[Paper Review] Asymptotic Safety and Black Hole Thermodynamics

D. Becker, Martin Reuter|arXiv (Cornell University)|Dec 18, 2012

Black Holes and Theoretical Physics3 references4 citations

TL;DR

This paper investigates quantum gravity corrections to black hole thermodynamics within the Asymptotic Safety program using a non-perturbative renormalization group approach on manifolds with boundaries. It introduces a bi-metric ansatz with running couplings for bulk and boundary terms, showing that the surface Newton coupling $G_k^{(0,lat)}$ governs a running ADM mass $M_k$, which vanishes near the Planck scale, implying quantum gravity stabilizes black holes via negative specific heat and zero entropy at high energy.

ABSTRACT

We present recent results on the non-perturbative renormalization group flow of Quantum Einstein Gravity (QEG) on spacetime manifolds with boundaries. As an application, novel quantum gravity corrections to the thermodynamics of black holes are discussed.

Motivation & Objective

• To extend the Asymptotic Safety program of Quantum Einstein Gravity to spacetimes with non-empty boundaries, particularly for black hole spacetimes.
• To investigate how boundary terms—specifically surface terms in the effective average action—modify the renormalization group flow and quantum gravity corrections.
• To define a physically meaningful, scale-dependent ADM mass $M_k$ that incorporates quantum corrections via the boundary Newton coupling $G_k^{(0,lat)}$.
• To analyze the implications of the running $M_k$ for semi-classical black hole thermodynamics, including entropy and specific heat.
• To clarify the conceptual distinction between bulk and boundary Newton constants and their roles in background-independent quantum gravity.

Proposed method

• Adopting a bi-metric truncation in the effective average action $\Gamma_k$, separating background $\bar{g}_{\mu\nu}$ from dynamical metric $g_{\mu\nu}$, to preserve background independence.
• Including 17 running couplings—11 for bulk terms and 6 for boundary terms—associated with Einstein-Hilbert, scalar, and surface invariants.
• Using the induced gravity approximation (large $N$ limit) to compute beta-functions and identify fixed points in theory space.
• Defining a scale-dependent ADM mass $M_k = -1/(8\pi G_k^{(0,\flat)}) \oint (K - K_0)$ using the boundary Newton coupling $G_k^{(0,\flat)}$.
• Applying the self-consistent background condition $\delta \Gamma_k / \delta \bar{h} \big|_{\bar{h}=0} = 0$ to relate level-(1) couplings to effective interactions.
• Deriving thermodynamic quantities (entropy, specific heat) by replacing the classical mass $M$ with the running $M_k$ in standard semi-classical formulas.

Experimental results

Research questions

• RQ1 Does a non-Gaussian fixed point exist in the Asymptotic Safety scenario when spacetime has a non-empty boundary, such as in black hole spacetimes?
• RQ2 How do the different Newton-type couplings—$G_k^{(0)}$, $G_k^{(0,\flat)}$, $G_k^{(1)}$, $G_k^{(1,\flat)}$—evolve under the renormalization group, and what is their physical interpretation?
• RQ3 Can a consistent, scale-dependent definition of the ADM mass be derived from the boundary term in the effective action, and how does it affect black hole thermodynamics?
• RQ4 What are the quantum gravity corrections to black hole entropy and specific heat when the Newton coupling $G_k^{(0,\flat)}$ runs toward the UV?
• RQ5 How does the running $M_k$ affect the final state of Hawking evaporation and the information paradox in the context of asymptotically safe quantum gravity?

Key findings

• A non-Gaussian UV fixed point exists in the gravitational sector even when spacetime has a boundary, supporting the Asymptotic Safety conjecture in this extended setting.
• The surface Newton coupling $G_k^{(0,\flat)}$ runs in the opposite direction to the bulk coupling $G_k^{(0)}$: while $G_k^{(0)}$ decreases in the UV (gravitational antiscreening), $G_k^{(0,\flat)}$ increases.
• The running ADM mass $M_k \propto 1/G_k^{(0,\flat)}$ decreases toward zero as $k \to m_{\text{Pl}}$, implying the black hole mass vanishes at the Planck scale.
• Black hole entropy $S = A / (4 G_k^{(0,\flat)})$ tends to zero in the UV limit, suggesting a loss of degrees of freedom and a potential thermodynamic stabilization.
• The specific heat capacity becomes positive near the Planck scale, indicating a possible thermodynamical stabilization of the final state of Hawking evaporation.
• The standard semi-classical laws of black hole thermodynamics remain valid when the classical mass $M$ is replaced by the running $M_k$, with corrections encoded in $G_k^{(0,\flat)}$.