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Emergence of critical phenomena from the black hole interior

Caiying Shao, Junqi Guo, Yu Tian, Hongbao Zhang

Abstract

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 ${p_*}$, we observe the areal radius at the tip follows a power law scaling, ${r_S } \propto {| {p - {p_*}}|^γ}$, with a universal critical exponent $γ\approx 0.5$. 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.

Emergence of critical phenomena from the black hole interior

Abstract

The emergence of the 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 singularity emerges inside of the black hole at the tip of the causal diamond. By varying the parameter of the initial profile for the scalar field towards the critical value , we observe the areal radius at the tip follows a power law scaling, , with a universal critical exponent . 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 singularity inside black holes, offering new insights into the nonlinear dynamics of strong gravitational field.

Paper Structure

This paper contains 1 section, 14 equations, 4 figures, 2 tables.

Table of Contents

  1. Acknowledgements

Figures (4)

  • Figure 1: The Penrose diagram is depicted in light colors for the pre-existing RN black hole, which will be perturbed by our scalar field at $t=0$ between $-x_0$ and $x_0$. Our numerical extrapolations for the later boundary data at $x=\pm x_0$ do not influence the real dynamics of the causal diamond, which is shaded in yellow.
  • Figure 2: The density plots of $r$ in the $t-x$ plane for the perturbed geometry of the RN black hole by the initial scalar profile $\varphi(x) = A \tanh(x/B)$ with $B=1$. The upper panel is for $A=0.3$, where $r$ at the tip of the causal diamond remains a finite non-zero value, while the lower panel is for the critical $A=0.3938351802555$, where the $r=0$ singularity emerges at the tip.
  • Figure 3: The diverging behavior of the Kretschmann scalar $K$ towards the $r=0$ singularity along $x=0$ for the initial scalar profile $\varphi = A\tanh(x/B)$ with $B = 1$ and $A=0.3938351802555$. A power law fit $K \propto {r^{ - \alpha }}$ to the numerical data near $r=0$ gives $\alpha = 6.020753$.
  • Figure 4: The power law scaling behavior of the areal radius at the tip of the causal diamond with respect to the amplitude $A$ for the initial scalar profile $\varphi=A\tanh(x/B)$ with $B=1$ near the threshold for the emergence of the $r=0$ singularity. The power law fit ${r_S} \propto |A - {A_*}{|^\gamma }$ to the numerical data near the threshod $A_*$ gives $\gamma = {\rm{0}}{\rm{.500029}}$.