Dirty black hole supported by a uniform electric field in Einstein-nonlinear electrodynamics-Dilaton theory

S. Habib Mazharimousavi

doi:10.1140/epjc/s10052-023-11544-5

2024 Impact factor 4.8

Particles and Fields

Open Access

This article has an erratum: [https://doi.org/10.1140/epjc/s10052-023-11723-4]

Eur. Phys. J. C (2023) 83: 406
https://doi.org/10.1140/epjc/s10052-023-11544-5

Regular Article - Theoretical Physics

Dirty black hole supported by a uniform electric field in Einstein-nonlinear electrodynamics-Dilaton theory

S. Habib Mazharimousavi^a

Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Türkiye

^a habib.mazhari@emu.edu.tr

Received: 9 January 2023
Accepted: 23 April 2023
Published online: 14 May 2023

Abstract

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., $L \sim \sqrt{- F} .$ ${\mathcal {L}}\sim \sqrt{-{\mathcal {F}}}.$ 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 $d s^{2} = - (1 - \frac{2 M}{ρ^{η^{2}}}) ρ^{2 (η^{2} - 1)} d τ^{2} + {(1 - \frac{2 M}{ρ^{η^{2}}})}^{- 1} d ρ^{2} + ϰ^{2} ρ^{2} d Ω^{2}$ $ds^{2}=-\left( 1-\frac{2\,M}{\rho ^{\eta ^{2}}} \right) \rho ^{2\left( \eta ^{2}-1\right) }d\tau ^{2}+\left( 1-\frac{2\,M}{ \rho ^{\eta ^{2}}}\right) ^{-1}d\rho ^{2}+\varkappa ^{2}\rho ^{2}d\Omega ^{2}$ with two parameters – namely the mass M and the Dilaton parameter $η^{2} > 1$ $\eta ^{2}>1$ ( $ϰ^{2} = \frac{1}{η^{2}}$ $\varkappa ^{2}=\frac{1}{\eta ^{2}}$ ) – which may be simply considered as the dirty Schwarzschild black hole. This is because with $η^{2} \to 1$ $\eta ^{2}\rightarrow 1$ 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 $η^{2} > 2$ $\eta ^{2}>2$ . Furthermore, the tidal force of this black hole behaves the same as a Schwarzschild black hole, however, its magnitude depends on $η^{2}$ $\eta ^{2}$ such that its minimum is not corresponding to $η^{2} = 1$ $\eta ^{2}=1$ (Schwarzschild limit).

The original online version of this article was revised: In this article the wrong figure appeared as Fig.5 with the wrong caption. The correct caption reads: The radial tidal force in terms of $\frac{R}{R +}$ $\frac{R}{R+}$ and $η$ $\eta$ . This figure implies the radial tidal force is always positive and in terms of $\frac{R}{R +}$ $\frac{R}{R+}$ for a given $η$ $\eta$ , it is a monotonic decreasing function that approaches zero. On the other hand for a given $\frac{R}{R +}$ $\frac{R}{R+}$ , the radial tidal force first decreases in terms of η and then increases. Figure 6 has been published with the wrong caption as well. The correct caption reads: The angular tidal force in terms of $\frac{R}{R +}$ $\frac{R}{R+}$ and $η$ $\eta$ . The angular tidal force is always negative and in terms of $\frac{R}{R +}$ $\frac{R}{R+}$ for a given $η$ $\eta$ , it is a monotonic increasing function that approaches zero. On the other hand for a given $\frac{R}{R +}$ $\frac{R}{R+}$ , the angular tidal force first decreases in terms of $η$ $\eta$ and then increases.

An erratum to this article is available online at https://doi.org/10.1140/epjc/s10052-023-11723-4.

Copyright comment corrected publication 2023

Erratum to this article: https://doi.org/10.1140/epjc/s10052-023-11723-4

© The Author(s) 2023. corrected publication 2023

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