Strong cosmic censorship: The nonlinear story

A satisfactory formulation of the laws of physics entails that the future evolution of a physical system should be determined from appropriate initial conditions. The existence of Cauchy horizons in solutions of the Einstein field equations is therefore problematic and expected to be an unstable artifact of general relativity. This is asserted by the strong cosmic censorship conjecture, which was recently put into question by an analysis of the linearized equations in the exterior of charged black holes in an expanding universe. Here, we numerically evolve the nonlinear Einstein-Maxwell-scalar field equations with a positive cosmological constant, under spherical symmetry, and provide strong evidence that mass inflation indeed does not occur in the near extremal regime. This shows that nonlinear effects might not suffice to save the strong cosmic censorship conjecture.

Figure 1

Radius function for constant-$u$ slices in a configuration with $M_0=1.0$, $Q=0.9$, $\mathrm{\Lambda }=0.06$, $\mu =0$, $A=0.4$, $v_c=3.0$, and $w=0.25$. Dashed-dotted green lines reach infinity, full blue lines hit the CH and red dashed lines hit the singularity at $r=0$.

Figure 2

Mass function (10) and Kretschmann scalar as functions of $\stackrel{\circ }{v}$ for configurations $\mathbf{A}\mathbf{1}$ (red solid line) and $\mathbf{A}\mathbf{2}$ (blue dashed line). Thin lines are evaluated at $u=u_{\mathrm{EH}}+1$ and thick lines are evaluated at $u=u_{\mathrm{EH}}+2$. These results are consistent with the existence of mass inflation leading to a weak singularity.

Figure 3

Mass function (10) and Kretschmann scalar as functions of $\stackrel{\circ }{v}$ for configurations $\mathbf{B}\mathbf{1}$ (red solid line) and $\mathbf{B}\mathbf{2}$ (blue dashed line). Thin lines are evaluated at $u=u_{\mathrm{EH}}+1$ and thick lines are evaluated at $u=u_{\mathrm{EH}}+2$.

Figure 4

Same as Fig. 3, for configurations $\mathbf{C}\mathbf{1}$ (red solid) and $\mathbf{C}\mathbf{2}$ (blue dashed).

Figure 5

Massless scalar field along the event horizon with corresponding “local power” for a configuration with $M_0=1.0$, $Q=0.95$, $\mathrm{\Lambda }=0$, $\mu =0$, $A=0.01$, $v_c=6.0$, and $w=0.25$. The power-law decay $\mathrm{\Phi }\sim v^{-3}$ matches to a very good precision the one expected from linearized analysis [32] and reproduces well previous nonlinear results [20].

Figure 6

${\delta }_{N,64}(K)$ at $u=18.2$ for configuration $\mathbf{B}\mathbf{1}$. 20 domains were employed in the $v$ direction, where each domain has $N$ points. The plot clearly shows exponential convergence until $N\approx 40$.

Figure 7

Constraint violations during our evolutions of configurations $\mathbf{B}$.

Figure 8

Scalar field derivative $\partial \mathrm{\Phi }/\partial \stackrel{\circ }{v}$ as a function of $\stackrel{\circ }{v}$ for configurations $\mathbf{A}$ (left plot), $\mathbf{B}$ (middle plot), and $\mathbf{C}$ (right plot) with initial profiles 1 (red solid lines) and 2 (blue dashed lines). $\partial \mathrm{\Phi }/\partial \stackrel{\circ }{v}$ evaluated at $u=u_{\mathrm{EH}}$.