Critical behavior in a massless scalar field collapse with self-interaction potential

We examine a one-parameter family of analytical solutions representing spherically symmetric collapse of a nonlinear massless scalar field with self-interaction in an asymptotically flat spacetime. The time evolution exhibits a type of critical behavior. Depending on the scalar charge parameter $q$ as compared to a critical value $q^{*}$, the incoming scalar wave collapses either to a globally naked central singularity if $q<q^{*}$ (weak field) or to a scalar-hairy black hole if $q>q^{*}$ (strong field), both having finite asymptotic masses. Near the critical evolution, the black hole mass follows a product-logarithmic scaling law: $-M^2\ln M\sim q-q^{*}$ with $0<M\ll 1$ and $q>q^{*}$. The solution admits no self-similarity and satisfies the null and the strong energy conditions.

Figure 1

The scalar potential $V(\varphi )$ given by (2) with $\lambda =1$.

Figure 2

The Misner-Sharp mass function $M(R,v)$ in the static limit $v\to +\infty$ and with $\lambda =1$. The critical case corresponds to $q=1$. Event horizons are located at intersections with $R/2$.

Figure 3

The Penrose diagram for subcritical evolutions with $0<q<1/\sqrt{\lambda }$. The incoming scalar wave from the past null infinity ${\mathcal{I}}^{-}$ is turned on at $v=0$ before which the spacetime is flat and empty. A timelike naked singularity at $R=0$ forms and persists as the scalar wave collides at the center.

Figure 4

The Penrose diagram for the critical evolution with $q=1/\sqrt{\lambda }$. The initial configuration is similar to the subcritical case. The collapse now proceeds to a null massless singularity at $R=0$.

Figure 5

The Penrose diagram for supercritical evolutions with $q>1/\sqrt{\lambda }$. The influx of the scalar field $(v\ge 0)$ generates a black hole which contains a spacelike singularity at the center. The apparent horizon (AH) is spacelike and approaches the event horizon (EH) from the inside as $v\to +\infty$.

Figure 6

Numerical determination of the apparent horizon $R_{\mathrm{AH}}(v)$ with $\lambda =1$ and $q=1.1$. As $v$ grows, the curve reaches a plateau that corresponds to the event horizon.