Topological Thermodynamics of Black Holes: Revisiting the methods of winding numbers calculation

TL;DR

The paper investigates two topological approaches to black-hole thermodynamics—the $\phi$-mapping topological current and a residue method—via the off-shell free energy framework. It establishes that the methods are equivalent when $M''S'-S''M'\neq0$, with Schwarzschild and RN illustrating both agreement and limitations near critical points. Extending the analysis to AdS black strings (neutral and charged) reveals a universal global topological charge $W=+1$, indicating charge independence in cylindrical symmetry. This suggests a potential universality of topological classifications for cylindrically symmetric AdS black holes, akin to prior BTZ results. The work highlights subtle links between mass, entropy, and topology, and points to broader implications for rotating or higher-dimensional cylindrical solutions.

Abstract

In this paper, the equivalence between two methods for computing winding numbers is established: the approach of $φ$-mapping topological current and the residue method. The methods are shown to be equivalent when the condition $M'' S' - S'' M' \neq 0$ holds, while deviations appear when this relation fails, signaling subtle connections between mass $M(r_h)$, entropy $S(r_h)$, and topological structure, with $r_h$ being the horizon radius. We first verify this equivalence to Schwarzschild and Reissner-Nordstr"om black holes, recovering known classifications and confirming the consistency of our approach with respect to the validity of the above condition. We then extend the analysis to four-dimensional black strings, regarded as cylindrically symmetric black hole solutions in asymptotically AdS spacetimes. Our results show that both neutral and charged black strings possess the same global topological number, $W = +1$, implying that electric charge does not influence their topological classification. This insensitivity to charge mirrors earlier findings for BTZ black holes in three dimensions, suggesting that it may represent a universal property of cylindrically symmetric black holes in AdS backgrounds.

Figures (3)

• Figure 1: The integral over $C$ is equal to the sum over $C_i$
• Figure 2: The arrows represent the unit vector field $n$ in a region of the plane $r_h-\Theta$. For this we have considered $\tau=4\pi r_0$ and $\alpha^2r_0=0.005$, where $r_0$ is the radius of the cavity that surrounds the black string. The zero-point(ZP) is marked with a red point and is located in $(r_h/r_0,\Theta)=(2.67,\pi/2)$.
• Figure 3: The unit vector field $n$ in a portion of the plane $r_h-\Theta$. We have considered $\alpha^2r_0^2=0.005$, $c^2=0.005$ and $\tau=4\pi r_0$. The zero-point(ZP) is marked with a red point located in $(r_h/r_0,\Theta)=(66.7,\pi/2)$. The blue contour C is enclosing the zero point.