# Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss–Bonnet Gravity

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## Abstract

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## 1. Introduction

## 2. The Gauss–Bonnet Gravity in Spherical Symmetry

#### Spherical Symmetry

## 3. The Noether Symmetry Approach

## 4. Noether Symmetries in Gauss–Bonnet Gravity

- Case 1: in $d\mathcal{G}eq3$ dimensions, we have $f\left(\mathcal{G}\right)={f}_{0}{\mathcal{G}}^{n}$ with $n\ne 1$. The Noether symmetry of this model is given by$$X={c}_{1}r{\partial}_{r}-4{c}_{1}\mathcal{G}{\partial}_{\mathcal{G}}+(4n-d){c}_{1}P{\partial}_{P}\phantom{\rule{0.166667em}{0ex}}$$$$\begin{array}{cc}\hfill I=& \frac{1}{{Q}^{3}}{c}_{1}{f}_{0}{r}^{d-4}{\mathcal{G}}^{n-2}[(1-n){r}^{4}{\mathcal{G}}^{2}P{Q}^{4}-4n(n-1)(d-2)(d-1)r\left(\right)open="("\; close=")">{Q}^{2}-1\hfill & \left(\right)open="("\; close=")">r{P}^{\prime}+(d-4n)P\\ {\mathcal{G}}^{\prime}-\end{array}]-{c}_{2}.$$
- Case 2: in $d=4$ dimensions, there is also the possibility of having a linear model of the form $f\left(\mathcal{G}\right)={f}_{0}\mathcal{G}$. Its Noether symmetry reads$$X={c}_{1}r{\partial}_{r}+{c}_{2}{\partial}_{P}\phantom{\rule{0.166667em}{0ex}},$$$$I=-\frac{8{f}_{0}}{{Q}^{3}}\left(\right)open="("\; close=")">\frac{6r\left(\right)open="("\; close=")">{Q}^{2}-3}{\left(\right)}{Q}^{\prime}+{c}_{2}\left(\right)open="("\; close=")">3{Q}^{2}-5$$

#### Spherically Symmetric Solutions

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Exact static and spherically symmetric solutions in $f\left(G\right)$ gravity, for $f\left(\mathcal{G}\right)={f}_{0}{\mathcal{G}}^{n}$ in arbitrary $d+1$ dimensions.

P(r)^{2} | Q(r)^{2} | d | n |
---|---|---|---|

$1+{e}^{-2{c}_{2}}\sqrt{{c}_{1}-4r}{r}^{\frac{3}{2}-\frac{d}{2}}$ | $1/P{\left(r\right)}^{2}$ | $d\mathcal{G}e3$ | $n>0,\ne 1$ |

$P}_{0}^{2}\left(\right)open="("\; close=")">1-\frac{{k}_{3}}{{r}^{\frac{d}{2}-2}$ | $1/P{\left(r\right)}^{2}$ | $d>3$ | $n=1$ |

$1\pm {r}^{2-\frac{d}{2}}\sqrt{\frac{4{k}_{1}d}{120\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{d+1}{d-4}}}\pm {r}^{2}\sqrt{\frac{{\mathcal{G}}_{0}(d-3)}{120\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{d+1}{d-4}}}$ | $1/P{\left(r\right)}^{2}$ | $d>3$ | $n=\frac{d+1}{4}$ |

$\forall P\left(r\right)$ | $\frac{1}{3}\left(\right)open="("\; close=")">A\left(r\right)-{e}^{{q}_{0}}{P}^{\prime}\left(r\right)+\frac{{e}^{2{q}_{0}}}{A\left(r\right)}{P}^{\prime}{\left(r\right)}^{2}$ | $d=3$ | $n>0$ |

$-\frac{1}{2}\mathrm{exp}\left(\right)open="["\; close="]">tan{h}^{-1}\left(\right)open="("\; close=")">\sqrt{\frac{{\mathcal{G}}_{0}}{30}}\frac{{r}^{2}}{2}$ | $1+\frac{\sqrt{{\mathcal{G}}_{0}}{r}^{2}}{2\sqrt{30}}$ | $d=4$ | $n=5/4$ |

1 | 1 | $d=4$ | $\forall n$ |

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**MDPI and ACS Style**

Bajardi, F.; Dialektopoulos, K.F.; Capozziello, S.
Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss–Bonnet Gravity. *Symmetry* **2020**, *12*, 372.
https://doi.org/10.3390/sym12030372

**AMA Style**

Bajardi F, Dialektopoulos KF, Capozziello S.
Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss–Bonnet Gravity. *Symmetry*. 2020; 12(3):372.
https://doi.org/10.3390/sym12030372

**Chicago/Turabian Style**

Bajardi, Francesco, Konstantinos F. Dialektopoulos, and Salvatore Capozziello.
2020. "Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss–Bonnet Gravity" *Symmetry* 12, no. 3: 372.
https://doi.org/10.3390/sym12030372