Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\widetilde{W}^{1+}$ subclass: Extending the topological classification of black hole thermodynamics

Wangyu Ai, Di Wu

TL;DR

This work reveals a novel thermodynamic topological subclass, $\widetilde{W}^{1+}$, in higher-odd-dimensional, multiply rotating Kerr-AdS black holes, expanding the established five-class/two-subclass classification. Using a generalized off-shell Helmholtz free energy $\mathcal{F}=M-\frac{S}{\tau}$ and a two-component phi-field within Duan’s phi-mapping framework, the authors define a conserved topological current and winding numbers to classify black hole states by stability. In 5D (and across 7D and 9D as shown in appendices), $\widetilde{W}^{1+}$ exhibits a low-temperature limit with a single stable small black hole and a high-temperature regime hosting three coexisting states (one stable large, one stable small, one unstable small), resulting in a global topological number $W=+1$ despite distinct asymptotic behavior from known classes. The universality across dimensions and the proposed symmetric candidate $\widetilde{W}^{1-}$ indicate that the thermodynamic topology of black holes is richer than previously understood, motivating an expansion of the classification framework.

Abstract

In this paper, we identify a novel topological subclass, dubbed $\widetilde{W}^{1+}$, in the thermodynamics of higher odd-dimensional, multiply rotating Kerr-AdS black holes. This discovery extends the established topological classification beyond the five classes and two subclasses previously known. The $\widetilde{W}^{1+}$ subclass exhibits a unique and previously unreported stability profile: it admits a thermodynamically stable small black hole state in the low-temperature limit, while in the high-temperature limit, the phase space simultaneously contains one stable large black hole, one stable small black hole, and one unstable small black hole state. Our analysis, which treats black hole solutions as topological defects, reveals a richer landscape of black hole thermodynamics than previously understood and necessitates an expansion of the topological classification scheme to accommodate this new phenomenology.

$\widetilde{W}^{1+}$ subclass: Extending the topological classification of black hole thermodynamics

TL;DR

This work reveals a novel thermodynamic topological subclass, , in higher-odd-dimensional, multiply rotating Kerr-AdS black holes, expanding the established five-class/two-subclass classification. Using a generalized off-shell Helmholtz free energy and a two-component phi-field within Duan’s phi-mapping framework, the authors define a conserved topological current and winding numbers to classify black hole states by stability. In 5D (and across 7D and 9D as shown in appendices), exhibits a low-temperature limit with a single stable small black hole and a high-temperature regime hosting three coexisting states (one stable large, one stable small, one unstable small), resulting in a global topological number despite distinct asymptotic behavior from known classes. The universality across dimensions and the proposed symmetric candidate indicate that the thermodynamic topology of black holes is richer than previously understood, motivating an expansion of the classification framework.

Abstract

In this paper, we identify a novel topological subclass, dubbed , in the thermodynamics of higher odd-dimensional, multiply rotating Kerr-AdS black holes. This discovery extends the established topological classification beyond the five classes and two subclasses previously known. The subclass exhibits a unique and previously unreported stability profile: it admits a thermodynamically stable small black hole state in the low-temperature limit, while in the high-temperature limit, the phase space simultaneously contains one stable large black hole, one stable small black hole, and one unstable small black hole state. Our analysis, which treats black hole solutions as topological defects, reveals a richer landscape of black hole thermodynamics than previously understood and necessitates an expansion of the topological classification scheme to accommodate this new phenomenology.

Paper Structure

This paper contains 7 sections, 27 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A Brief review of the thermodynamic topological approach
  3. Five topological classes and two topological subclasses
  4. Newly identified topological subclass: $\widetilde{W}^{1+}$
  5. Conclusions and discussions
  6. Seven-dimensional triple-rotating Kerr-AdS black hole case
  7. Nine-dimensional four-rotating Kerr-AdS black hole case

Figures (5)

  • Figure 1: The zero points of $\phi^{r_h}$ are shown in the $r_h-\beta$ plane for $W^{1-}$, $W^{0+}$, $W^{0-}$, $W^{1+}$, $W^{0-\leftrightarrow 1+}$, $\overline{W}^{1+}$, $\hat{W}^{1+}$ (sub)classes, respectively. The red line corresponds to a thermodynamically unstable black hole branch with $w = -1$, while the purple line corresponds to a thermodynamically stable black hole branch with $w = 1$. The black dot represents the generation point (GP) within the degenerate point (DP), and the pink dot represents the annihilation point (AP) within the DP. (a) Typical case: four-dimensional Schwarzschild black hole. (b) Typical case: four-dimensional Kerr black hole. (c) Typical case: four-dimensional Schwarzschild-AdS black hole. (d) Typical case: four-dimensional Kerr-AdS black hole. (e) Typical case: four-dimensional static two-charge AdS black hole when $q_1 < q_{1c} = 3/(8\pi Pq_2)$. (f) Typical case: four-dimensional static two-charge AdS black hole when $q_1 \ge q_{1c} = 3/(8\pi Pq_2)$. (g) Typical case: four-dimensional dyonic AdS black hole.
  • Figure 2: Zero points of the vector $\phi^{r_h}$ on the $r_h$-$\tau$ plane for the five-dimensional doubly-rotating Kerr-AdS black hole, with parameters $a_1/r_0 = 0.5$, $a_2/r_0 = 1/3$, and $l/r_0 = 1$. Here, the thermodynamically unstable ($w = -1$, red line) and stable ($w = +1$, purple line) branches intersect at the AP (pink dot). Counting the number of stable and unstable states reveals two of the former and one of the latter, which yields a topological number $W = -1 +1 +1 = 1$.
  • Figure 3: The schematic diagram of the zero points of $\phi^{r_h}$ is shown in the $r_h-\beta$ plane for the possible new topological subclass $\widetilde{W}^{1-}$.
  • Figure 4: Plot of the vector field $\phi^{r_h}$ zero points in the $r_h$-$\tau$ plane for the seven-dimensional triple-rotating Kerr-AdS black hole. The thermodynamically unstable and stable branches, identified by their winding numbers $w = -1$ (red curve) and $w = +1$ (purple curve) respectively, meet at the AP (pink dot). The total topological number is $W = +1$, obtained from the sum of two stable and one unstable state. The chosen parameters are $a_1/r_0 = 0.5$, $a_2/r_0 = 0.25$, $a_3 = 0.2$, and $l/r_0 = 1/3$.
  • Figure 5: Zero points of the vector field $\phi^{r_h}$ in the $r_h-\tau$ plane for the nine-dimensional, four-rotating Kerr-AdS black hole. The thermodynamically unstable and stable branches, characterized by their respective winding numbers $w = -1$ (red curve) and $w = 1$ (purple curve), converge at the AP (pink dot) and the GP (black dot). The global topological number is $W = 1$, which is the sum of contributions from two stable (+1) and one unstable (-1) states. Parameters are fixed as $a_1/r_0 = 1/3$, $a_2/r_0 = 0.25$, $a_3/r_0 = 0.2$, $a_3/r_0 = 1/6$ and $l/r_0 = 1$.