[Paper Review] Higher-Derivative Corrections to Entropy and the Weak Gravity Conjecture in Anti-de Sitter Space

Sera Cremonini, Callum R. T. Jones+2 more|arXiv (Cornell University)|Dec 24, 2019

Black Holes and Theoretical Physics54 references4 citations

TL;DR

This paper computes four-derivative corrections to the geometry, extremality bound, and thermodynamics of AdS-Reissner-Nordström black holes in general dimensions and horizon topologies. It confirms a universal relationship between the shift in the extremality bound and the shift in microcanonical entropy, linking the Weak Gravity Conjecture to positivity of entropy corrections, and derives bounds on Wilson coefficients implying $c - a > 0$ in the dual CFT.

ABSTRACT

We compute the four-derivative corrections to the geometry, extremality bound, and thermodynamic quantities of AdS-Reissner-Nordstr{ö}m black holes for general dimensions and horizon geometries. We confirm the universal relationship between the extremality shift at fixed charge and the shift of the microcanonical entropy, and discuss the consequences of this relation for the Weak Gravity Conjecture in AdS. The thermodynamic corrections are calculated using two different methods: first by explicitly solving the higher-derivative equations of motion and second, by evaluating the higher-derivative Euclidean on-shell action on the leading-order solution. In both cases we find agreement, up to the addition of a Casimir energy in odd dimensions. We derive the bounds on the four-derivative Wilson coefficients implied by the conjectured positivity of the leading corrections to the microcanonical entropy of thermodynamically stable black holes. These include the requirement that the coefficient of Riemann-squared is positive, meaning that the positivity of the entropy shift is related to the condition that $c - a$ is positive in the dual CFT. We discuss implications for the deviation of $η/s$ from its universal value and a potential lower bound.

Motivation & Objective

• To compute four-derivative corrections to the geometry, extremality bound, and thermodynamic quantities of AdS-Reissner-Nordström black holes in general dimensions and horizon geometries.
• To confirm the universal relationship between the shift in the extremality bound at fixed charge and the shift in microcanonical entropy, extending known flat-space results to Anti-de Sitter space.
• To derive bounds on four-derivative Wilson coefficients from the conjectured positivity of entropy corrections in thermodynamically stable black holes.
• To explore the implications of these corrections for the deviation of $\eta/s$ from its universal value and the existence of a potential lower bound.

Proposed method

• Explicitly solving the higher-derivative equations of motion for the corrected black hole geometry in AdS.
• Evaluating the higher-derivative Euclidean on-shell action on the leading-order solution to compute thermodynamic corrections.
• Using two independent methods—direct solution of equations of motion and on-shell action evaluation—to cross-verify results, with agreement found except for a Casimir energy contribution in odd dimensions.
• Deriving the entropy shift via corrections to the horizon radius and temperature, with distinct treatments for the 'away from extremality' and 'very-near extremal' limits.
• Applying thermodynamic relations to connect the extremality shift $\Delta M$ to the temperature times the entropy shift $T \Delta S$, showing $\Delta M \propto T \Delta S$ in both regimes.
• Using the extremal limit to extract non-analytic scaling behavior, confirming $T \sim \sqrt{\alpha'}$ and $\Delta S \sim \sqrt{\alpha'}$ in the very-near extremal regime.

Experimental results

Research questions

• RQ1 Does the universal relationship between extremality shift and entropy shift, known in flat space, hold in Anti-de Sitter space?
• RQ2 What are the four-derivative corrections to the mass, charge, and entropy of AdS-Reissner-Nordström black holes with general horizon geometry?
• RQ3 How do higher-derivative corrections affect the extremality bound, and what constraints do they impose on Wilson coefficients?
• RQ4 What is the role of the Casimir energy in odd-dimensional AdS black holes, and how does it affect the on-shell action computation?
• RQ5 Can the positivity of the entropy shift be linked to the $c - a > 0$ condition in the dual CFT, and what does this imply for the Weak Gravity Conjecture?

Key findings

• The entropy shift is proportional to the extremality shift at fixed charge, confirming the universal relation $\Delta M \propto T \Delta S$ in AdS, with $\Delta M = -T \Delta S$ away from extremality and $\Delta M = -\frac{1}{2}T \Delta S$ in the very-near extremal limit.
• The entropy shift is positive when the coefficient of Riemann-squared in the higher-derivative Lagrangian is positive, which corresponds to $c - a > 0$ in the dual CFT.
• The two independent methods—solving equations of motion and evaluating the on-shell action—yield consistent results, with agreement broken only by a Casimir energy term in odd dimensions.
• In the very-near extremal limit, the entropy shift scales as $\Delta S \sim \sqrt{\alpha'}$, and the temperature scales as $T \sim \sqrt{\alpha'}$, leading to a non-analytic behavior in the corrections.
• The extremality shift is of order $\alpha'$, and the proportionality between $\Delta M$ and $T \Delta S$ holds in both the away-from-extremality and very-near-extremal regimes.
• The positivity of the entropy shift implies a lower bound on $\eta/s$, suggesting a potential universal lower bound in higher-derivative gravity theories.