Testing Cosmic Censorship Conjecture for Extremal and Near-extremal (2+1)-dimensional MTZ Black Holes

We test the validity of the weak cosmic censorship conjecture for the ($2+1$)-dimensional charged anti-de Sitter black hole solution, which was derived by Martinez, Teitelboim, and Zanelli (MTZ). We first construct a thought experiment by throwing test charged particles on an extremal MTZ black hole. We derive that extremal ($2+1$) dimensional black holes can be overcharged by test particles, unlike their analogues in $4$ and higher dimensions. Nearly-extremal black holes can also be overcharged, by a judicious choice of energy and charge for the test particles in the case when we ignore the second order effects. Contrary to this, nearly-extremal black holes cannot be overcharged by the second order perturbation, obeying the weak cosmic censorship conjecture.