# Critical Solutions of Scalarized Black Holes

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## Abstract

**:**

## 1. Introduction

## 2. EMs Theory

## 3. Limit of Cold Black Holes

#### 3.1. Branches of Black Holes

#### 3.2. Approach to the Critical Solution for $\alpha =200$

#### 3.3. $\alpha $-Dependence of the Critical Solution

## 4. Excited Solutions

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Phase diagram of scalarized black holes: mass to charge ratio q vs. coupling constant $\alpha $. (

**b**) Reduced horizon area ${a}_{H}$ (inset: reduced temperature ${t}_{H}$) vs. q for $\alpha =200$: cold branch (dotted blue), hot branch (dashed red) and RN branch (solid black).

**Figure 2.**Approach to the critical solution for $\alpha =200$: (

**a**) metric function $g\left(r\right)=N\left(r\right){e}^{-2\delta \left(r\right)}$, (

**b**) metric function $N\left(r\right)=1-\frac{2m\left(r\right)}{r}$, (

**c**) electromagnetic function ${A}_{t}\left(r\right)$ and (

**d**) scalar function $\Phi \left(r\right)$ vs. the compactified radial coordinate $x=1-\frac{{r}_{H}}{r}$. The insets highlight the vicinity of the critical radius $r={r}_{cr}$.

**Figure 3.**Approach to the critical solution for $\alpha =200$: (

**a**) radial coordinate r, (

**b**) metric function $N\left(r\right)=1-\frac{2m\left(r\right)}{r}$, (

**c**) electromagnetic function ${A}_{t}\left(r\right)$ and (

**d**) scalar function $\Phi \left(r\right)$ vs. the radial distance $l\left(r\right)$ (Equation (19)).

**Figure 4.**Critical solutions for a set of couplings $\alpha $: (

**a**) metric function $N\left(r\right)=1-\frac{2m\left(r\right)}{r}$ and (

**b**) scalar function $\Phi \left(r\right)$ vs. the compactified radial coordinate $x=1-\frac{{r}_{H}}{r}$. Note that the key applies to both figures.

**Figure 5.**Properties of critical solutions: (

**a**) ${Q}_{cr}\left(\alpha \right)$ (solid red) and limit for $\alpha \to \infty $ according to Equation (20) (dashed green), (

**b**) $\Phi \left({r}_{H}\right)\left(\alpha \right)$ (solid red) and relation from Equation (21) (dashed green), (

**c**) ${\Phi}^{\prime}\left({r}_{H}\right)\left(\alpha \right)$ (solid red) and expansion coefficient ${\Phi}_{1}\left({r}_{H}\right)\left(\alpha \right)$ from Equation (14) (dashed green), and (

**d**) ${m}^{\prime}\left({r}_{H}\right)\left(\alpha \right)$ (solid red) and expansion coefficient ${m}_{1}\left({r}_{H}\right)\left(\alpha \right)$ from Equation (14) (dashed green).

**Figure 6.**(

**a**) Reduced horizon area ${a}_{H}$ vs. q for $\alpha =200$: In addition to the RN branch (solid black), the ($n=0$) cold branch (finely dotted blue) and the ($n=0$) hot branch (coarsely dotted red), the $n=1$ excited solutions (solid green) are shown. (

**b**) A similar figure for the scalar field at the horizon ${\Phi}_{H}$ vs. reduced temperature ${t}_{H}$.

**Figure 7.**Critical solutions with $n=1$ for a set of couplings $\alpha $: (

**a**) metric function $N\left(r\right)=1-\frac{2m\left(r\right)}{r}$ and (

**b**) scalar function $\Phi \left(r\right)$ vs. the compactified radial coordinate $x=1-\frac{{r}_{H}}{r}$.

**Figure 8.**Critical solutions with $n=2$ for a set of couplings $\alpha $: (

**a**) metric function $N\left(r\right)=1-\frac{2m\left(r\right)}{r}$ and (

**b**) scalar function $\Phi \left(r\right)$ vs. the compactified radial coordinate $x=1-\frac{{r}_{H}}{r}$.

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Blázquez-Salcedo, J.L.; Kahlen, S.; Kunz, J.
Critical Solutions of Scalarized Black Holes. *Symmetry* **2020**, *12*, 2057.
https://doi.org/10.3390/sym12122057

**AMA Style**

Blázquez-Salcedo JL, Kahlen S, Kunz J.
Critical Solutions of Scalarized Black Holes. *Symmetry*. 2020; 12(12):2057.
https://doi.org/10.3390/sym12122057

**Chicago/Turabian Style**

Blázquez-Salcedo, Jose Luis, Sarah Kahlen, and Jutta Kunz.
2020. "Critical Solutions of Scalarized Black Holes" *Symmetry* 12, no. 12: 2057.
https://doi.org/10.3390/sym12122057