# On the Traversable Yukawa–Casimir Wormholes

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## Abstract

**:**

## 1. Introduction

## 2. A Review of Casimir Wormhole

- (i)
- the non-singularity condition, $b\left(r\right)/r<1$ for $r>{r}_{0}$;
- (ii)
- the existence of the throat, $b\left(r\right)={r}_{0}$ at $r={r}_{0}$;
- (iii)
- the asymptotic flat limit, $b\left(r\right)/r\to 0$ as $r\to \infty $;
- (iv)
- the flare-out condition, ${b}^{\prime}\left(r\right)r-b\left(r\right)<0$.

## 3. Yukawa–Casimir Wormholes

- (i)
- a global modification: $b\left(r\right)=\left(\frac{2{r}_{0}}{3}+\frac{{r}_{0}^{2}}{3r}\right){e}^{-\mu (r-{r}_{0})}$;
- (ii)
- a modification only in the constant term: $b\left(r\right)=\frac{2{r}_{0}}{3}{e}^{-\mu (r-{r}_{0})}+\frac{{r}_{0}^{2}}{3r}$;
- (iii)
- a modification only in the variable term: $b\left(r\right)=\frac{2{r}_{0}}{3}+\frac{{r}_{0}^{2}}{3r}{e}^{-\mu (r-{r}_{0})}$.

#### 3.1. The Global Correction

#### 3.2. Constant Term Correction

#### 3.3. Correction in the Variable Term

#### 3.4. Stability Analysis

- (i)
- ${v}_{s-\mathrm{global}}^{2}>0\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}r\in [{r}_{0},\infty )$ if $\mu <2.56$,
- (ii)
- ${v}_{s-\mathrm{constant}}^{2}>0\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}r\in [{r}_{0},\infty )$ if $\mu <1.01$,
- (iii)
- ${v}_{s-\mathrm{radial}}^{2}>0\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}r\in [{r}_{0},\infty )$ for all values of $\mu $.

#### 3.5. Correction for Small Parameters

#### 3.5.1. First Order Correction

#### 3.5.2. Second Order Correction

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Behavior of the flare-out condition and of the moment-energy tensor components associated with the global modification. In all graphs, we have $\overline{\mu}=0.0$ (solid line), $\overline{\mu}=0.1$ (dashdot line), $\overline{\mu}=0.5$ (dashed line) and $\overline{\mu}=1.0$ (dotted line). (

**a**) flare-out condition, $[{b}^{\prime}\left(r\right)r-b\left(r\right)]/{b}^{2}\left(r\right)<0$; (

**b**) energy density, $\rho \left(r\right)$; (

**c**) radial pressure, ${p}_{r}\left(r\right)$; (

**d**) tangential pressure, ${p}_{t}\left(r\right)$.

**Figure 2.**In (

**a**), we have $\mu =0.0$ (solid line), $\mu =0.1$ (dashdot line), $\mu =0.276$ (dashed line), $\mu =0.5$ (dotted line) and $\mu =1.0$ (long dashed line). In (

**b**), the same notations as in Figure 1 are employed. (

**a**) redshift function, $\mathsf{\Phi}\left(r\right)$; (

**b**) embedding diagram, $z\left(r\right)$.

**Figure 3.**Behavior of the flare-out condition and of the moment-energy tensor components associated with the constant term correction. In all graphs, we have $\mu =0.0$ (solid line), $\mu =0.1$ (dashdot line), $\mu =0.5$ (dashed line) and $\mu =1.0$ (dotted line). (

**a**) flare-out condition, $[{b}^{\prime}\left(r\right)r-b\left(r\right)]/{b}^{2}\left(r\right)<0$; (

**b**) energy density, $\rho \left(r\right)$; (

**c**) radial pressure, ${p}_{r}\left(r\right)$; (

**d**) tangential pressure, ${p}_{t}\left(r\right)$.

**Figure 4.**In (

**a**), we have $\mu =0.0$ (solid line), $\mu =0.1$ (dashdot line), $\mu =0.28646$ (dashed line), $\mu =0.5$ (dotted line) and $\mu =1.0$ (long dashed line). In (

**b**), the same notations as in Figure 1 are employed. (

**a**) redshift function, $\mathsf{\Phi}\left(r\right)$; (

**b**) embedding diagram, $z\left(r\right)$.

**Figure 5.**Behavior of the flare-out condition and of the moment-energy tensor components associated with the correction in the variable term. In all graphs, we have $\mu =0.0$ (solid line), $\mu =0.1$ (dashdot line), $\mu =0.5$ (dashed line) and $\mu =1.0$ (dotted line). (

**a**) flare-out condition, $[{b}^{\prime}\left(r\right)r-b\left(r\right)]/{b}^{2}\left(r\right)$; (

**b**) energy density, $\rho \left(r\right)$; (

**c**) radial pressure, ${p}_{r}\left(r\right)$; (

**d**) tangential pressure, ${p}_{t}\left(r\right)$.

**Figure 6.**In all graphs we have $\mu =0.0$ (solid line), $\mu =0.1$ (dashdot line), $\mu =0.5$ (dashed line) and $\mu =1.0$ (dotted line); (

**a**) redshift function, $\mathsf{\Phi}\left(r\right)$; (

**b**) embedding diagram, $z\left(r\right)$.

**Figure 7.**Plot of ${v}_{s}^{2}\left(r\right)$ versus r. In all graphs, we have $\mu =0.0$ (solid line), $\mu =0.5$ (dashdot line) and $\mu =1.0$ (long dashed line). In (

**a**), we have $\mu =2.56$ (dashed line) and $\mu =3.0$ (dotted line). In (

**b**), we have $\mu =1.01$ (dashed line) and $\mu =1.5$ (dotted line). In (

**c**), we have $\mu =1.5$ (dashed line) and $\mu =3.0$ (dotted line). (

**a**) global correction; (

**b**) constant correction; (

**c**) correction in the variable term.

**Figure 8.**Behavior of the second-order modified redshift function on the parameter $\mu $ for Yukawa–Casimir WH.

**Figure 9.**Quantum Weak Energy Condition (QWEC) check. In all graphs, we have $\mu =0.0$ (solid line), $\mu =0.1$ (dashdot line), $\mu =0.5$ (dashed line) and $\mu =1.0$ (dotted line). (

**a**) global correction; (

**b**) constant correction; (

**c**) correction in the variable term.

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**MDPI and ACS Style**

Oliveira, P.H.F.d.; Alencar, G.; Jardim, I.C.; Landim, R.R.
On the Traversable Yukawa–Casimir Wormholes. *Symmetry* **2023**, *15*, 383.
https://doi.org/10.3390/sym15020383

**AMA Style**

Oliveira PHFd, Alencar G, Jardim IC, Landim RR.
On the Traversable Yukawa–Casimir Wormholes. *Symmetry*. 2023; 15(2):383.
https://doi.org/10.3390/sym15020383

**Chicago/Turabian Style**

Oliveira, Pedro Henrique Ferreira de, Geová Alencar, Ivan Carneiro Jardim, and Ricardo Renan Landim.
2023. "On the Traversable Yukawa–Casimir Wormholes" *Symmetry* 15, no. 2: 383.
https://doi.org/10.3390/sym15020383