On the critical behaviour of gapped gravitational collapse in confined spacetime

The gravitational collapse of a massless scalar field enclosed with a perfectly reflecting wall in a spacetime with a cosmological constant $\Lambda$ is investigated. The mass scaling for the gapped collapse $ M_{AH}-M_g \propto (\epsilon_c-\epsilon)^\xi$ is confirmed and a new time scaling for the gapped collapse $T_{AH}-T_g\propto(\epsilon_c-\epsilon)^\zeta$ is found. We find that both of these two critical exponents depend on the combination $\Lambda R^2$, where $R$ is the radial position of the reflecting wall. Especially, we find an evolution of the critical exponent $\xi$ from $0.37$ in the confined asymptotic dS case with $\Lambda R^2=1.5$ to $0.7$ in asymptotic AdS case ($\Lambda R^2\rightarrow-\infty$), while the critical exponent $\zeta$ varies from $0.10$ to $0.26$, which shows the new critical behavior for the gapped collapse is essentially different from the one in the Choptuik's case.