Statistics, holography, and black hole entropy in loop quantum gravity

In loop quantum gravity the quantum states of a black hole horizon consist of pointlike discrete quantum geometry excitations (or punctures) labeled by spin $j$. The excitations possibly carry other internal degrees of freedom, and the associated quantum states are eigenstates of the area $A$ operator. The appropriately scaled area operator $A/(8\pi \ell )$ can also be interpreted as the physical Hamiltonian associated with the quasilocal stationary observers located at a small distance $\ell$ from the horizon. Thus, the local energy is entirely accounted for by the geometric operator $A$. Assuming that: Close to the horizon the quantum state has a regular energy momentum tensor and hence the local temperature measured by stationary observers is the Unruh temperature. Degeneracy of matter states is exponential with the area $\exp (\lambda A/{\ell }_p^2)$, which is supported by the well-established results of QFT in curved spacetimes, which do not determine $\lambda$ but assert an exponential behavior. The geometric excitations of the horizon (punctures) are indistinguishable. And finally that the semiclassical limit the area of the black hole horizon is large in Planck units. It follows that: Up to quantum corrections, matter degrees of freedom saturate the holographic bound, viz., $\lambda$ must be equal to $\frac{1}{4}$. Up to quantum corrections, the statistical black hole entropy coincides with Bekenstein-Hawking entropy $S=A/(4{\ell }_p^2)$. The number of horizon punctures goes like $N\propto \sqrt{A/{\ell }_p^2}$; i.e., the number of punctures $N$ remains large in the semiclassical limit. Fluctuations of the horizon area are small $\mathrm{\Delta }A/A\propto {({\ell }_p^2/A)}^{1/4}$, while fluctuations of the area of an individual puncture are large (large spins dominate). A precise notion of local conformal invariance of the thermal state is recovered in the $A\to \infty$ limit where the near horizon geometry becomes Rindler. We also show how the present model (constructed from loop quantum gravity) provides a regularization of (and gives a concrete meaning to) the formal Gibbons-Hawking Euclidean path-integral treatment of the black hole system. These results offer a new scenario for semiclassical consistency of loop quantum gravity in the context of black hole physics, and suggest a concrete dynamical mechanism for large spin domination leading simultaneously to semiclassicality and continuity.

Figure 1

In the infinite area limit, one recovers flat spacetime in the local formulation. The path integral is defined in the portion of spacetime between the horizon and the world sheet of stationary observers at distance $\ell$.