Detailed black hole state counting in loop quantum gravity

We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced number-theoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step toward obtaining the behavior of entropy in the asymptotic (large horizon area) regime.

Figure 1

Plot of the black hole degeneracy spectrum $D^{\mathrm{DL}}$ (in units of ${10}^{19}$) in terms of the area (in units of $4\pi \gamma {\ell }_P^2$) for a range of area values. The periodicity can be traced all the way back to the smaller values of the area.

Figure 2

Plot of $S_{\le }(a)$ in terms of the area (in units of $4\pi \gamma {\ell }_P^2$). The computation of $S_{\le }(a)$ has been done by using the algorithm based in the number-theoretical method discussed in the paper.

Figure 3

Plot of the black hole degeneracy spectrum $D^{\mathrm{ENP}}$ (in units of ${10}^{21}$) in terms of the area (in units of $4\pi \gamma {\ell }_P^2$) for a range of area values.

Figure 4

Plot of $S_{\le }^{\mathrm{ENP}}(a)$ in terms of the area (in units of $4\pi \gamma {\ell }_P^2$). This plot is essentially the same for $S_{\le }^{\mathrm{GM}}(a)$.

Figure 5

Plot of $\log D_{*}^{\mathrm{DL}}(a)$ and $\log D^{\mathrm{DL}}(a)$ in terms of the area (in units of $4\pi \gamma {\ell }_P^2$). The figure shows the values associated with area eigenvalues multiples of $\sqrt{3}$, $\sqrt{2}$, and $\sqrt{15}$, respectively. The solid lines correspond to the asymptotic approximations given by straight lines with the slopes $s_2$, $s_1$, and $s_{10}$ appearing in Table . When the projection constraint is taken into account (lower plot) the asymptotic approximations have also a logarithmic correction $-(\log a)/2$.

Figure 6

Plot of $\log D^{\mathrm{DL}}(a)$ for area eigenvalues of the form $q_1\sqrt{2}+q_2\sqrt{2}$ with $q_1$, $q_2\in \mathbb{N}\cup \{0\}$. The solid line with the largest slope $s_{\{1,2\}}=0.645008\dots$ [determined by Eq. (6.2) for $\mathcal{I}=\{1,2\}$] gives the asymptotic approximation for their growth. The other solid lines correspond to the $\sqrt{2}$ and $\sqrt{3}$ subfamilies. As it can be seen the envelope of the points grows faster than these subfamilies. In all the cases the logarithmic correction $-(\log a)/2$ has been included.

Figure 7

Plot of the logarithm of the black hole degeneracy spectrum, $\log D^{\mathrm{DL}}(a)$, in terms of the area (in units of $4\pi \gamma {\ell }_P^2$). The figure shows the values associated with area eigenvalues that can be written as integer linear combinations of $\sqrt{3}$, $\sqrt{2}$, and $\sqrt{15}$. The arrows mark the position of the bands as predicted by Eq. (6.4), taking into account that only even values of $P$ contribute when the projection constraint is considered. Notice that beyond $a=5$ there are bands at each of these positions. Notice also that there are many values of the area spectrum for which $D^{\mathrm{DL}}(a)=0$ and, hence, do not contribute to the band structure shown here.

Figure 8

Plot of $D^{\mathrm{DL}}$ (in units of ${10}^{30}$) versus area (in units of $4\pi \gamma {\ell }_P^2$) considering only areas involving the squarefrees numbers 2, 3, and 15.

Figure 9

Plot of $D_m^{\mathrm{DL}}(a)≔\sum _{c\in \mathcal{C}(a)}{d}_m^{\mathrm{DL}}(c)$ (in units of ${10}^{11}$) versus area (in units of $4\pi \gamma {\ell }_P^2$).

Figure 10

Plot of $r$ degeneracy $D_r(a)=\sum _{c\in \mathcal{C}(a)}{d}_r(c){\delta }_m(c)$ (in units of ${10}^{12}$) versus area $a$ expressed in units of $4\pi \gamma {\ell }_P^2$. Here ${\delta }_m(c)$ is one if the configuration $c$ satisfies the projection constraint and zero otherwise. Without this factor the period of the peaks decreases by a factor of 2.

Figure 11

This is a new version of Fig. 1 where we have highlighted one of the peaks singled out by the peak counter $P$. We also show the position of the peaks according to formula (6.4).

Figure 12

Plot of the degeneracy distribution given by formula (6.3) (dashed line) compared to the maximum degeneracy distribution (6.5) (solid line).

Figure 13

Plots of the value of $S_{\le }^{*}(a)$ for the points in the area spectrum and all the areas smaller than 18 (in units of $4\pi \gamma {\ell }_p^2$), respectively. A detailed view of the framed parts can be seen in Fig. 14.

Figure 14

Detail of Fig. 13 for areas $a\in [14,18]$. Notice the steps that appear for the area values considered in this plot.

Figure 15

Plot of the value of $S_{\le }(a)$ for both the points in the area spectrum and all the values of the area smaller than 18 (in units of $4\pi \gamma {\ell }_p^2$). We also show the entropy values corresponding to prequantized values of the area.

Figure 16

Logarithmic plot of $D^{\mathrm{DL}}(a)$ for all the points in the area spectrum smaller than 18 (compare this figure with Fig. 7) and a detail of the last two peaks. The arrows mark the position of the bands as predicted by Eq. (6.4). Notice that there are many points in the area spectrum for which $D^{\mathrm{DL}}(a)=0$.