Dirty black hole supported by a uniform electric field in Einstein-Nonlinear Electrodynamics-Dilaton theory

In this study, we present an exact dirty/hairy black hole solution in the context of gravity coupled minimally to a nonlinear electrodynamic (NED) and a Dilaton field. The NED model is known in the literature as the square-root (SR) model i.e., $\mathcal{L}\sim \sqrt{-\mathcal{F}}.$ The black hole solution which is supported by a uniform radial electric field and a singular Dilaton scalar field is non-asymptotically flat and singular with the singularity located at its center. An appropriate transformation results in an interesting line element $ds^{2}=-\left( 1-\frac{2M}{\rho ^{\eta ^{2}}} \right) \rho ^{2\left( \eta ^{2}-1\right) }d\tau ^{2}+\left( 1-\frac{2M}{ \rho ^{\eta ^{2}}}\right) ^{-1}d\rho ^{2}+\varkappa ^{2}\rho ^{2}d\Omega ^{2}$ with two parameters - namely the mass $M$ and the Dilaton parameter $\eta ^{2}>1$ ($\varkappa ^{2}=\frac{1}{\eta ^{2}}$) - which may be simply considered as the dirty Schwarzschild black hole. This is because with $\eta ^{2}\rightarrow 1$ the spacetime reduces to the Schwarzschild black hole. We show that although the causal structure of the above spacetime is similar to the Schwarzschild black hole, it is thermally stable for $\eta ^{2}>2$. Furthermore, the tidal force of this black hole behaves the same as a Schwarzschild black hole, however, its magnitude depends on $\eta ^{2}$\ such that its minimum is not corresponding to $\eta ^{2}=1$\ (Schwarzschild limit).