Cover Story (Issue6, 2024) |Recent development on critical collapse

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

The discovery of the critical phenomena in gravitational collapseby Choptuikisa breakthrough in numerical relativity.Choptuik studiedtheimplosionof a massless scalar field in spherical symmetry.There are two extremities in this model. At the first extremity, when the initial value of the scalar field is weak enough, the field bounces at the center and then is dispersed to infinity: a flat spacetime remains. At the other one, when the initial value is strong enough, the field will collapse to form a blackhole. Critical collapse occursin the intermediate case between these two extremities. Analytic expressions are very important for understanding the dynamics of gravitational collapse. However, the high nonlinearity of the Einstein equationsmakes it verychallengingto seek the analytic solutions to collapse.

Ina recent article [1], the authorsstudied the dynamics of critical collapse of the same model as worked with by Choptuik. Approximateanalyticexpressions for the metric functions and matter field in the large-radius region were obtained,agreeingwell with the numerical results.

It wasfound that,in the central region,owingto the boundary conditions, the equation of motion for the scalar field is reduced to the flat-spacetime form.Specifically, the smoothness requirement at the centermakesthe first-order derivativesof the metric functions with respect to the areal radius asymptote to zero.Consequently,the terms related to gravity in the equationof motionfor the scalar fieldarenegligible.It is true thatthe Ricci curvature scalarin the central region can be very large. However,this quantity is mainlyattributedtothe second-order derivatives of the metricfunctionsand other terms, rather than to the first-order ones.

References:

[1] Jun-Qi Guo, Yu Hu, Pan-Pan Wang, and Cheng-Gang Shao, ChinesePhysicsC48, 065104 (2024). [arXiv:2307.04372[gr-qc]]