[Paper Review] The scaling of black hole entropy in loop quantum gravity

Amit Ghosh, Alejandro Pérez|arXiv (Cornell University)|Oct 8, 2012

Black Holes and Theoretical Physics26 references11 citations

TL;DR

This paper proposes a renormalization mechanism in loop quantum gravity (LQG) that explains how the semiclassical black hole entropy $A/(4G_{\text{Newton}})$ emerges from the UV Planck-scale discreteness of area eigenvalues. By introducing a local formulation of quantum horizons and a thermal Hamiltonian based on area quanta, it shows that entropy consistency with Hawking's formula requires a holographic bound on non-geometric degrees of freedom and a scale-dependent Newton constant, resolving the species problem and enabling IR agreement via RG flow.

ABSTRACT

We discuss some general properties of black hole entropy in loop quantum gravity from the perspective of local stationary observers at distance l from the horizon. The present status of the theory indicates that black hole entropy differs from the low energy (IR) expected value A/(4G) (in natural units) in the deep Planckian regime (UV). The partition function is well defined if the number of non-geometric degrees of freedom g_M (encoding the degeneracy of the area a_p eigenvalue at a puncture p) satisfy the holographic bound g_M < exp(ap/(4G)). Our framework provides a natural renormalization mechanism such that S_UV ---> S_IR=A/(4 G) as the scale l flows.

Motivation & Objective

• To understand how the semiclassical black hole entropy $A/(4G_{\text{Newton}})$ arises from the UV-discrete area spectrum in loop quantum gravity.
• To address the long-standing issue of the Immirzi parameter fine-tuning by introducing a dynamical renormalization mechanism.
• To resolve the species problem in LQG by ensuring matter contributions to entropy do not scale with field count, preserving consistency with Hawking's formula.
• To establish a connection between the local quantum horizon formalism and the continuum limit of LQG through renormalization group flow.
• To propose that Hawking entropy acts as a gravitational measurement fixing renormalization conditions for couplings in the UV regime.

Proposed method

• Adopt a local formulation of quantum horizons in equilibrium, treating the horizon as a thermodynamic system with local degrees of freedom.
• Introduce a local Hamiltonian based on area quanta, modeling the energy of the horizon as a function of the area eigenvalue at each puncture.
• Apply a thermal partition function with a chemical potential tied to the number of punctures, reflecting the polymer-like nature of quantum states.
• Enforce a microholographic bound $g_{\scriptscriptstyle M} < \exp(a_p/(4G))$ to ensure convergence of the partition function and avoid divergence.
• Use renormalization group flow to relate UV couplings ($\gamma_*, G_*$) to IR Newton's constant $G_{\text{Newton}}$, allowing $S_{\text{UV}} \to S_{\text{IR}} = A/(4G_{\text{Newton}})$.
• Analyze the interplay between matter and gravity at the Planck scale via the quantum Hamiltonian constraint, ensuring consistency with the holographic bound and entropy scaling.

Experimental results

Research questions

• RQ1 How can the semiclassical black hole entropy $A/(4G_{\text{Newton}})$ be derived from the UV-discrete area spectrum in loop quantum gravity without fine-tuning the Immirzi parameter?
• RQ2 What conditions on the UV couplings $\gamma_*$ and $G_*$ are required for the entropy to flow from the Planck-scale discrete value to the IR Hawking value?
• RQ3 How can the matter field contribution to black hole entropy be consistent with the Bekenstein-Hawking formula, especially when the number of species increases?
• RQ4 What role does the microholographic bound $g_{\scriptscriptstyle M} < \exp(a_p/(4G))$ play in ensuring the finiteness and consistency of the partition function?
• RQ5 Can the semiclassical limit of LQG be understood as a renormalization group flow where Hawking entropy serves as a boundary condition fixing UV couplings?

Key findings

• The partition function in LQG is well-defined only if the number of non-geometric degrees of freedom $g_{\scriptscriptstyle M}$ at each puncture satisfies the holographic bound $g_{\scriptscriptstyle M} < \exp(a_p/(4G))$, ensuring finiteness and consistency.
• The semiclassical black hole entropy $S_{\text{IR}} = A/(4G_{\text{Newton}})$ emerges via renormalization group flow from the UV Planck-scale discreteness, provided the UV couplings $\gamma_*$ and $G_*$ are properly related to $G_{\text{Newton}}$.
• The species problem is resolved: matter contributions to entropy do not scale with the number of fields when the microholographic bound is satisfied, preserving consistency with Hawking's formula.
• Two viable scenarios allow IR agreement: (1) low spins dominate with $G_{\text{Newton}}/(\gamma_* G_*) \ll 1$, or (2) large spins dominate with $a_0 \gg 1$, provided matter couplings are constrained by the Hamiltonian constraint.
• The requirement for consistency with Hawking entropy imposes nontrivial constraints on the RG flow of $G(\ell)$ and matter couplings, suggesting that Hawking entropy can act as a renormalization condition fixing UV parameters.