Universal thermodynamic topological classes of black holes in perfect fluid dark matter background

Abstract: In this paper, we study the universal thermodynamic topological classes of a family of black holes in a perfect fluid dark matter (PFDM) background. Recent research on black hole thermodynamics suggests that all black holes can be classified into four universal thermodynamic classes, denoted by $W^{1-}$, $W^{0+}$, $W^{0-}$, and $W^{1+}$. Our study reveals that the Schwarzschild black hole in PFDM belongs to the $W^{1-}$ class, and independence of black hole size thermodynamically unstable at both low- and high-temperature limits. The Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes in the PFDM background belong to the same universal thermodynamic class, $W^{0+}$, which represents small, stable black holes and large, unstable black holes at low-temperature limits, whereas no black hole state exists at high temperatures. The AdS black holes behave differently compared to their counterparts in PFDM. The Schwarzschild-AdS black hole belongs to the $W^{0-}$ class, indicating no black hole state at low temperatures, but small, unstable and large, stable black hole states at high temperatures. Furthermore, the Kerr-AdS black hole belongs to the $W^{1+}$ class, characterized by small, stable black holes at low temperatures, large, stable black holes at high temperatures, and unstable, intermediate-sized black holes at both low and high temperatures. These findings uncover the universal topological classifications underlying black hole thermodynamics, offering profound insights into the fundamental principles of quantum gravity.

• The paper introduces a generalized off-shell free energy framework to classify black holes in a perfect fluid dark matter background.
• It employs topological defect analysis to categorize black holes into four distinct thermodynamic classes based on temperature variations.
• Results highlight how black hole stability transitions correlate with varying thermal regimes and dark matter interactions.

Universal Thermodynamic Topological Classes of Black Holes in Perfect Fluid Dark Matter Background

This paper explores the thermodynamic topology of various black hole solutions embedded in a perfect fluid dark matter (PFDM) background. By leveraging the principles of topological thermodynamics, the paper provides insights into the classification of different types of black holes based on their stability at varying temperature regimes. This analytical approach offers an enhancement to our understanding of black holes in contexts close to quantum gravitational thresholds.

Topological Classification Methodology

The foundation of this research builds upon the introduction of generalized off-shell free energy and topological defect analysis, which categorizes black holes into four distinct thermodynamic classes: $W^{1-}$, $W^{0+}$, $W^{0-}$, and $W^{1+}$. These classes are characterized by the behavior and stability of black holes under various temperature conditions, which correspond to their winding numbers, indicating thermodynamic stability.

Mathematical Framework

The generalized off-shell free energy $\mathcal{F}$ is developed as follows:

$\mathcal{F} = M - \frac{S}{\tau}$

where $M$ is mass, $S$ is entropy, and $\tau$ is the inverse temperature parameter. This framework utilizes a vector field $\phi$, denoted by:

$$\phi = (\phi^{r_h}, \phi^\Theta) = \left(\frac{\partial \tilde{\mathcal{F}}}{\partial r_h}, \frac{\partial \tilde{\mathcal{F}}}{\partial \Theta}\right)$$

Here, $\tilde{\mathcal{F}}$ accounts for the generalized free energy including angular considerations. The asymptotic behavior of $\phi$ along predefined contours (as shown by figures in the original work) informs the topological classification by marking zero points or defects.

Schwarzschild Black Hole in PFDM

The Schwarzschild black hole's behavior in a PFDM background reveals it belongs to the $W^{1-}$ class. This indicates an unstable thermal nature at both low and high-temperature limits with a consistent negative winding number.

Figure 1: The $n$-vector field $\phi$ plot for Schwarzschild black hole in a PFDM background, marking zero points of instability.

Reissner-Nordström and Kerr-Newman Black Holes in PFDM

For Reissner-Nordström and Kerr-Newman black holes, the $W^{0+}$ class emerges as the dominant categorization. These configurations feature a distinct small, stable state and a large, unstable state, which dissolve at sufficiently high temperatures. This classification is affirmed by zero topological numbers over the parameter space considered.

Figure 2: The $n$-vector field for the Reissner–Nordström black hole depicting discrete topological transitions in PFDM background.

AdS Black Hole Solutions in PFDM

In contrast to their non-AdS counterparts, Schwarzschild-AdS black holes map to the $W^{0-}$ class. This classification reflects the transition from stable small to unstable large black holes with topological characteristics that include annihilation points that guide the transitions.

Conversely, Kerr-AdS black holes fall under the $W^{1+}$ class, characterized by multiple stable regions at varying temperatures, highlighting their dynamic topological intensity across a broader parameter space.

Figure 3: Contour analysis of Schwarzschild-AdS illustrating change across topological domains in PFDM.

Conclusion

This research has systematically classified black holes in PFDM environments using topological defects of thermodynamic properties. It provides a compelling unified model to understand various black holes' stability across temperatures, significantly contributing to the theoretical modeling near quantum gravity regimes. The outlined topological classes encapsulate complex interactions, allowing further exploration into other spacetimes and dark matter interactions. The potential to evolve Probes into more intricate quantum fields remains vast, suggesting pathways for future research into relativistic and quantum domains.