[Paper Review] Information Theoretical Approach to Detecting Quantum Gravitational Corrections

Behnam Pourhassan, Xiaoping Shi+6 more|arXiv (Cornell University)|Oct 19, 2023

Statistical Mechanics and Entropy4 citations

TL;DR

This paper proposes an information-theoretical framework to detect quantum gravitational corrections in higher-dimensional black holes by analyzing the Kullback-Leibler divergence and Fisher information of particle emission probability distributions derived from the Parikh-Wilczek formalism. It finds that quantum gravitational effects are most detectable at an intermediate mass scale, where both divergence and Fisher information peak before declining due to quantum fluctuation dominance, with detectability decreasing in higher dimensions.

ABSTRACT

One way to test quantum gravitational corrections is through black hole physics. In this paper, We investigate the scales at which quantum gravitational corrections can be detected in a black hole using information theory. This is done by calculating the Kullback-Leibler divergence for the probability distributions obtained from the Parikh-Wilczek formalism. We observe that the quantum gravitational corrections increase the Kullback-Leibler divergence as the mass of the black hole decreases, which is expected as quantum gravitational corrections can be neglected for larger black holes. However, we further observe that after a certain critical value, quantum gravitational corrections tend to decrease again as the mass of the black hole decreases. To understand the reason behind this behavior, we explicitly obtain Fisher information about such quantum gravitational corrections and find that it also increases as the mass decreases, but again, after a critical value, it decreases. This is because at such a scale, quantum fluctuations dominate the system and we lose information about the system. We obtain these results for higher-dimensional black holes and observe this behavior for Kullback-Leibler divergence and Fisher information depending on the dimensions of the black hole. These results can quantify the scale dependence and dimension dependence of the difficulty in detecting quantum gravitational corrections.

Motivation & Objective

• To quantify the scale and dimension dependence of quantum gravitational correction detection in black holes.
• To investigate how quantum gravitational corrections alter the probability distribution of particles emitted during black hole evaporation.
• To determine the conditions under which quantum gravitational effects become experimentally detectable using information-theoretic measures.
• To analyze the stability of higher-dimensional Schwarzschild black holes under quantum-corrected geometry.
• To explore the role of the novel quantum mass in modifying thermodynamic and information-theoretic properties of black holes.

Proposed method

• Calculates the Kullback-Leibler divergence between the original and quantum-corrected probability distributions of emitted particles in the Parikh-Wilczek formalism.
• Applies the Fisher information measure to quantify the amount of information available about quantum gravitational corrections in the emission process.
• Uses an effective quantum-corrected metric with an exponential correction term parameterized by η to model quantum gravitational effects.
• Introduces a novel quantum mass M_Q to describe the modified thermodynamics of higher-dimensional black holes.
• Derives the quantum-corrected specific heat C_Q in terms of the quantum mass and entropy, analyzing its sign for thermodynamic stability.
• Performs numerical analysis across different spacetime dimensions (D=4,5) to study the dependence of divergence and Fisher information on dimensionality and black hole mass.

Figure 1: $D_{KL}$ as a function of $M$ and $\eta$ for $D=4$ , $D=5$ , $D=6$ , and $D=11$ .

Experimental results

Research questions

• RQ1 At what mass scale is the Kullback-Leibler divergence between corrected and uncorrected emission distributions maximized, indicating peak detectability of quantum gravitational effects?
• RQ2 How does the Fisher information about quantum gravitational corrections vary with decreasing black hole mass, and what does this imply about information loss at small scales?
• RQ3 How does the dimensionality of spacetime influence the detectability of quantum gravitational corrections as measured by information-theoretic quantities?
• RQ4 What is the critical mass at which quantum fluctuations begin to dominate, causing a decline in both Kullback-Leibler divergence and Fisher information?
• RQ5 Under what conditions does the quantum-corrected specific heat remain positive, indicating thermodynamic stability at small black hole radii?

Key findings

• The Kullback-Leibler divergence between corrected and uncorrected emission probability distributions increases with decreasing black hole mass up to a critical point, after which it decreases due to dominant quantum fluctuations.
• Fisher information about quantum gravitational corrections also increases with decreasing mass up to a critical value, then declines, confirming that information about corrections is lost at very small scales.
• The critical mass at which divergence and Fisher information peak depends on the number of spacetime dimensions, with higher dimensions reducing the range of detectable quantum gravitational effects.
• For D=4 and D=5, the quantum-corrected specific heat remains positive when the correction coefficient η ≥ 1, indicating thermodynamic stability at small black hole radii.
• The stability condition 1 + [(D−2)S₀ − 1]ηe^−S₀ ≤ 0 is satisfied for η ≈ 1 in the small-radius limit, suggesting that quantum corrections can stabilize otherwise unstable Schwarzschild black holes.
• The maximum detectability of quantum gravitational corrections occurs at an intermediate mass scale, not at the smallest possible black hole sizes, due to information loss from quantum fluctuations.

Figure 2: $FI$ as a function of $M$ i.e size of the blackhole for $D=4$ , $D=5$ , $D=6$ , and $D=11$ .