Cover Story (Issue 6, 2024) | Recent development on critical collapse

Author: Prof. Zhou-Jian Cao (Beijing Normal University)

    The discovery of the critical phenomena ingravitational collapse by Choptuik is a breakthrough innumerical relativity. Choptuik studied the implosion of amassless scalar field in spherical symmetry. There are two extremities inthis model. At the first extremity, when the initial value of the scalar fieldis weak enough, the field bounces at the center and then is dispersed toinfinity: a flat spacetime remains. At the other one, when the initial value isstrong enough, the field will collapse to form a black hole. Criticalcollapse occurs in the intermediate case between these two extremities.Analytic expressions are very important for understanding the dynamics ofgravitational collapse. However, the high nonlinearity of the Einsteinequations makes it very challenging to seek the analyticsolutions to collapse.

       In a recent article[1], the authors studied the dynamics of critical collapseof the same model as worked with by Choptuik. Approximate analytic expressionsfor the metric functions and matter field in the large-radius region wereobtained, agreeing well with the numerical results.

       It was foundthat, in the central region, owing to the boundary conditions,the equation of motion for the scalar field is reduced to the flat-spacetimeform. Specifically, the smoothness requirement at the center makes thefirst-order derivatives of the metric functions with respect to the arealradius asymptote to zero. Consequently, the terms related to gravityin the equation of motion for the scalar field are negligible. Itis true that the Ricci curvature scalar in the central region can bevery large. However, this quantity is mainly attributed to thesecond-order derivatives of the metric functions and other terms,rather than to the first-order ones.

References: [1] Jum-Qi Guo, Yu Hu, Pan-Pan Wang, and Cheng-Gang Shao, Chinese Physics C 48, 065104, 2024