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Topological Complex Analysis of Kerr--Newman Black Hole Microstructure in f(R) Gravity

Wen-Xiang Chen

TL;DR

This work develops a topological, complex-analytic framework to characterize Kerr–Newman black hole microstructure in $f(R)$ gravity by mapping microstates to singularities of an analytically continued partition function and extracting a topological index $W$ from residues. It shows that $W$ classifies horizon structure and thermodynamic stability: $W=+1$ for single-horizon configurations (e.g., Schwarzschild or extremal KN), and $W=0$ for two-horizon KN cases, with $W$ remaining invariant under Starobinsky-type deformations $f(R)=R+\alpha R^2$. The Starobinsky example demonstrates quantitative entropy shifts via $S_h=\frac{f'(R_0)A_h}{4G}$ while preserving the topological class, highlighting a form of phase protection under smooth changes of the gravitational sector. The framework offers a complementary perspective to holographic counting and thermodynamic geometry, linking microstate structure to a global topological invariant that could reflect deeper quantum-gravitational constraints on black hole evolution and information. These insights pave the way for applying the residue-based topology to other gravity theories and multi-horizon spacetimes, potentially informing our understanding of black hole stability and transitions in a broad quantum-gravitational context.

Abstract

We investigate the microstructure of Kerr Newman black holes in modified gravity of the f(R) type using a topological complex analytic framework inspired by holography. In this approach, black hole microstates are identified with singularities of an analytically continued partition function, and the entropy is obtained from residues weighted by winding numbers. We show that the microstructure is characterized by a discrete topological index, which encodes both horizon structure and thermodynamic stability. Non extremal Kerr Newman black holes with both inner and outer horizons correspond to a vanishing topological index, while single horizon configurations correspond to a positive unit topological index. An explicit Starobinsky type modified gravity model demonstrates that this classification is robust under changes to the gravitational sector. We further discuss the limitations of the analytic continuation procedure and suggest that this topological classification may indicate a form of phase protection in black hole thermodynamics.

Topological Complex Analysis of Kerr--Newman Black Hole Microstructure in f(R) Gravity

TL;DR

This work develops a topological, complex-analytic framework to characterize Kerr–Newman black hole microstructure in gravity by mapping microstates to singularities of an analytically continued partition function and extracting a topological index from residues. It shows that classifies horizon structure and thermodynamic stability: for single-horizon configurations (e.g., Schwarzschild or extremal KN), and for two-horizon KN cases, with remaining invariant under Starobinsky-type deformations . The Starobinsky example demonstrates quantitative entropy shifts via while preserving the topological class, highlighting a form of phase protection under smooth changes of the gravitational sector. The framework offers a complementary perspective to holographic counting and thermodynamic geometry, linking microstate structure to a global topological invariant that could reflect deeper quantum-gravitational constraints on black hole evolution and information. These insights pave the way for applying the residue-based topology to other gravity theories and multi-horizon spacetimes, potentially informing our understanding of black hole stability and transitions in a broad quantum-gravitational context.

Abstract

We investigate the microstructure of Kerr Newman black holes in modified gravity of the f(R) type using a topological complex analytic framework inspired by holography. In this approach, black hole microstates are identified with singularities of an analytically continued partition function, and the entropy is obtained from residues weighted by winding numbers. We show that the microstructure is characterized by a discrete topological index, which encodes both horizon structure and thermodynamic stability. Non extremal Kerr Newman black holes with both inner and outer horizons correspond to a vanishing topological index, while single horizon configurations correspond to a positive unit topological index. An explicit Starobinsky type modified gravity model demonstrates that this classification is robust under changes to the gravitational sector. We further discuss the limitations of the analytic continuation procedure and suggest that this topological classification may indicate a form of phase protection in black hole thermodynamics.
Paper Structure (23 sections, 23 equations, 2 figures, 1 table)

This paper contains 23 sections, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background: $f(R)$ Gravity and Kerr--Newman Black Holes
  3. Field Equations and Entropy in $f(R)$ Gravity
  4. Kerr--Newman Black Hole Solutions in $f(R)$ Gravity
  5. Topological Complex Analysis Framework
  6. Partition Function, Spectral Zeta Function, and Analytic Continuation
  7. Interpretive summary.
  8. Contour Integration on the Complex Plane and Residue Sum for $\rho(E)$
  9. Interpretive summary.
  10. Topological Index from Residues and Connection to Chern Number
  11. Interpretive summary.
  12. Application to Kerr--Newman Black Hole Microstructure in $f(R)$ Gravity
  13. Horizon Structure and Thermodynamics of Kerr--Newman in $f(R)$
  14. Interpretive summary.
  15. Microstructure Interactions and Comparison with Ruppeiner Geometry
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Illustration of a two‐sheet Riemann surface for an analytically continued function (e.g. the spectral zeta function). The jagged line denotes a branch cut connecting Sheet 1 and Sheet 2. Black dots mark simple poles: one on Sheet 1 may correspond to the outer horizon and the one on Sheet 2 to the inner horizon of a black hole. Integrating the function along the red contour $\mathcal{C}$ on Sheet 1 picks up the residue of the upper pole, contributing to the density of states or entropy. Summing over all such residues gives the total microstate count.
  • Figure 2: Schematic of a Kerr--Newman black hole with outer ($r_+$) and inner ($r_-$) horizons. The outer horizon (solid circle) contributes a topological charge $+1$ (associated with a stable thermodynamic branch), while the inner horizon (dashed circle) contributes $-1$ (an unstable branch). In $f(R)$ gravity, the horizon radii and entropy are modified by the function $f(R)$ (e.g. scaled by $f'(R_0)$), but the topological charges remain the same. The net topological index $W$ is the sum of contributions: here $W=+1 + (-1) = 0$ for a non-extremal Kerr--Newman black hole, indicating a cancellation between the two horizon contributions.