein equations. Then the desired micro-canonical ensemble consists of states in $ℋBH = ⊕n ℋBHn$ . Note that, because there is no contribution to the symplectic structure from matter terms, surface states in $ℋBH$ refer only to the gravitational sector.

The next step is to calculate the entropy of this quantum, micro-canonical ensemble. Note first that what matters are only the surface states. For, the ‘bulk-part’ describes, e.g., states of gravitational radiation and matter fields far away from $Δ$ and are irrelevant for the entropy $SΔ$ of the WIH. Heuristically, the idea then is to ‘trace over’ the bulk states, construct a density matrix $ρ BH$ describing a maximum-entropy mixture of surface states and calculate $tr ρBH lnρBH$ . As is usual in entropy calculations, this translates to the evaluation of the dimension $𝒩$ of a well-defined sub-space $ℋBHS$ of the surface Hilbert space, namely the linear span of those surface states which occur in $ℋBH$ . Entropy $SΔ$ is given by $ln𝒩$ .

A detailed calculation [83 , 149 ] leads to the following expression of entropy:

where $𝒪 (ln(a0∕ℓ2Pl))$ is a term which, when divided by $a0∕ℓ2Pl$ , tends to zero as $a0∕ℓ2p$ tends to infinity. Thus, for large black holes, the leading term in the expression of entropy is proportional to area. This is a non-trivial result. For example if, as in the early treatments [171 , 35 , 162 , 163 , 135 , 136

] one ignores the horizon boundary conditions and the resulting Chern–Simons term in the symplectic structure, one would find that $S Δ$ is proportional to $√ --- a0$ . However, the theory does not have a unique prediction because the numerical coefficient depends on the value of the Barbero–Immirzi parameter $γ$ . The appearance of $γ$ can be traced back directly to the fact that, in the $γ$ -sector of the theory, the area eigenvalues are proportional to $γ$ .

One adopts a ‘phenomenological’ viewpoint to fix this ambiguity. In the infinite dimensional space of geometries admitting $Δ$ as their inner boundary, one can fix one space-time, say the Schwarzschild space-time with mass $M ≫ M 0 Pl$ , (or, the de Sitter space-time with the cosmological constant $2 Λ0 ≪ 1∕ ℓPl$ , or, …). For agreement with semi-classical considerations in these cases, the leading contribution to entropy should be given by the Hawking–Bekenstein formula (87 ). This can happen only in the sector $γ = γ0$ . The quantum theory is now completely determined through this single constraint. We can go ahead and calculate the entropy of any other type I WIH in this theory. The result is again $S = S Δ BH$ . Furthermore, in this $γ$ -sector, the statistical mechanical temperature of any type I WIH is given by Hawking’s semi-classical value $κ ℏ∕(2π)$ [28 , 136 ]. Thus, we can do one thought experiment – observe the temperature of one large black hole from far away – to eliminate the Barbero–Immirzi ambiguity and fix the theory. This theory then predicts the correct entropy and temperature for all WIHs with $a0 ≫ ℓ2$ , irrespective of other parameters such as the values of the electric or dilatonic charges or the cosmological constant. An added bonus comes from the fact that the isolated horizon framework naturally incorporates not only black hole horizons but also the cosmological ones for which thermodynamical considerations are also known to apply [99 ]. The quantum entropy calculation is able to handle both these horizons in a single stroke, again for the same value $γ = γ 0$ of the Barbero–Immirzi parameter. In this sense, the prediction is robust.

Finally, these results have been subjected to further robustness tests. The first comes from non-minimal couplings. Recall from Section 6 that in presence of a scalar field which is non-minimally coupled to gravity, the first law is modified [124 , 183 , 123 ]. The modification suggests that the Hawking–Bekenstein formula $2 SBH = a0∕ (4ℓPl)$ is no longer valid. If the non-minimal coupling is dictated by the action

where $R$ is the scalar curvature of the metric $g ab$ and $V$ is a potential for the scalar field, then the entropy should be given by [21 ]

An immediate question arises: Can the calculation be extended to this qualitatively new situation? At first, this seems very difficult within an approach based on quantum geometry, such as the one described above, because the relation is non-geometric. However, the answer was shown to be in the affirmative [19

] for type I horizons. It turns out that the metric $¯qab$ on $M$ is no longer coded in the gravitational momentum $P ai$ alone but depends also on the scalar field. This changes the surface term $Ω S$ in the symplectic structure (90

) as well as the horizon boundary condition (89

) just in the right way for the analysis to go through. Furthermore, state counting now leads to the desired expression (96

) precisely for the same value $γ = γ0$ of the Barbero–Immirzi parameter [19 ].

Next, one can consider type II horizons which can be distorted and rotating. In this case, all the (gravitational, electro-magnetic, and scalar field) multipoles are required as macroscopic parameters to fix the system of interest. Therefore, now the appropriate ensemble is determined by fixing all these multipoles to lie in a small range around given values. This ensemble can be constructed by first introducing multipole moment operators and then restricting the quantum states to lie in the subspace of the Hilbert space spanned by their eigenvectors with eigenvalues in the given intervals. Again recent work shows that the state counting yields the Hawking–Bekenstein formula (87 ) for minimally coupled matter and its modification (96 ) for non-minimally coupled scalar field, for the same value $γ = γ0$ of the Barbero–Immirzi parameter [8 , 24 ].

To summarize, the isolated horizon framework serves as a natural point of departure for a statistical mechanical calculation of black hole entropy based on quantum geometry. How does this detailed analysis compare with the ‘It from Bit’ scenario [187 ] with which we began? First, the quantum horizon boundary conditions play a key role in the construction of a consistent quantum theory of the horizon geometry. Thus, unlike in the ‘It from Bit’ scenario, the calculation pertains only to those 2-spheres $S$ which are cross-sections of a WIH. One can indeed divide the horizon into elementary cells as envisaged by Wheeler: Each cell contains a single puncture. However, the area of these cells is not fixed but is dictated by the $j$ -label at the puncture. Furthermore, there are not just 2 but rather $2j + 1$ states associated with each cell. Thus, the complete theory is much more subtle than that envisaged in the ‘It from Bit’ scenario.