# Thin-Shell Wormholes in Einstein and Einstein–Gauss–Bonnet Theories of Gravity

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## Abstract

**:**

## 1. Introduction

#### 1.1. Einstein–Rosen Bridge

- (a)
- Photons $\alpha $ and $\beta $ initially are in the lower sheet. They go to the center $r=0$. The values $u=-2.67$ and $u=-2.08$ correspond to $r=2M$ and $r=0$, respectively. The vertical bold line is the curvature singularity $r=0$. At this moment, the singularity is in between two quasi-Euclidean spaces.
- (b)
- Both photons go to the center. A throat is going to appear.
- (c)
- The throat just opened. The circumference of the throat is smaller than $4\pi M$.
- (d)
- The maximal throat, $2\pi r=4\pi M$. The photon $\alpha $ has passed though the throat.
- (e)
- The throat is shrinking. Both photons have passed though the throat.
- (f)
- The moment of the throat closing. In this stage, both photons are still in the upper sheet, while the photon $\beta $ approaches the central singularity.
- (g)
- Photon $\beta $ is just caught. Then, $\beta $ disappears in the singularity and stops existing.
- (h)
- Photon $\alpha $ keeps escaping to the null infinity of the upper sheet.

#### 1.2. Wormhole Properties in Brief

#### 1.2.1. Embedding Wormholes and Asymptotic Flatness

#### 1.2.2. Flaring-Out Condition

#### 1.2.3. Absence of the Horizon

#### 1.2.4. Magnitude of the Tension at the Throat

#### 1.2.5. Exotic Matter

#### 1.2.6. Other Properties

#### 1.3. Simple Exact Solutions and Their Stability

## 2. Thin-Shell Wormholes

#### 2.1. Junction Conditions

#### 2.2. Construction

#### 2.3. Equation of Motion for the Shell

#### 2.4. Simplest Thin-Shell Wormhole

#### 2.5. Stability

#### 2.5.1. Global Stability

#### 2.5.2. Local Stability

- (1)
- There are stable solutions in ${\beta}_{0}\ge (3+\sqrt{3})/2$ or ${\beta}_{0}<-1/2$.
- (2)
- No solution in $(3+\sqrt{3})/2>{\beta}_{0}>-1/2$ is stable.
- (3)
- The solution at ${a}_{0}=3M$ is unstable regardless of the value of ${\beta}_{0}$.

#### 2.5.3. Pure Tension

## 3. Generalized Thin-Shell Wormholes

#### 3.1. Charged Generalization

#### 3.2. Presence of a Cosmological Constant

#### 3.2.1. Schwarzschild–de Sitter Thin-Shell Wormhole: $\Lambda >0$

#### 3.2.2. Schwarzschild–Anti de Sitter Thin-Shell Wormhole: $\Lambda <0$

#### 3.2.3. (Anti) de Sitter Thin-Shell Wormhole

#### 3.3. Non-${Z}_{2}$ Symmetric Case

#### 3.4. In Higher Dimensions

## 4. Pure Tension Wormholes in Einstein Gravity

#### 4.1. Einstein Gravity

#### 4.2. Advantage of Use of Pure Tension

#### 4.3. Pure Tension Wormholes in Einstein Gravity

#### 4.4. Effective Potential

#### 4.5. Static Solutions and Stability Criterion

## 5. Pure Tension Wormholes in Einstein–Gauss–Bonnet Gravity

#### 5.1. Einstein–Gauss–Bonnet Gravity

#### 5.2. Pure Tension Wormholes in Einstein–Gauss–Bonnet Gravity

#### 5.3. Bulk Solution

#### 5.4. Equation of Motion for a Thin-Shell

#### 5.5. Effective Potential for the Shell

#### 5.6. Negative Energy Density of the Shell

#### 5.7. Static Solutions

#### 5.8. Stability Criterion

#### 5.8.1. Einstein Gravity

#### 5.8.2. Einstein–Gauss–Bonnet Gravity

#### 5.9. Effect of the Gauss–Bonnet Term on the Stability

#### 5.10. Stability Analysis

## 6. Conclusions

#### 6.1. In Einstein Gravity

#### 6.2. In Einstein–Gauss–Bonnet Gravity

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | Recently, the instability of the present wormhole was reported by [76]. |

**Figure 1.**(

**Left**) The plot of (6). (

**Right**) The surface is obtained by rotating the function around z axis in the $\varphi $ direction. The narrowest surface $z=0$ corresponds to $r=2M$.

**Figure 3.**The red dotted line is the geodesic of photon $\alpha $ from Region IV, while the blue one is that of photon $\beta $ from Region III. After passing through the anti horizon $v=-u$, photon $\alpha $ goes to Region I and never across the event horizon $v=u$. The geodesic of photon $\beta $ must terminate at the singularity $r=0$ in a finite time in Region II.

**Figure 4.**The dynamics of the throat and the motion of photons. A throat emerges instantaneously and connects two asymptotically flat space-sheets. After that, it expands and then starts to contract. Finally, it pinches off the connection between the two space-sheets.

**Figure 5.**A Penrose diagram for a two-way traversable spacetime. The center vertical line describes a wormhole throat which connects the left and the right regions. From the above picture, a timelike traveller clearly can pass through the throat to go to the other region and also come back to the original region.

**Figure 6.**The embedding of the Morris–Thorne type metric. In general, wormholes do not have a mirror symmetry (such as the Einstein–Rosen bridge) as long as the flaring out condition is satisfied.

**Figure 8.**If ${\beta}_{0}$ is given, one knows the range of ${a}_{0}$ for the stable static throat solution. Region ${a}_{0}<2M$ is unphysical since such region does not exist in the wormhole. The upper broken line is ${\beta}_{0}=(3+\sqrt{3})/2$ while lower is ${\beta}_{0}=-1/2$. Shaded regions correspond to stable regions. We draw the horizontal line with ${\beta}_{0}=4$ for an example. In this case, wormhole can be stable if $2.19M<{a}_{0}<2.59M$.

**Figure 9.**Charged thin-shell wormholes. Shaded regions correspond to stable regions. The outer horizon ${r}_{+}$ corresponds to ${r}_{+}/M=1+\sqrt{1-{\left|Q\right|}^{2}/{M}^{2}}$; therefore, the regions inside of the outer horizons (vertical lines) have no physical meaning in these figures when $0\le \left|Q\right|/M\le 1$.

**Figure 11.**Schwarzschild–anti de Sitter thin-shell wormhole. The shaded regions indicate the stability.

**Figure 12.**De-Sitter wormholes. A stable region is below the curve. Each vertical line represents the cosmological horizon. Hence, the right side of the vertical line is an unphysical region.

**Figure 13.**Anti de Sitter wormholes. Stable region is below the curve. Figure (1) describes a wormhole in the Minkowski spacetime.

**Figure 14.**Non-${Z}_{2}$ symmetry. The shaded regions represent stability. The area of stable regions decreases with increasing the difference between ${M}_{+}$ and ${M}_{-}$. The horizontal axis is normalized by ${M}_{+}$.

**Figure 16.**(1) The potential $\tilde{V}\left(a\right)$ for $d=4$, $k=+1$ and $M>0$. The dashed line is the potential for the critical value defined in Equation (87). (2) The potential for $d=5$, $k=+1$ and $M>0$.

**Figure 17.**The potential $\overline{V}\left(a\right)$ for $d=5,6,7$ in Einstein and Einstein–Gauss–Bonnet (EGB) gravity with $k=1$, $\alpha =0.02$, $m=1$, $\Lambda =1$ and $\sigma =-0.1$.

Static Solution | Horizon Avoidance | Stability | |
---|---|---|---|

$k-M+{Q}^{2}=0$ | $\forall {a}_{0}>0$ | Satisfied | Marginally stable |

$k-M+{Q}^{2}\ne 0$ | None | – | – |

**Table 2.**The existence and stability of static wormholes in spherical symmetry in four and higher dimensions.

Static Solution | Horizon Avoidance | Stability | ||
---|---|---|---|---|

$q=0$ | ${[(d-1)M/2]}^{1/(d-3)}$ | $\lambda <{H}_{+}(d,0)$ | Unstable | |

${a}_{0-}$: Stable | ||||

$M>0$ | $0<q<{q}_{c}$ | ${a}_{0\pm}$ | $\lambda <{H}_{\pm}(d,q)$ | ${a}_{0+}$: Unstable |

$q={q}_{c}$ | ${[(d-1)M/4]}^{1/(d-3)}$ | $\lambda <R\left(d\right)$ | Unstable | |

${q}_{c}<q$ | None | – | – | |

$M\le 0$ | None | – | – |

**Table 3.**The existence and stability of static wormholes in planar symmetry in four and higher dimensions.

Static Solution | Horizon Avoidance | Stability | ||
---|---|---|---|---|

$M>0$ | $q=0$ | None | – | – |

$q>0$ | ${[2(d-2){q}^{2}M/(d-1)]}^{1/(d-3)}$ | $\lambda <J(d,q)$ | Stable | |

$M<0$ | $q\ge 0$ | None | – | – |

Marginally | ||||

$M=0$ | $Q=0$ | $\forall {a}_{0}>0$ | Satisfied | stable |

$\left|Q\right|>0$ | None | – | – |

**Table 4.**The existence and stability of static wormholes in hyperbolic symmetry in four and higher dimensions.

Static Solution | Horizon Avoidance | Stability | ||
---|---|---|---|---|

$M>0$ | $q=0$ | None | – | – |

$q>0$ | ${a}_{0-}$ | $\lambda <I(d,q)$ | Stable | |

$M<0$ | $q=0$ | ${\left[(d-1)\right|M|/2]}^{1/(d-3)}$ | $\lambda <N(d,0)$ | Stable |

$q>0$ | ${a}_{0+}$ | $\lambda <N(d,q)$ | Stable | |

$M=0$ | $Q=0$ | None | – | – |

$\left|Q\right|>0$ | $[\sqrt{d-2}{\left|Q\right|]}^{1/(d-3)}$ | $\Lambda /3<S(d,q)$ | Stable |

Existence | Possible Range of ${\mathit{a}}_{0}$ | Stability | ||
---|---|---|---|---|

$k=1$ | $\Lambda >0$ | $0<m<{m}_{\mathrm{c}}^{\left(\mathrm{GR}\right)}$ | $0<{a}_{0}<{a}_{\mathrm{c}}^{\left(\mathrm{GR}\right)}$ | Unstable |

$\Lambda \le 0$ | $m>0$ | ${a}_{0}>0$ | Unstable | |

$k=0$ | $\Lambda \ge 0$ | None | – | – |

$\Lambda <0$ | $m=0$ | ${a}_{0}>0$ | Marginally stable | |

$k=-1$ | $\Lambda \ge 0$ | None | – | – |

$\Lambda <0$ | $m<{m}_{\mathrm{c}}^{\left(\mathrm{GR}\right)}(<0)$ | ${a}_{0}>{a}_{\mathrm{c}}^{\left(\mathrm{GR}\right)}$ | Stable |

**Table 6.**The existence and stability of Z${}_{2}$ symmetric static thin-shell wormholes made of pure negative tension in the GR branch with $\tilde{\alpha}>0$ and $1+4\tilde{\alpha}\tilde{\Lambda}>0$. “S“, “M” and “U” stand for “Stable”, “Marginally stable” and “Unstable”, respectively.

Static Solutions Exist? | Stability | ||
---|---|---|---|

$k=1$ | $m>0$ | Yes | U |

$m\le 0$ | No | – | |

$k=0$ | $m=0$ | $\Lambda \ge 0$: No | – |

$\Lambda <0$: Yes | M | ||

$m\ne 0$ | No | – | |

$k=-1$ | $m\ge 0$ | No | – |

$m<0$ | $\Lambda \ge 0$: No | – | |

$-(2d-5)/(2d-1)<4\tilde{\alpha}\tilde{\Lambda}<0$: Yes | S | ||

$4\tilde{\alpha}\tilde{\Lambda}=-(2d-5)/(2d-1)$: Yes | S or M | ||

$-1<4\tilde{\alpha}\tilde{\Lambda}<-(2d-5)/(2d-1)$ with $d=5$: Yes | S or M | ||

$-1<4\tilde{\alpha}\tilde{\Lambda}<-(2d-5)/(2d-1)$ with $d\ge 6$: Yes | S, M, or U |

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Kokubu, T.; Harada, T.
Thin-Shell Wormholes in Einstein and Einstein–Gauss–Bonnet Theories of Gravity. *Universe* **2020**, *6*, 197.
https://doi.org/10.3390/universe6110197

**AMA Style**

Kokubu T, Harada T.
Thin-Shell Wormholes in Einstein and Einstein–Gauss–Bonnet Theories of Gravity. *Universe*. 2020; 6(11):197.
https://doi.org/10.3390/universe6110197

**Chicago/Turabian Style**

Kokubu, Takafumi, and Tomohiro Harada.
2020. "Thin-Shell Wormholes in Einstein and Einstein–Gauss–Bonnet Theories of Gravity" *Universe* 6, no. 11: 197.
https://doi.org/10.3390/universe6110197