# Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries

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## Abstract

**:**

## 1. Introduction

## 2. Determining Equations

## 3. Five Noether Symmetries

## 4. Six Noether Symmetries

## 5. Seven Noether Symmetries

## 6. Eight Noether Symmetries

## 7. Nine Noether Symmetries

## 8. Eleven Noether Symmetries

$[{X}_{1},{X}_{5}]={X}_{6}$, | $[{X}_{1},{X}_{6}]={X}_{5}$, | $[{X}_{1},{X}_{7}]={X}_{8}$, | $[{X}_{1},{X}_{8}]={X}_{7}$, | $[{X}_{1},{X}_{9}]={X}_{10}$, |

$[{X}_{1},{X}_{10}]={X}_{9}$, | $[{X}_{5},{X}_{2}]={X}_{7}$, | $[{X}_{6},{X}_{2}]={X}_{8}$, | $[{X}_{2},{X}_{7}]={X}_{5}$, | $[{X}_{2},{X}_{8}]={X}_{6}$, |

$[{X}_{5},{X}_{3}]={X}_{9}$, | $[{X}_{6},{X}_{3}]={X}_{10}$, | $[{X}_{3},{X}_{9}]={X}_{5}$, | $[{X}_{3},{X}_{10}]={X}_{6}$, | $[{X}_{7},{X}_{4}]={X}_{9}$, |

$[{X}_{8},{X}_{4}]={X}_{10}$, | $[{X}_{4},{X}_{9}]={X}_{7}$, | $[{X}_{4},{X}_{10}]={X}_{8}$, | $[{X}_{5},{X}_{6}]={X}_{1}$, | $[{X}_{7},{X}_{5}]={X}_{2}$, |

$[{X}_{9},{X}_{5}]={X}_{3}$, | $[{X}_{6},{X}_{8}]={X}_{2}$, | $[{X}_{6},{X}_{10}]={X}_{3}$, | $[{X}_{7},{X}_{8}]={X}_{1}$, | $[{X}_{9},{X}_{7}]={X}_{4}$, |

$[{X}_{8},{X}_{10}]={X}_{4}$, | $[{X}_{9},{X}_{10}]={X}_{1}$, |

## 9. Summary and Discussion

**7(i)**–

**7(v)**, we have six KVs and one Noether symmetry ${\partial}_{u},$ with a trivial gauge function. For metric

**7(vi)**, we have the minimal set of NS along with two extra NS, given by ${X}_{5}$ and ${X}_{6}.$ The gauge function corresponding to ${X}_{5}$ is found to be $F=\frac{t({c}_{1}t+2{c}_{2})}{2{c}_{1}}$. It can bee seen that the Noether symmetry ${X}_{6}$ for this metric corresponds to an HV $\frac{{c}_{1}t+{c}_{2}}{2{c}_{1}}{\partial}_{t},$ with homothetic constant $\psi =\frac{1}{2}.$ Similar results are obtained for the metric

**7(vii)**. Here, the number of KVs is five.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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No. | $\mathit{\lambda}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{v}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ |
---|---|---|

3a | ${\lambda}^{\prime}\ne 0$ | ${v}^{\u2033}\ne 0$ and $v\ne cosht$ |

3b | $\lambda \ne sinht$ | $v=cosht$ |

3c | ${\left(\lambda {\lambda}^{\prime}\right)}^{\u2033}\ne 0$ | $v=$ const. $=\xi $ |

3d | $\lambda =\sqrt{a{t}^{2}+2bt+c}$; $a\ne 0$, | $v=$ const. $=\xi $ |

3e | $\lambda =\sqrt{2bt+c}$ | $v=$ const. $=\xi $ |

No. | $\mathit{\lambda}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{\nu}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | Noether Symmetries | Invariants | Lie Algebra |
---|---|---|---|---|---|

7(i) | $sin\left(\beta t\right)$; $\beta \ne 0$ | $\gamma \ne 0$ | ${X}_{5}=cosh\left(\beta r\right){\partial}_{t}-cot\left(\beta t\right)sinh\left(\beta r\right){\partial}_{r}$ | ${I}_{5}=2cosh\left(\beta r\right)\dot{t}+2cos\left(\beta t\right)sin\left(\beta t\right)sinh\left(\beta r\right)\dot{r}$ | $[{X}_{1},{X}_{5}]=\beta {X}_{6}$, $[{X}_{1},{X}_{6}]=\beta {X}_{5}$, |

${X}_{6}=sinh\left(\beta r\right){\partial}_{t}-cot\left(\beta t\right)cosh\left(\beta r\right){\partial}_{r}$ | ${I}_{6}=\frac{1}{\beta}\frac{\partial {I}_{5}}{\partial r}$ | $[{X}_{6},{X}_{5}]=\beta {X}_{1}$. | |||

7(ii) | $cos\left(\beta t\right)$ | $\gamma \ne 0$ | ${X}_{5}=cosh\left(\beta r\right){\partial}_{t}+tan\left(\beta t\right)sinh\left(\beta r\right){\partial}_{r}$ | ${I}_{5}=2cosh\left(\beta r\right)\dot{t}-2cos\left(\beta t\right)sin\left(\beta t\right)sinh\left(\beta r\right)\dot{r}$ | Same as in case 7(i) |

${X}_{6}=sinh\left(\beta r\right){\partial}_{t}+tan\left(\beta t\right)cosh\left(\beta r\right){\partial}_{r}$ | ${I}_{6}=\frac{1}{\beta}\frac{\partial {I}_{5}}{\partial r}$ | ||||

7(iii) | $sinh\left(\beta t\right)$; $\beta \ne 0$ | $\gamma \ne 0$ | ${X}_{5}=cosh\left(\beta r\right){\partial}_{t}-coth\left(\beta t\right)sinh\left(\beta r\right){\partial}_{r}$ | ${I}_{5}=2cosh\left(\beta r\right)\dot{t}+2cosh\left(\beta t\right)sinh\left(\beta t\right)sinh\left(\beta r\right)\dot{r}$ | Same as in case 7(i) |

${X}_{6}=sinh\left(\beta r\right){\partial}_{t}-coth\left(\beta t\right)cosh\left(\beta r\right){\partial}_{r}$ | ${I}_{6}=\frac{1}{k}\frac{\partial {I}_{5}}{\partial r}$ | ||||

7(iv) | $cosh\left(\beta t\right)$ | $\gamma \ne 0$ | ${X}_{5}=cos\left(\beta r\right){\partial}_{t}-tanh\left(\beta t\right)sin\left(\beta r\right){\partial}_{r}$ | ${I}_{5}=2cos\left(\beta r\right)\dot{t}+2cosh\left(\beta t\right)sinh\left(\beta t\right)sin\left(\beta r\right)\dot{r}$ | Same as in case 7(i) |

${X}_{6}=sin\left(\beta r\right){\partial}_{t}+tanh\left(\beta t\right)cos\left(\beta r\right){\partial}_{r}$ | ${I}_{6}=\frac{1}{\beta}\frac{\partial {I}_{5}}{\partial r}$ | ||||

7(v) | ${e}^{\beta t}$ | $\gamma \ne 0$ | ${X}_{5}={\partial}_{t}-\beta r{\partial}_{r}$ | ${I}_{5}=2\dot{t}+2\beta r{e}^{2\beta t}\dot{r}$ | $[{X}_{5},{X}_{1}]=\beta {X}_{1}$, $[{X}_{1},{X}_{6}]={X}_{5}$, |

${X}_{6}=r{\partial}_{t}-\frac{{e}^{-2\beta t}}{2\beta}{\partial}_{r}-\frac{\beta {r}^{2}}{2}{\partial}_{r}$ | ${I}_{6}=2r\dot{t}+\frac{\dot{r}}{\beta}+\beta {r}^{2}{e}^{2\beta t}\dot{r}$ | $[{X}_{6},{X}_{5}]=\beta {X}_{6}$ | |||

7(vi) | ${c}_{1}t+{c}_{2}$ | $\nu =\lambda $ | ${X}_{5}=\frac{{u}^{2}}{2}{\partial}_{s}+\frac{u({c}_{1}t+{c}_{2})}{2{c}_{1}}{\partial}_{t}$; $F=\frac{t({c}_{1}t+2{c}_{2})}{2{c}_{1}}$ | ${I}_{5}=\frac{{u}^{2}}{2}L+\frac{u({c}_{1}t+{c}_{2})\dot{t}}{{c}_{1}}-\frac{{t}^{2}}{2}-\frac{{c}_{2}t}{{c}_{1}}$ | $[{X}_{0},{X}_{5}]={X}_{6}$, $[{X}_{0},{X}_{6}]={X}_{0}$, |

${V}_{6}=u{\partial}_{u}+\frac{{c}_{1}t+{c}_{2}}{2{c}_{1}}{\partial}_{t}$ | ${I}_{6}=uL+\frac{({c}_{1}t+{c}_{2})\dot{t}}{{c}_{1}}$ | $[{X}_{6},{X}_{5}]={X}_{5}.$ | |||

7(vii) | ${({c}_{1}t+{c}_{2})}^{\frac{\alpha -2\beta}{\alpha}}$; | ${c}_{1}t+{c}_{2}$ | ${X}_{5}=u{\partial}_{u}+\frac{{c}_{1}t+{c}_{2}}{2{c}_{1}}{\partial}_{t}$ | ${I}_{5}=uL+\frac{({c}_{1}t+{c}_{2})\dot{t}}{{c}_{1}}$ | $[{X}_{0},{X}_{5}]={X}_{0}$, $[{X}_{1},{X}_{6}]={X}_{1}$, |

$\alpha \ne 0$ | ${X}_{6}=r{\partial}_{r}$ | ${I}_{6}=-2r{({c}_{1}t+{c}_{2})}^{\frac{2(\alpha -2\beta )}{\alpha}}\dot{r}$ | $[{X}_{6},{X}_{1}]={X}_{5}$ |

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**MDPI and ACS Style**

Akhtar, S.S.; Hussain, T.; Bokhari, A.H.
Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries. *Symmetry* **2018**, *10*, 712.
https://doi.org/10.3390/sym10120712

**AMA Style**

Akhtar SS, Hussain T, Bokhari AH.
Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries. *Symmetry*. 2018; 10(12):712.
https://doi.org/10.3390/sym10120712

**Chicago/Turabian Style**

Akhtar, Sumaira Saleem, Tahir Hussain, and Ashfaque H. Bokhari.
2018. "Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries" *Symmetry* 10, no. 12: 712.
https://doi.org/10.3390/sym10120712