Emergence of critical phenomena from the black hole interior

The emergence of the $r=0$ singularity inside a spherically symmetric charged black hole, is studied numerically within the Einstein-Maxwell-real scalar model. When the scalar field reaches a critical strength, the $r=0$ singularity emerges inside of the black hole at the tip of the causal diamond. By varying the parameter $p$ of the initial profile for the scalar field towards the critical value $p_{\ast }$, we observe the areal radius at the tip follows a power law scaling, $r_S\propto \mathrm{\mid }p-p_{\ast }{\mathrm{\mid }}^{\gamma }$, with a universal critical exponent $\gamma \approx 0.5$. This remarkable discovery, analogous to Choptuik’s critical phenomena for the black hole formation, provides the first evidence of the universality and scaling for the emergence of the $r=0$ singularity inside black holes, offering new insights into the nonlinear dynamics of strong gravitational field.