Universal topological classes of black holes surrounded by quintessence

TL;DR

The paper extends the thermodynamic topology framework to black holes embedded in quintessence, systematically classifying their global stability structures into universal topological types. By employing a generalized off-shell free energy and Duan's phi-mapping, it assigns winding numbers to black-hole branches across Schwarzschild, Kerr, AdS, and Einstein–Gauss–Bonnet spacetimes, revealing distinct classes such as W^{1−}, W^{0+}, W^{0−}, and W^{1+}. Quintessence shifts the temperature ranges where stable or unstable branches appear but does not fundamentally alter the topological class, whereas rotation and Gauss–Bonnet corrections can change the class (notably Kerr vs Schwarzschild, and 4D/5D EGB cases). The results illuminate how curvature, rotation, and dark-energy fields shape the global stability landscape of black holes in diverse backgrounds. This topology-driven perspective offers a unified lens for understanding black-hole thermodynamics in cosmologically relevant fields.

Abstract

This work explores the universal topological classes of various black hole configurations immersed in a quintessence field. The results indicate that the Schwarzschild black hole surrounded by quintessence is of the $W^{1-}$ type, exhibiting thermodynamic instability in both extremal temperature limits. The introduction of rotation alters its topological nature to the $W^{0+}$ class, where the Kerr black hole displays coexistence of a stable small black hole and an unstable large black hole at low temperatures. In comparison, anti de Sitter (adS) black holes display different thermodynamic topologies. The Schwarzschild-adS solution corresponds to the $W^{0-}$ class, where an unstable small branch and a stable large branch emerge in the high temperatures. Meanwhile, the Kerr-adS classified as $W^{1+}$, maintains stability in both high- and low-temperature limits. Furthermore, the 4D/5D quintessential black hole belongs to the $W^{0+}$ class within the Einstein-Gauss-Bonnet framework. Overall, parameters such as rotation and Gauss-Bonnet in quintessence background field affect the topological classifications of the black holes, while the quintessence field primarily modifies the stability range without changing the topological class itself.

Figures (16)

• Figure 1: The zero points of $\phi ^{r_+}$ in the $r_+-\beta$ plane for Sch-BH-$\omega_q$, we set $a/r_0=0.1$, $c r_0^2=0.1$.
• Figure 2: The unit vector field $n$ on the $r_+-\Theta$ plane for the Sch-BH-$\omega_q$, we set $\beta/r_0=20$.
• Figure 3: The asymptotic behavior of the $n$-vector field $\phi$ at the boundary ($C=I_{1}\cup I_{2}\cup I_{3}\cup I_{4}$).
• Figure 4: Contour $\Phi_{1}$ depicts the variation of the vector field $\phi$ components along the pat $C_1$ show in Fig. 2 for the Sch-BH-$\omega_q$ case.
• Figure 5: The zero points of $\phi ^{r_+}$ in the $r_+-\beta$ plane for Kerr-BH-$\omega_q$, we set $a/r_0=0.1$, $c r_0^2=0.1$.