Logarithmic Corrections to Black Hole Entropy: From Macroscopic to Microscopic Analysis
Ashoke Sen's paper "Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions" leverages the Euclidean gravity approach to elucidate the logarithmic corrections to the entropy of non-extremal black holes. The examination focuses on diverse dimensions and non-extremal black holes, encompassing both rotating and non-rotating configurations, while meticulously considering integration over zero modes. This paper builds on previous insights derived from extremal cases and compares the macroscopic results with existing predictions in loop quantum gravity, highlighting potential discrepancies, and exploring the constraints these results impose on ultraviolet (UV) completions of gravity.
Summary of Key Findings
- Methodological Insights: The paper adopts the Euclidean gravity framework to compute the logarithmic corrections to non-extremal black hole entropy. The corrections are derived from the one-loop contribution to the black hole's partition function. Higher-loop contributions are shown, via power-counting arguments, to introduce no further logarithmic terms, attributing the correction origins to massless field loops and low-energy data.
- Numerical Discrepancies: For Schwarzschild black holes in four dimensions, a notable disagreement surfaces between macroscopic predictions and loop quantum gravity, particularly due to missing contributions from massless graviton loops in the latter.
- Entropy in Different Ensembles: The paper distinguishes between the microcanonical entropy (related to the specific mass interval) and other ensemble entropies. The microcanonical entropy, which relates most directly to microstate counting, receives logarithmic corrections that incorporate the conformal anomaly, perturbative loop corrections, and quantum corrections to near-horizon geometry.
- Comparison with Loop Quantum Gravity: The findings underscore a discrepancy between the predicted logarithmic entropy corrections from macroscopic Euclidean methods and those anticipated by loop quantum gravity frameworks. Particularly, the macroscopic results suggest a correction term that includes contributions absent in some loop quantum gravity models. The analysis from SU(2) Chern-Simons theory and U(1) theory in loop quantum gravity, although differing slightly in outcomes, both contrast with Sen's macroscopic predictions.
- BTZ Black Hole: The exercise is extended to the BTZ black holes, where both microscopic and macroscopic results align well with one another regarding logarithmic corrections. This consistency underscores the robustness of the Euclidean gravity approach in contexts where concrete microscopic theories, like conformal field theory (CFT), are present.
Implications and Future Directions
The delicate interplay between infrared and ultraviolet characteristics of gravity captured through these computations lays down stringent benchmarks for any theory aiming at a UV completion. By achieving alignment, or highlighting mismatches, between the macroscopic and microscopic evaluation of black hole entropy, this work sets up a rigorous testbed for theories of quantum gravity. Exploring these constraints could guide the refinement of quantum gravity models or inspire new insights into the holographic nature of gravitational entropy.
In future investigations, researchers could focus on resolving the discrepancies with loop quantum gravity predictions or exploring further dimensions and configurations to extend these constraints and corrections. The implications of these corrections on the broader field of gravitational holography and quantum thermodynamics remain profound and invite additional theoretical scrutiny and computational evaluation. The foundation laid by Sen’s work offers a rich avenue for future theoretical and ontological engagement with non-extremal black hole entropy.