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Entropy calculation for a toy black hole

Hanno Sahlmann

TL;DR

The paper introduces an exactly solvable toy model for black hole entropy in loop quantum gravity by adopting an equidistant area spectrum, replacing the true nonuniform spectrum to enable analytic progress. A generating function $G(g,z)$ is constructed to count horizon states, yielding exact formulas for $N(a,j)$ and related sums, and allowing detailed asymptotic analysis. The results show exponential growth of state counts with area, with the leading behavior governed by base 2 and a universal subleading $-\tfrac{1}{2}\log a$ correction for the physical $j=0$ sector, while joint asymptotics in area and spin reveal a more intricate structure than a simple Kerr-like Smarr relation. The toy model thus provides a useful laboratory that reproduces the qualitative features of the full spectrum and clarifies which aspects are robust under the equidistant approximation, while highlighting differences in numerical coefficients and the limitations of the simplification.

Abstract

In this note we carry out the counting of states for a black hole in loop quantum gravity, however assuming an equidistant area spectrum. We find that this toy-model is exactly solvable, and we show that its behavior is very similar to that of the correct model. Thus this toy-model can be used as a nice and simplifying `laboratory' for questions about the full theory.

Entropy calculation for a toy black hole

TL;DR

The paper introduces an exactly solvable toy model for black hole entropy in loop quantum gravity by adopting an equidistant area spectrum, replacing the true nonuniform spectrum to enable analytic progress. A generating function is constructed to count horizon states, yielding exact formulas for and related sums, and allowing detailed asymptotic analysis. The results show exponential growth of state counts with area, with the leading behavior governed by base 2 and a universal subleading correction for the physical sector, while joint asymptotics in area and spin reveal a more intricate structure than a simple Kerr-like Smarr relation. The toy model thus provides a useful laboratory that reproduces the qualitative features of the full spectrum and clarifies which aspects are robust under the equidistant approximation, while highlighting differences in numerical coefficients and the limitations of the simplification.

Abstract

In this note we carry out the counting of states for a black hole in loop quantum gravity, however assuming an equidistant area spectrum. We find that this toy-model is exactly solvable, and we show that its behavior is very similar to that of the correct model. Thus this toy-model can be used as a nice and simplifying `laboratory' for questions about the full theory.

Paper Structure

This paper contains 11 sections, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Counting
  3. The asymptotics of $N(a,0)$ and $N_\leq(a,0)$
  4. The asymptotics of unrestricted states
  5. Asymptotics in both variables
  6. Conclusions
  7. An alternative modified area spectrum
  8. The asymptotics of $N(a,0)$
  9. The asymptotics of unrestricted states
  10. Extremal states
  11. Asymptotics in both variables

Figures (4)

  • Figure 1: $\ln(N(a,0))$ and $\ln(N_\leq(a,0))$ (dots) and the corresponding asymptotic result (solid line)
  • Figure 2: $\ln(N(a,j))$ (dots) and the asymptotic result (solid line) compared for two different values of $a$
  • Figure 3: Comparison of $\ln(N(a,j))$ (dots) and the function $s$ from \ref{['eq_s']} for $a=100$
  • Figure 4: $\ln(N(a,j))$ (dots) and the function $s$ from \ref{['eq_s2']} compared for two different values of $a$