Zero-point length as a topological protection of black hole regularity

Abstract

We investigate the thermodynamic topology of regular black holes with zero-point length using an extended first law that includes the zero-point length stored in the geometry. By treating the regularization scale $l_0$ as a thermodynamic variable, we analyze the Hessian geometry of the thermodynamic manifold and demonstrate that the vector field $\vecφ = (T, Ψ)$, where $T$ is the temperature and $Ψ$ is the conjugate to $l_0$, never vanishes in the physical parameter space for $l_0 > 0$. This implies the absence of Morse critical points and a vanishing winding number ($W = 0$), indicating topological protection against the formation of naked singularities. Crucially, we show that in the singular limit $l_0 \to 0$, a non-zero winding number ($W = 1$) emerges, characterizing the Schwarzschild singularity as a topological defect. The conservation of this topological invariant under smooth evolution provides a rigorous topological formulation of the weak cosmic censorship conjecture: the presence of zero-point length not only regularizes the spacetime background but also enforces topological protection against the formation of singularities, preventing black hole-to-naked singularity transitions.

Figures (2)

• Figure 1: Plot of the vector field $(T,\Psi)$ in the parametric space $(r_+, l_0)$. The region below $l_0<\sqrt{2}/r_+$ represents the case of regular black hole with $l_0>0$ and $r_+>0$. The dashed red line is the extremal black hole case $r_+=\sqrt{2} l_0$.
• Figure 2: Plot of the vector field $(T, \Psi)$ in the parametric space $(r_+, l_0)$. The region represents the case of positive and negative region for $l_0$ and $r_+$. There is a topological defect in the central region $(r_+=0,l_0=0)$.