Challenging the Weak Cosmic Censorship with Phantom Fields

Abstract

Penrose's weak cosmic censorship conjecture asserts that spacetime singularities produced by gravitational collapse are generically hidden behind event horizons, thus preventing them from causally influencing distant observers and preserving the predictability of the exterior region. In this work, we probe this conjecture in a setup that deliberately violates one of its central assumptions - the dominant energy condition - by considering the spherical collapse of a phantom scalar field with negative energy density. In principle, such a field could produce a Schwarzschild geometry with negative mass and therefore no event horizon. Our aim is to assess whether, once the dominant energy condition is abandoned, the fully coupled evolution of matter and geometry can dynamically generate or expose naked singularities, thereby probing the robustness of cosmic censorship. To this end, we perform high-accuracy numerical relativity simulations based on fourth-order finite-difference schemes. Starting from smooth, asymptotically flat initial data representing regular phantom scalar wave packets, we follow their fully nonlinear evolution through collapse or dispersion. While an ordinary (positive-energy) scalar field exhibits the standard Choptuik critical behavior at the threshold of black-hole formation, the phantom field displays qualitatively different dynamics. For all amplitudes considered, we find no evidence for trapped surfaces, naked singularities, or alternative stationary end states. Instead, the phantom scalar field always disperses, suggesting that cosmic censorship remains dynamically preserved even in the presence of negative-energy matter.

Figures (19)

• Figure 1: Representative initial profiles of the scalar field $\xi(r)$ used in the numerical simulations. The Gaussian profile (red curve) corresponds to a localized pulse, while the tanh-type profile (yellow curve) produces a smoothed step with an approximately constant plateau.
• Figure 2: Initial profiles for $e^\alpha$, $e^\beta$ and $v_{\rm char}$, describing a nearly flat spacetime locally perturbed by a Gaussian pulse of the phantom field centered at $r_0=10\tilde{M}$. The amplitude of the wave packet is $A=0.1\tilde{M}$ and its width is $\sigma=\tilde{M}$.
• Figure 3: Nonlinear evolution of a standard scalar field (red solid curve) and of the outgoing characteristic velocity $v_{+}$ (yellow dashed curve) for a Gaussian initial profile with $\sigma=\tilde{M}$, $r_0=10\tilde{M}$. Each small panel corresponds to a different time snapshot. Left panels: Initial amplitude $A=0.2\tilde{M}$; the field disperses and no horizon forms. Right panels: Initial amplitude $A=\tilde{M}$; the vanishing of $v_{+}$ within a finite region signals the formation of a trapped surface and an apparent horizon.
• Figure 4: Critical amplitude $A^*$ separating dispersion from black-hole formation as a function of spatial resolution $N_p$. Different colors denote different CFL factors. As the resolution increases, $A^*$ converges toward $0.566\tilde{M}$, demonstrating numerical robustness.
• Figure 5: Nonlinear evolution of a phantom scalar field $\xi$ (red solid curve) and of the outgoing characteristic velocity $v_{+}$ (yellow dashed curve) for Gaussian initial data with $\sigma=\tilde{M}$ and $r_0=10\tilde{M}$. Each small panel shows a different time snapshot of the evolution. Left panels: initial amplitude $A=1.5\tilde{M}$ with CFL $=0.5$. The evolution develops irregular features at the origin and eventually breaks down. Right panels: same initial data but with CFL $=0.25$, while the spatial resolution $\Delta r$ and the Kreiss-Oliger dissipation coefficient $\eta_{\rm KO}$ are kept fixed. In this case the evolution remains well behaved throughout the simulation (note that the times shown in the bottom-right panels differ from those in the corresponding panels on the left).