Differential topology and micro-structure of black hole in Einstein-Euler-Heisenberg spacetimes with exponential entropy

TL;DR

This work investigates the thermodynamic topology and microstructure of exact black holes in the Einstein–Euler–Heisenberg theory using a $\ abla$-mapping framework with exponential entropy. It analyzes canonical, mixed, and grand canonical ensembles to classify topological charges via critical points and examines stability through temperature and heat capacity, while comparing thermodynamic geometries (Weinhold, Ruppeiner, HPEM, GTD) in the exponential entropy setting. An exact magnetically charged BH solution is derived with metric components $B(r)=1+rac{Q_m^2}{r^2}+rac{2(\,\alpha-\beta)Q_m^4}{5 r^6}-\frac{2M}{r}$ and $R(r)=r$ under $\,\phi=0, f(\phi)=1$, highlighting the role of higher-order invariants. The paper also formulates the energy emission rate modified by the exponential framework and shows how the parameter $\beta$ can enhance or suppress Hawking radiation in certain regimes. Together, these results connect topological stability, geometric thermodynamics, and quantum effects in nonlinear electrodynamics, offering insights into BH phase structure and microstructure in strong-field regimes.

Abstract

Exact black holes in the Einstein Euler-Heisenberg theory are explored with an exponential entropy framework by using the topological current $Ψ$-mapping theory. The topology classes are investigated through the canonical, mixed, and grand canonical ensembles. In particular, the magnetic charge is fixed for the canonical ensemble, whereas the magnetic potential is included for the mixed ensemble and the grand canonical ensemble with maintaining its consistency through the magnetic potential. The topological charges are analyzed for each ensemble through critical points. As a result, it is found that the canonical, mixed, and grand canonical ensembles lead to either $1$, $-1$, or no generation/annihilation points. Moreover, it is shown how temperature and heat capacity depend on the horizon radius in order to verify the stability of a black hole. Furthermore, the behavior of the thermodynamic curvatures of a black hole is investigated through the geometric methods.

Figures (9)

• Figure 1: The field of unit vectors with fixed values of $\alpha =1.1$, $Q_m=1$, and $\tau =30$.
• Figure 2: The field of normalized vector with fixed values of $\alpha =1.1$, $Q_m=1$, $\tau =30$ and $\Psi_m=1$.
• Figure 3: Weinhold curvature scalar (green curve) $R^{\mathrm{W}}$ and heat capacity (red curve) with fixed values of $\alpha =3$, $\beta =2.1$ and ${Q_m}=1.5$.
• Figure 4: GTD curvature scalar (green curve) $R^{\mathrm{GTD}}$ and heat capacity (red curve) with fixed values of $\alpha =3$, $\beta =2.1$ and ${Q_m}=1.5$.
• Figure 5: Rate of energy emission with fixed values of $\alpha =0.6$, ${Q_m}=0.1$, $\beta$ = 2.00 (green curve), 2.20 (blue dashed curve), 2.40 (red dashed curve), 2.60 (black dashed curve), and 2.80 (orange dashed curve).