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

We show that the mimetic theory with the constraint $g^{ρσ}\partial_ρϕ\partial_σϕ=1$ cannot realize the black hole geometry with the horizon(s). To overcome such issue, we may change the mimetic constraint a little bit by $ω(ϕ) g^{ρσ}\partial_ρϕ\partial_σϕ=-1,$ 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 {\it JCAP 01 (2019) 058}. 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.

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