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

In this paper, we consider the mimetic-like field equations coupled with the Lagrange multiplier and the potential to derive non-trivial spherically symmetric black hole (BH) solutions. We divided this study into three cases: in the first one, we choose the Lagrange multiplier and the potential to vanish and derive a BH solution that coincides with the BH of the Einstein general relativity despite the non-vanishing value of the mimetic-like scalar field. The first case is consistent with the previous studies in the literature where the mimetic theory coincides with GR [1]. In the second case, we derive a solution with a constant value of the potential and a dynamical value of the Lagrange multiplier. This solution has no horizon, and therefore, the obtained space-time does not correspond to the BH. In this solution, there appears a 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, potential, and 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 studying 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-Nordstrom black hole space-time, 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, the region of the Euclidian signature or the region with two times appears. We give a condition that such unphysical region(s) is hidden inside the black hole horizon and the existence of the region(s) becomes less unphysical. We also study the thermodynamics of the multi-horizon BH and consider the extremal case, where the radii of two horizons coincide with each other. We observe that the Hawking temperature and the heat capacity vanish in the extremal limit. Finally, we would like to stress the fact that in spite that the field equations we use have no cosmological constant, our BH solutions of the second and third case behave asymptotically as AdS/dS.