Thermodynamic topology of black holes and an invariant of spacetime

Technical Summary: Thermodynamic Topology of Black Holes and an Invariant of Spacetime

Problem Statement
Previous investigations into the thermodynamic topology of black holes (e.g., Wei, Liu, and Mann [15]) rely on a formalism that introduces an auxiliary variable $\Theta \in [0, \pi]$. This variable is a mathematical tool required to construct a two-component vector field necessary for defining a winding number in a $\phi$-mapping construction. However, $\Theta$ is not a physical thermodynamic state variable, and its introduction amounts to an auxiliary completion of a one-dimensional problem. This raises a fundamental question: can the thermodynamic topology be formulated directly in terms of physical thermodynamic state variables, without relying on non-physical auxiliary parameters solely to satisfy the requirements of the topological mapping?

Methodology
The author proposes a new formalism that eliminates the auxiliary variable $\Theta$ by constructing the topological vector field directly from the gradient of the off-shell grand free energy ($\hat{F}$). This potential is appropriate for thermodynamic systems exchanging both energy and matter with their environment, allowing for thermal and chemical equilibrium.

1. Vector Field Construction: For a black hole characterized by entropy $S$ and generalized charges $\{\tilde{Q}_2, \dots, \tilde{Q}_N\}$, the off-shell grand free energy is defined as $\hat{F} = M - S/\tau_1 - \sum \tilde{Q}_m/\tau_m$, where $\tau_1$ is the inverse temperature and $\tau_m$ are inverse chemical potentials. The vector field $\vec{\phi}$ is defined as the gradient of $\hat{F}$ with respect to the thermodynamic state variables:
   $$\vec{\phi} = \nabla \hat{F} = \left( \frac{\partial \hat{F}}{\partial S}, \frac{\partial \hat{F}}{\partial \tilde{Q}_2}, \dots, \frac{\partial \hat{F}}{\partial \tilde{Q}_N} \right)$$
   The zeros of this vector field correspond to equilibrium configurations (black hole solutions) where $T = 1/\tau_1$ and $\mu_m = 1/\tau_m$.

2. Topological Tensor and Flux: Instead of a simple vector field mapping to a circle ($S^1$), the gradient of the $N$-dimensional state space defines a map to an $(N-1)$-sphere ($S^{N-1}$). The author constructs a conserved, rank-$N$ topological tensor $j^{\nu_1 \dots \nu_N}$. The topological flux $Q_{\{\tau_m\}}$ through a hypersurface at fixed ensemble parameters is calculated as the sum of the local topological indices of the isolated zeros (defects) enclosed within that surface.
   $$Q_{\{\tau_m\}} = \sum_{x_i \in \Sigma} \text{index}_{x_i}(\vec{\phi})$$
   The local index is determined by the sign of the Jacobian of the transformation, which is shown to correlate with the sign of the heat capacity (stability) of the black hole solution.

3. Asymptotic Topological Flux: The author defines the asymptotic topological flux ($Q_\infty$) as the limit of the topological flux through a family of finite hypersurfaces as the ensemble parameters $\{\tau_m\}$ approach infinity. In this limit, the thermodynamic state variables vanish, and the black hole geometry approaches the asymptotic/background spacetime geometry.

Key Results
The formalism was applied to several black hole solutions to demonstrate its validity and utility:

• Reissner-Nordström (RN) Black Hole (Minkowski Background): Possesses a unique zero with a negative heat capacity. The local index is $-1$, yielding a total topological flux $Q = -1$.
• RN-AdS Black Hole: Depending on the ensemble parameter region (specifically the inverse chemical potential $\tau_2$), the system exhibits either one or two zeros.
  • In the region $\tau_2 > 1$, there are two zeros (one stable, one unstable) with indices $+1$ and $-1$, summing to $Q = 0$.
  • In the region $\tau_2 < 1$, there is one stable zero with index $+1$, yielding $Q = 1$.
  • Crucially, the asymptotic topological flux $Q_\infty$ is determined by the connected sector of the parameter space leading to the asymptotic limit. For RN-AdS, the asymptotic sector corresponds to $\tau_2 > 1$, resulting in $Q_\infty = 0$.
• RN-dS and Kerr-Newman Black Holes: Both yield an asymptotic topological flux of $Q_\infty = -1$.
• Classification of Spacetimes: Based on $Q_\infty$, the author classifies maximally symmetric constant-curvature spacetimes into two distinct universal classes:
  • Class 1 ($Q_\infty = -1$): Minkowski and de Sitter (dS) spacetimes.
  • Class 2 ($Q_\infty = 0$): Anti-de Sitter (AdS) spacetime.

Significance and Claims
The paper claims that the asymptotic topological flux $Q_\infty$ serves as an invariant associated with the asymptotic/background spacetime geometry, rather than the specific properties of the black hole solution. This is supported by the observation that distinct black hole solutions (e.g., RN and Kerr-Newman) approaching the same Minkowski background yield the same $Q_\infty$.

A primary finding is that the flat-space limit of AdS is nontrivial within this classification. Even as the AdS curvature radius $l \to \infty$ (or $\Lambda \to 0^-$), the asymptotic topological flux remains $Q_\infty = 0$, distinct from the $Q_\infty = -1$ of Minkowski space. The author suggests this provides a complementary geometric perspective on the AdS distance conjecture, which posits that the $\Lambda \to 0^-$ limit involves an infinite tower of light states and a breakdown of the low-energy effective description. The topological distinction implies that the AdS geometry does not continuously deform into the Minkowski geometry in the thermodynamic topological sense, even when the curvature vanishes.

The work establishes a direct correspondence between topological structures and thermodynamic variables without auxiliary mathematical tools, offering a higher-rank generalization of previous winding-number approaches and a new method for classifying spacetime backgrounds based on their thermodynamic topological invariants.