Trapping a geon: Black hole formation by an imploding gravitational wave

We describe the formation of a black hole via the implosion of an axisymmetric gravitational wave. Finite difference simulations of the vacuum Einstein equations are used to obtain these results. The initial data consist of nearly linear solutions to the vacuum constraint equations that represent even-parity, ingoing wave packets with quadrupole angular dependence. A black hole is demonstrated to form as a result of imploding a wave packet with a sufficiently large value of a strength parameter, $\Theta \equiv \frac{2\pi M_p}{\lambda }=1.06>{\Theta }_{\mathrm{crit}}\simeq 0.80$, where $2\lambda$ is the radial width of the wave packet and $M_p$ denotes its mass. Black hole formation is verified by observing (i) the exponential collapse of the central value of the lapse function $\alpha$, (ii) the formation of a trapped region and marginally outer-trapped surfaces, and (iii) the emission of quasi-normal-mode radiation. For the $\Theta =1.06\mathrm{}$ case, just over 2% of the mass emerges in normal-mode radiation.