Weak cosmic censorship with $SU(2)$ gauge field and bound on charge-to-mass ratio

Abstract

We numerically construct the stationary solutions of $SU(2)$ Einstein-Yang-Mills theory in four dimensional anti-de Sitter spacetime. When the $t$ component of $SU(2)$ gauge field is taken to the only nonzero component, we construct a class of counterexamples to the weak cosmic censorship conjecture in Einstein-Maxwell theory. However, including a nonzero $\phi$ component of $SU(2)$ gauge field, we can argue that there is a minimum value $q_W$, when the charge carried by the $\phi$ component is larger than this minimum value, for sufficiently large boundary electric amplitude $a$, the original counterexamples can be removed and cosmic censorship is preserved.