# Static Einstein–Maxwell Magnetic Solitons and Black Holes in an Odd Dimensional AdS Spacetime

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## Abstract

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## 1. Introduction

## 2. Magnetic Fields in an Odd-Dimensional AdS Spacetime

## 3. Einstein–Maxwell Solutions: the Formalism

#### 3.1. The ansatz and Equations

#### 3.2. The Asymptotics

#### 3.3. The Mass Computation

#### 3.4. Other Quantities

## 4. Einstein–Maxwell Solutions: The Results

#### 4.1. The Solitons

#### 4.2. The Black Holes

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) the profiles of a typical soliton with ${c}_{m}=2$ and (

**b**) a typical black hole with ${r}_{H}=1$ and ${c}_{m}=4$. Both profiles correspond to $D=5$ and $L=1$.

**Figure 2.**(

**a**) Mass M vs. ${c}_{m}$ and (

**b**) μ vs. ${c}_{m}$ for static solitons in $D=5$ (

**red**), $D=7$ (

**blue**) and $D=9$ (

**orange**). In both figures, the dots mark the endpoints of the branch of solitons.

**Figure 3.**(

**a**) the mass M is shown vs. temperature ${T}_{H}$ for static $D=5$ black holes with several different values of the magnetic parameter ${c}_{m}$. For ${c}_{m}<{c}_{m}^{*}$, one finds both type I solutions (continuous lines) and type II solutions (dashed lines). Type II black holes can be deformed into solitons in the limit ${T}_{H}\to \infty $. For ${c}_{m}>{c}_{m}^{*}$, only Type I black holes are present and the limit ${T}_{H}\to 0$ is singular; (

**b**) the horizon area ${A}_{H}$ is shown vs. temperature ${T}_{H}$ for the same solutions. Note that for type II black holes (${c}_{m}<{c}_{m}^{*}$), the horizon shrinks to zero as ${T}_{H}\to \infty $, a limit which corresponds to a soliton deformation of the AdS background. For ${c}_{m}>{c}_{m}^{*}$, only Type I solutions are found, while the limit ${T}_{H}\to 0$ has ${A}_{H}\to 0$, being singular.

**Figure 4.**(

**a**) Deformation parameter ϵ vs. ${T}_{H}$ for static black holes in $D=5$ for different values of ${c}_{m}$ in various colors. Type I solutions are plotted with continuous lines and type II with dashed lines. Note that, in type II BHs (${c}_{m}<{c}_{m}^{*}$), the horizon becomes spherical as it shrinks when ${T}_{H}\to \infty $, forming a soliton. For ${c}_{m}>{c}_{m}^{*}$, only Type I is present, and in the limit ${T}_{H}\to 0$ the solution becomes singular; (

**b**) mass vs. ${c}_{m}$ for static solitons and black holes in $D=5$. In

**red**, we plot the solitons, and the

**red**dot marks the endpoint. The

**blue**line marks the singular limit of black holes. The

**black**dashed line separates the two types of static BHs: Type I (

**purple**area) and Type II (

**green**area). Type II black holes can be contracted to form a soliton.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Blázquez-Salcedo, J.L.; Kunz, J.; Navarro-Lérida, F.; Radu, E.
Static Einstein–Maxwell Magnetic Solitons and Black Holes in an Odd Dimensional AdS Spacetime. *Entropy* **2016**, *18*, 438.
https://doi.org/10.3390/e18120438

**AMA Style**

Blázquez-Salcedo JL, Kunz J, Navarro-Lérida F, Radu E.
Static Einstein–Maxwell Magnetic Solitons and Black Holes in an Odd Dimensional AdS Spacetime. *Entropy*. 2016; 18(12):438.
https://doi.org/10.3390/e18120438

**Chicago/Turabian Style**

Blázquez-Salcedo, Jose Luis, Jutta Kunz, Francisco Navarro-Lérida, and Eugen Radu.
2016. "Static Einstein–Maxwell Magnetic Solitons and Black Holes in an Odd Dimensional AdS Spacetime" *Entropy* 18, no. 12: 438.
https://doi.org/10.3390/e18120438