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

TL;DR

This paper investigates black hole solutions in (2+1)-dimensional mimetic gravity, modifying the mimetic constraint to successfully realize black hole geometries and exploring three classes of solutions with different scalar field configurations.

Contribution

It introduces a modified mimetic constraint allowing black hole solutions in lower-dimensional gravity, expanding the understanding of mimetic gravity's capabilities.

Findings

Successfully constructed black hole solutions with modified mimetic constraint.

Demonstrated solutions with various scalar field, Lagrange multiplier, and potential configurations.

Showed the formalism remains valid with only one horizon in the solutions.

Abstract

We show that the mimetic theory with the constraint $g^{\rho \sigma}\partial_\rho\phi \partial_\sigma\phi=1$ cannot realize the black hole geometry with the horizon(s). To overcome such issue, we may change the mimetic constraint a little bit by $\omega(\phi) g^{\rho \sigma}\partial_\rho\phi \partial_\sigma\phi=-1,$ where $\omega(\phi)$ is a function of the scalar field $\phi$. 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.