# Scalarized Nutty Wormholes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Setting

#### 2.1. Action and Equations of Motion

#### 2.2. Throats, Equators, and Boundary Conditions

#### 2.3. Junction Conditions

#### 2.4. Energy Conditions

## 3. Results

#### 3.1. Numerics

#### 3.2. Solutions

#### 3.3. Domain of Existence

#### 3.4. Throat Properties

#### 3.5. Junction Conditions and Critical Polar Angle

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**Examples of nutty wormhole solutions (left plots: Gauss–Bonnet with parameters $\alpha /{M}^{2}=2.5$, $D/M=2$ and $n=N/M=3$, right plots: Chern–Simons with $\alpha /{M}^{2}=4$, $D/M=1$ and $n=N/M=3$): (

**a**,

**b**) metric profile functions ${e}^{{f}_{0}}$, ${e}^{{F}_{1}}$, scalar field function $\varphi $, and scaled circumferential radius ${R}_{c}/{\eta}_{0}$ vs radial coordinate $\eta $; (

**c**,

**d**) stress-energy tensor components ${T}_{t}^{t}$, ${T}_{\varphi}^{t}$, ${T}_{\eta}^{\eta}$, and ${T}_{\theta}^{\theta}$ vs. radial coordinate $\eta /M$; (

**e**,

**f**) NEC conditions $-{T}_{t}^{t}+{T}_{\eta}^{\eta}\ge 0$ and $-{T}_{t}^{t}+{T}_{\theta}^{\theta}\ge 0$ vs radial coordinate $\eta $.

**Figure 2.**Domain of existence ((

**left**) plot: Gauss–Bonnet, (

**right**) plot: Chern–Simons) for several values of the scaled NUT charge $n=N/M$: scaled coupling constant $\alpha /{M}^{2}$ vs scaled scalar charge $D/M$. The solid red curves represent the black hole limit, the dashed green curves the degenerate wormhole limit, and the dotted blue curves the singular limit.

**Figure 3.**Properties at the throat (left plots: Gauss–Bonnet, right plots: Chern–Simons) for several values of the scaled NUT charge $n=N/M$: (

**a**,

**b**) scaled circumferential radius ${R}_{\mathrm{th}}/M$, (

**c**,

**d**) metric function ${e}^{{f}_{0}}$, (

**e**,

**f**) scalar field ${\varphi}_{\mathrm{th}}$ vs scaled scalar charge $D/M$. The solid red curves represent the black hole limit, the dashed green curves the degenerate wormhole limit, and the dotted blue curves the singular limit.

**Figure 4.**Properties at the throat (left plots: Gauss–Bonnet, right plots: Chern–Simons) for scaled NUT charge $n=N/M=1$ and several values of the scaled scalar charge $D/M$: (

**a**,

**b**) energy density ${\u03f5}_{\mathrm{th}}$, (

**c**,

**d**) critical polar angle ${\theta}_{\mathrm{th}}$ vs scaled coupling constant $\alpha /{M}^{2}$. The dashed green curves represent the degenerate wormhole limit, and the dotted blue curves the singular limit.

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Ibadov, R.; Kleihaus, B.; Kunz, J.; Murodov, S.
Scalarized Nutty Wormholes. *Symmetry* **2021**, *13*, 89.
https://doi.org/10.3390/sym13010089

**AMA Style**

Ibadov R, Kleihaus B, Kunz J, Murodov S.
Scalarized Nutty Wormholes. *Symmetry*. 2021; 13(1):89.
https://doi.org/10.3390/sym13010089

**Chicago/Turabian Style**

Ibadov, Rustam, Burkhard Kleihaus, Jutta Kunz, and Sardor Murodov.
2021. "Scalarized Nutty Wormholes" *Symmetry* 13, no. 1: 89.
https://doi.org/10.3390/sym13010089