Area spectrum of horizon and black hole entropy

Abstract

We calculate the number of degrees of freedom in spin network states related to the general area spectrum in loop quantum gravity based on the ABCK framework. We find that a black hole entropy (the logarithm of the degrees of freedom) is proportional to its area but it provides us with a different value of the Barbero–Immirzi parameter (γ=0.9284… or 0.7274…). It means that one should be careful in choosing the area spectrum when we describe a horizon area in full quantum gravity.

Appendix:  The Laplace transformation

We introduce the Laplace transformation of N(a) by

$$P(L) := \int^{\infty}_{0} N(a) e^{-La} da, $$

where a=A/4πγ. Integrating (31) over a from 0 to ∞ we get one exact equation:

$$P(L) = \frac{1}{L (1 -G(L) )}, $$

where

with $\tilde{a}_{x}(n,s,t)=a_{x}(n,s,t)/4\pi\gamma$ and $\tilde{a}_{y}(n,s,t) =a_{y}(n,\allowbreak s,t)/4\pi\gamma$. In order to obtain N(a), we must perform the inverse Laplace transformation of P(L). We use the well-known fact that N(a) is determined by the poles of P(L). We find

$$N(a) = \sum_{L_i, \mathrm{Re}(L_i)>0} \mathrm{res}_{L_i} e^{L_ia} + \mbox{subleading terms}. $$

This is the form (21) which was used in Sect. 3. Now we search zeros L _i which satisfy G(L _i )−1=0.

We find that the only one real zero is given by

$$L_0 = 2.91673 \ldots=: \tilde{\gamma}_M. $$

Near this zero, the function P(L) behaves as

$$P(L) \approx\frac{C_M}{L-\tilde{\gamma}_M}, $$

where

$$C_M = -\frac{1}{\tilde{\gamma}_M G'(\tilde{\gamma}_M)} = 0.215979 \ldots. $$

The leading behavior for large a is given by

(A.1)

where $\gamma_{M}=\tilde{\gamma}_{M}/\pi(=0.9284\ldots)$.

It is easy to prove that all complex zeros L _i (i=1,2,…), i.e., the complex poles of P(L _i ), with no vanishing imaginary part must have smaller real part than $\tilde{\gamma}_{M}$ as follows: Since G(L _i ) must be 1 on the poles, let us focus on the real part of G(L _i ). Defining $L_{i}:=L_{i}^{(1)}+i L_{i}^{(2)}$, we find that $\mathrm{Re} [\mathrm{e}^{-L_{i}\tilde{a}_{x}} ]$= $\mathrm{e}^{-L_{i}^{(1)}\tilde{a}_{x}}\cos(L_{i}^{(2)} \tilde {a}_{x} )$. To satisfy G(L _i )=1, $L_{i}^{(1)}$ (i=1,2,…) must be smaller than $\tilde{\gamma}_{M}$.

We show the values of the poles of P(L) for Re[L _i ]>2 and |Im[L _i ]|<100 in Table 1, which are solved numerically. In comparison with the leading relation (A.1), the complex zeros’ contribution to N(a) is exponentially smaller.

Table 1 The poles of P(L). We show the values of poles for Re[L _i ]>2 and |Im[L _i ]|<100