Planckian charged black holes and their cosmological ramifications

TL;DR

The paper constructs Planck-scale regular, electrically charged black holes by coupling nonlinear electrodynamics with a minimal length, yielding ultraviolet-finite cores that recover Maxwell theory at large distances. Embedding these objects in AdS and de Sitter backgrounds reveals a Van der Waals-like thermodynamic structure in AdS and a rich set of cosmological instantons (lukewarm, Nariai, cold, ultracold) in de Sitter space. Using the no-boundary proposal, the authors compute Euclidean actions to assess quantum production rates, finding strong suppression for all charged channels both before and after inflation, in contrast to neutral Planck-scale black holes which could contribute to dark matter in CNT25. The results highlight a clear separation between UV-regular effects and late-time cosmological production, suggesting rotation and more general charge-momentum configurations as future avenues. The work advances the understanding of Planck-scale gravity with nonlinear electrodynamics and its cosmological implications, and provides a framework for exploring Planck-scale remnants and their observational consequences.

Abstract

The application of nonlinear electrodynamics at high energy scales has led to a variety of interesting phenomena in recent years, particularly within the context of non-singular spacetime geometries. Additionally, it is postulated that gravity near the Planck scale is governed by a minimal cut-off length, which acts as a renormalization scale against ultraviolet pathologies. Within this framework, we combine both concepts by introducing modifications to the electric and matter sectors of a black hole as its size approaches this minimal length. The result is an electrically charged black hole that is free from ultraviolet divergences and recovers the Maxwell limit at classical scales. We further explore the geometric and thermodynamic properties of the resulting solution within a cosmological anti-de Sitter background, revealing a chemical analogy with that of a Van der Waals fluid. Subsequently, we examine the charged black hole in de Sitter space and construct four corresponding gravitational instantons. We then study their cosmological quantum production using the formalism of the pair creation rate within the context of the no-boundary proposal.

Figures (14)

• Figure 1: The NED electric field in Planck units. The blue line stands for $Q=e\,$, the orange line for $Q=2e$ and the green line for $Q=3e\,$. The dashed lines represent the classical divergent Coulomb fields for each case.
• Figure 2: The classical Coulomb field (orange line) and the field improved by NED (blue line) outside and inside of a Planck-sized positively charged sphere with radius $r_{\rm s}\,$.
• Figure 3: The regular metric potential $f(r)$vs$r$ in Planck units.
• Figure 4: The plots are illustrated in Planck units, for $Q=e$ (blue lines), for $Q=15e$ (orange lines) and for $Q=25e$ (green lines). The uncharged non-singular case ($Q=0$) almost coincides with the blue lines, while the black dashed line corresponds to the semi-classical temperature of the Schwarzschild black hole.
• Figure 5: The function $F(r_+)$vs$r_+$ in Planck units for $N=1$ (blue line), for $N=2$ (orange line) and for $N=3$ (green line).