#
Relating Noncommutative SO(2,3)_{★} Gravity to the Lorentz-Violating Standard-Model Extension

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## Abstract

**:**

## 1. Introduction

## 2. Noncommutative SO(2,3)${}_{\u2605}$ Gravity

## 3. Gravitational Sector of the Lorentz-Violating Standard-Model Extension

#### 3.1. Covariant Match

#### 3.2. Linearized Lorentz-Violating Standard-Model Extension

## 4. Connecting NC SO(3,2)${}_{\u2605}$ Gravity to the SME

## 5. Conclusions, Prospects for Further Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Geometric Quantities to 2nd Order in the Metric Perturbation

**Metric:**- $$\begin{array}{ccc}{g}_{\mu \nu}\hfill & =\hfill & {\eta}_{\mu \nu}+{h}_{\mu \nu},\hfill \\ {g}^{\mu \nu}\hfill & =\hfill & {\eta}^{\mu \nu}-{h}^{\mu \nu}+{h}^{\mu \alpha}{{h}_{\alpha}}^{\nu}+o({h}^{3}).\hfill \end{array}$$
**Vierbein:**- $$\begin{array}{ccc}\hfill {{e}_{\mu}}^{a}& =\hfill & {{\delta}_{\mu}}^{a}+{\textstyle \frac{1}{2}}{{h}_{\mu}}^{a}-{\textstyle \frac{1}{8}}{h}_{\mu \lambda}{h}^{\lambda a}+o({h}^{3}),\hfill \\ \hfill {e}_{\mu a}& =\hfill & {\eta}_{\mu a}+{\textstyle \frac{1}{2}}{h}_{\mu a}-{\textstyle \frac{1}{8}}{h}_{\mu \lambda}{{h}^{\lambda}}_{a}+o({h}^{3}),\hfill \\ \hfill {e}^{\mu a}& =\hfill & {\eta}^{\mu a}-{\textstyle \frac{1}{2}}{h}^{\mu a}+{\textstyle \frac{3}{8}}{{h}^{\mu}}_{\lambda}{h}^{\lambda a}+o({h}^{3}),\hfill \\ \hfill {{e}^{\mu}}_{a}& =\hfill & {{\delta}^{\mu}}_{a}-{\textstyle \frac{1}{2}}{{h}^{\mu}}_{a}+{\textstyle \frac{3}{8}}{h}^{\mu \lambda}{h}_{\lambda a}+o({h}^{3}),\hfill \\ \hfill e:=det({{e}_{\mu}}^{a})& =\hfill & 1+{\textstyle \frac{1}{2}}{{h}_{\mu}}^{\mu}+{\textstyle \frac{1}{8}}({{h}_{\mu}}^{\mu}{{h}_{\nu}}^{\nu}-2{{h}_{\mu}}^{\nu}{{h}_{\nu}}^{\mu})+o({h}^{3}).\hfill \end{array}$$Note again that the expressions for the vierbein quantities cannot be related to each other simply by raising and lowering indices: ${e}^{\mu a}\ne {\eta}^{\mu \lambda}{{e}_{\lambda}}^{a}$, etc. Note also that the index placement in the definition of e is important: $det({{e}^{\mu}}_{a})={\textstyle \frac{1}{det({{e}_{\mu}}^{a})}}$.
**Connection coefficients:**- $$\begin{array}{ccc}{\Gamma}_{\alpha \mu \nu}\hfill & =\hfill & {\textstyle \frac{1}{2}}({\partial}_{\mu}{h}_{\nu \alpha}+{\partial}_{\nu}{h}_{\mu \alpha}-{\partial}_{\alpha}{h}_{\mu \nu}),\hfill \\ {{\Gamma}^{\alpha}}_{\mu \nu}\hfill & =\hfill & {\textstyle \frac{1}{2}}({\eta}^{\alpha \sigma}-{h}^{\alpha \sigma})({\partial}_{\mu}{h}_{\nu \sigma}+{\partial}_{\nu}{h}_{\mu \sigma}-{\partial}_{\sigma}{h}_{\mu \nu})+o({h}^{3}),\hfill \\ {{\omega}_{\mu}}^{ab}\hfill & =\hfill & \left[-{\textstyle \frac{1}{2}}{\partial}^{a}{{h}_{\mu}}^{b}+-{\textstyle \frac{1}{8}}{h}^{a\lambda}{\partial}_{\mu}{{h}_{\lambda}}^{b}+{\textstyle \frac{1}{4}}{h}^{a\lambda}{\partial}_{\lambda}{{h}_{\mu}}^{b}-{\textstyle \frac{1}{4}}{h}^{a\lambda}{\partial}^{b}{h}_{\lambda \mu}\right]-\left[a\rightleftharpoons b\right]+o({h}^{3}).\hfill \end{array}$$
**Derivative compatibility:**- $$\begin{array}{ccc}{\nabla}_{\gamma}{g}_{\mu \nu}\hfill & =\hfill & 0,\hfill \\ {\nabla}_{\gamma}{{e}_{\mu}}^{a}\hfill & =\hfill & 0.\hfill \end{array}$$
**Riemann tensor:**- $$\begin{array}{ccc}{R}_{\alpha \beta \mu \nu}\hfill & =\hfill & [(-{\textstyle \frac{1}{2}}{\partial}_{\alpha}{\partial}_{\mu}{h}_{\beta \nu}-{\textstyle \frac{1}{8}}{\partial}_{\alpha}{h}_{\mu \lambda}{\partial}_{\beta}{{h}_{\nu}}^{\lambda}-{\textstyle \frac{1}{8}}{\partial}_{\mu}{h}_{\alpha \lambda}{\partial}_{\nu}{{h}_{\beta}}^{\lambda}-{\textstyle \frac{1}{8}}{\partial}_{\lambda}{h}_{\alpha \mu}{\partial}^{\lambda}{h}_{\beta \nu}\hfill \\ & & \phantom{\rule{1.em}{0ex}}-{\textstyle \frac{1}{4}}{\partial}_{\alpha}{h}_{\mu \lambda}{\partial}_{\nu}{{h}_{\beta}}^{\lambda}+{\textstyle \frac{1}{4}}{\partial}_{\alpha}{h}_{\mu \lambda}{\partial}^{\lambda}{h}_{\beta \nu}+{\textstyle \frac{1}{4}}{\partial}_{\mu}{h}_{\alpha \lambda}{\partial}^{\lambda}{h}_{\beta \nu})-\left(\alpha \rightleftharpoons \beta \right)]-\left[\mu \rightleftharpoons \nu \right]+o({h}^{3}).\hfill \end{array}$$
**Ricci tensor:**- $$\begin{array}{ccc}{R}_{\alpha \mu}={g}^{\beta \nu}{R}_{\alpha \beta \mu \nu}\hfill & =\hfill & [{\textstyle \frac{1}{2}}{\partial}_{\alpha}{\partial}_{\lambda}{{h}_{\mu}}^{\lambda}-{\textstyle \frac{1}{4}}{\partial}_{\alpha}{\partial}_{\mu}{{h}_{\lambda}}^{\lambda}-{\textstyle \frac{1}{4}}{\partial}_{\lambda}{\partial}^{\lambda}{h}_{\alpha \mu}\hfill \\ & & \phantom{\rule{1.em}{0ex}}-{\textstyle \frac{1}{2}}{h}^{\lambda \rho}\left({\partial}_{\alpha}{\partial}_{\lambda}{h}_{\mu \rho}-{\textstyle \frac{1}{2}}{\partial}_{\alpha}{\partial}_{\mu}{h}_{\lambda \rho}-{\textstyle \frac{1}{2}}{\partial}_{\lambda}{\partial}_{\rho}{h}_{\alpha \mu}\right)\hfill \\ & & \phantom{\rule{1.em}{0ex}}+\left({\textstyle \frac{1}{4}}{\partial}_{\lambda}{{h}_{\rho}}^{\rho}-{\textstyle \frac{1}{2}}{\partial}^{\rho}{h}_{\rho \lambda}\right)\left({\partial}_{\alpha}{{h}_{\mu}}^{\lambda}-{\textstyle \frac{1}{2}}{\partial}^{\lambda}{h}_{\alpha \mu}\right)\hfill \\ & & \phantom{\rule{1.em}{0ex}}-{\textstyle \frac{1}{4}}({\partial}_{\lambda}{{h}_{\mu}}^{\rho})\left({\partial}_{\rho}{{h}_{\alpha}}^{\lambda}-{\textstyle \frac{1}{2}}{\partial}^{\lambda}{h}_{\alpha \rho}\right)+{\textstyle \frac{1}{8}}({\partial}_{\alpha}{h}_{\lambda \rho})({\partial}_{\mu}{h}^{\lambda \rho})]+\left[\alpha \rightleftharpoons \mu \right]+o({h}^{3}).\hfill \end{array}$$

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u | Weight C${}_{(\mathit{u})}$ | Geometric Quantity L${}_{\mathsf{\alpha}\mathsf{\beta}\mathsf{\gamma}\mathsf{\delta}}^{(\mathit{u})}$ |
---|---|---|

1 | $3{c}_{2}+16{c}_{3}$ | ${R}_{\alpha \beta \gamma \delta}$ |

2 | $-6-22{c}_{2}-36{c}_{3}$ | ${g}_{\beta \delta}{R}_{\alpha \gamma}$ |

3 | $\frac{1}{{\ell}^{2}}}(6+28{c}_{2}+56{c}_{3})$ | ${g}_{\alpha \gamma}{g}_{\beta \delta}$ |

4 | $-4-16{c}_{2}-32{c}_{3}$ | ${e}_{a}^{\mu}{e}_{\beta b}({\tilde{\nabla}}_{\gamma}{e}_{\alpha}^{a})({\tilde{\nabla}}_{\delta}{e}_{\mu}^{b}$) |

5 | $4+12{c}_{2}+32{c}_{3}$ | ${e}_{\delta a}{e}_{b}^{\mu}({\tilde{\nabla}}_{\alpha}{e}_{\gamma}^{a})({\tilde{\nabla}}_{\beta}{e}_{\mu}^{b})$ |

6 | $2+4{c}_{2}+8{c}_{3}$ | ${g}_{\beta \delta}{e}_{a}^{\mu}{e}_{b}^{\nu}[({\tilde{\nabla}}_{\alpha}{e}_{\nu}^{a})({\tilde{\nabla}}_{\gamma}{e}_{\mu}^{b})-({\tilde{\nabla}}_{\gamma}{e}_{\mu}^{a})({\tilde{\nabla}}_{\alpha}{e}_{\nu}^{b})]$ |

SME Coefficients | Young Projection |
---|---|

${s}^{(4)\mu \rho \alpha \nu \sigma \beta}$ | |

${s}^{(4,1)\mu \rho \nu \sigma \alpha \beta}$ | |

${s}^{(4,2)\mu \rho \alpha \nu \sigma \beta}$ | |

${k}^{(4,3)\mu \alpha \nu \beta \rho \sigma}$ |

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**MDPI and ACS Style**

Bailey, Q.G.; Lane, C.D.
Relating Noncommutative SO(2,3)_{★} Gravity to the Lorentz-Violating Standard-Model Extension. *Symmetry* **2018**, *10*, 480.
https://doi.org/10.3390/sym10100480

**AMA Style**

Bailey QG, Lane CD.
Relating Noncommutative SO(2,3)_{★} Gravity to the Lorentz-Violating Standard-Model Extension. *Symmetry*. 2018; 10(10):480.
https://doi.org/10.3390/sym10100480

**Chicago/Turabian Style**

Bailey, Quentin G., and Charles D. Lane.
2018. "Relating Noncommutative SO(2,3)_{★} Gravity to the Lorentz-Violating Standard-Model Extension" *Symmetry* 10, no. 10: 480.
https://doi.org/10.3390/sym10100480