Generating black holes in $4D$ Einstein-Gauss-Bonnet gravity

In recent times there is a surge of interest in constructing Einstein-Gauss-Bonnet (EGB) gravity, in the limit $D\to 4$, of the $D$-dimensional EGB gravity. Interestingly, the static spherically symmetric solutions in the various proposed $D\to 4$ regularized EGB gravities coincide, and incidentally some other theories also admit the same solution. We prove a theorem that characterizes a large family of nonstatic or radiating spherically symmetric solutions to the $4D$ EGB gravity, representing, in general, spherically symmetric Type II fluid. An extension of the theorem, given without proof as being similar to the original theorem, generates static spherically symmetric black hole solutions of the theory. It not only enables us to identify available known black hole solutions as particular cases but also to generate several new solutions of the $4D$ EGB gravity.