Violation of Cosmic Censorship in Einstein-Maxwell-Scalar Models with Fractional Coupling

Abstract

The weak cosmic censorship conjecture plays a foundational role in classical gravity by asserting that spacetime singularities are generically hidden behind event horizons. In this work, we explore its robustness in the Einstein-Maxwell-Scalar theory with fractional coupling by studying both static black hole solutions and their fully nonlinear dynamical evolution. We identify a class of scalarized black holes that develop negative energy density near the event horizon, indicating violations of the classical energy conditions. Numerical evolutions of perturbed configurations reveal that sufficiently strong fractional coupling drives rapid curvature growth and geometric degeneration in the near-horizon region, accompanied by persistent negative energy density. While the simulations do not resolve the ultimate end state, the observed dynamics consistently point toward a weakening of the horizon-supporting structure and are suggestive of incipient naked singularity formation. These results uncover a classical mechanism through which fractional coupling can challenge the validity of the weak cosmic censorship conjecture in asymptotically flat spacetimes.

Figures (6)

• Figure 1: Distribution of black hole solutions in the parameter space. The red dashed curve denotes the boundary between hairy and hairless black hole solutions, also referred to as the existence line. The purple solid curve marks the boundary of the physical region, corresponding to the divergence of the coupling function, which occurs when $1 + b \phi^2 = 0$, where $f(\phi)$ becomes singular. The blue solid curve indicates the boundary of scalar-hairy black hole solutions exhibiting negative energy density. For $q < 1$, the parameter space can be divided into four distinct regions according to the existence and physical properties of the black hole solutions: Region I lies below the existence line; Region II is located between the existence line and the physical boundary; Region III is the region above the physical boundary but below the scalar-hairy black hole boundary; and Region IV lies above the scalar-hairy black hole boundary.
• Figure 2: At a fixed charge parameter $q = 0.9$, The left panel shows the metric function $\zeta(z)$ as a function of the radial coordinate $z$, while the right panel displays the scalar field $\phi(z)$ as a function of $z$, where $z$ denotes the compactified radial coordinate. The five selected parameter sets are ordered by decreasing $b$ and are located, respectively, on the existence line separating bald and hairy solutions, in region II, on the divergence line of the coupling function, in region III, and on the existence boundary of hairy black hole solutions. As the parameters approach the existence boundary of the hairy solutions, the metric function $\zeta(z)$ exhibits a pronounced drop and tends toward zero at a finite radial position, while the radial profile of the scalar field $\phi(z)$ also undergoes significant changes, indicating a marked transition in the geometric and physical properties of the solutions in this region.
• Figure 3: The dashed curve represents the RN solution in Region I, while the solid curve corresponds to the hairy solution in Region II. Both classes of solutions are physically regular. The top-left panel shows the Kretschmann scalar, which remains finite from the immediate vicinity of the horizon to spatial infinity, indicating a well-behaved spacetime geometry. The top-right panel shows the spatial distribution of the energy density, exhibiting a monotonically decreasing trend. The bottom-left panel illustrates the spatial profile of the scalar field, which increases monotonically. The bottom-right panel presents the Misner-Sharp mass, growing monotonically from near the horizon to infinity, suggesting an energy distribution extending to infinity.
• Figure 4: The figure corresponds to the parameter choice $b=-10$. The top-left panel shows the Kretschmann scalar. The top-right panel shows the energy density. The inset shows a magnified view of the zero-crossing region. The bottom-left panel and bottom-right panel respectively represents the spatial distribution of the scalar field and Misner-Sharp mass. It is interesting that in the region with negative energy density, the Misner-Sharpe mass shows a decreasing trend. On the contrary, once the energy density becomes positive, it begins to increase.
• Figure 5: Spacetime evolution of the Kretschmann invariant and the energy density for $b=-100$. top-left panel: the temporal evolution of $K(z_{c})$ over $t\in[0,10]$. Its steady growth reflects the continuous enhancement of spacetime curvature. top-right panel: the spatial profiles of $K(z)$ in the radial interval $z\in[z_c,1)$ at evolution times $t_1$ to $t_8$ (here, $t_1 \approx 0.104$, and $\Delta t\approx 1.415$). The curvature peak continuously increases, indicating rapid amplification of curvature and the potential violation of WCCC. bottom-left panel: the temporal evolution of $\rho(z_{c})$ over $t\in[0,10]$. In the later stages, the curve shows a persistent downward trend and reaches significantly negative values, further confirming the breakdown of energy conditions in this region. bottom-right panel: the spatial profiles of the energy density $\rho(z)$ in the radial interval $z\in[z_c,1)$ at evolution times $t_1$ to $t_8$. The energy density becomes increasingly negative, revealing a deepening violation of the energy conditions.