Black holes with Lagrange multiplier and potential in mimetic-like gravitational theory: multi-horizon black holes

In this paper, we employ the {\bf mimetic-like} field equations coupled with the Lagrange multiplier and %mimetic {\bf potential} to derive non-trivial spherically symmetric black hole (BH) solutions. We divided this study into three cases: The first one in which we take the Lagrange multiplier and %mimetic {\bf the potential} to have vanishing value and derive a BH solution that completely coincides with the BH of the Einstein general relativity despite the non-vanishing value of the {\bf mimetic-like scalar field}. The first case is completely consistent with the previous studies in the literature that mimetic theory coincides with GR \cite{Nashed:2018qag}. In the second case, we derive a solution with a constant value of the {\bf potential} and a dynamical value of the Lagrange multiplier. This solution has no horizon and therefore the obtained spacetime does not correspond to the BH. In this solution, there appears the region of the Euclidian signature where the signature of the diagonal components of the metric is $(+,+,+,+)$ or the region with two times where the signature is $(+,+,-,-)$. Finally, we derive a BH solution with non-vanishing values of the Lagrange multiplier, {\bf potential}, and {\bf mimetic-like scalar field}. This BH shows a soft singularity compared with the Einstein BH solution. The relevant physics of the third case is discussed by showing their behavior of the metric potential at infinity, calculating their energy conditions, and study their thermodynamical quantities. We give a brief discussion on how our third case can generate a BH with three horizons as in the de Sitter-Reissner-Nordstr\"om black hole spacetime, where the largest horizon is the cosmological one and two correspond to the outer and inner horizons of the BH. Even in the third case, there appears the region of the Euclidian signature or the region with two times.