Consistency between black hole and mimetic gravity – Case of (2 + 1)-dimensional gravity

We show that the mimetic theory with the constraint gρσ∂ρϕ∂σϕ=1 cannot realize the black hole geometry with the horizon(s). To overcome such issue, we may change the mimetic constraint a little bit by ω(ϕ)gρσ∂ρϕ∂σϕ=−1, where ω(ϕ) is a function of the scalar field ϕ. As an example, we consider (2+1)-dimensional mimetic gravity with the mimetic potential and construct black hole (BH) solutions by using this modified constraint. We study three different classes: In the first class, we assume the Lagrange multiplier and mimetic potential are vanishing and obtain a BH solution that fully matches the BH of GR despite the non-triviality of the mimetic field which ensures the study presented in Nashed et al. (2019) 42. In the second class, we obtain a BH having constant mimetic potential and a non-trivial form of the Lagrange multiplier. In the third class, we obtain a new BH solution with non-vanishing values of the mimetic field, the Lagrange multiplier, and the mimetic potential. In any case, the solutions correspond to the space-time with only one horizon but we show that the formalism for the constraint works.