# The Standard-Model Extension and Gravitational Tests

## Abstract

**:**

## 1. Introduction

## 2. Symmetry Violation

#### 2.1. Rotations

#### 2.2. CPT

## 3. The SME

#### 3.1. Gravitationally Coupled Matter

#### 3.2. Pure Gravity

## 4. Gravitational Tests: Existing Results and Proposals

## 5. Gravitational Čerenkov

## 6. Conclusions

## Conflicts of Interest

## References

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System | Coefficients | Proposal | Constraints |
---|---|---|---|

gravitational Čerenkov radiation | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$, $\overline{s}{}_{\mu \nu {\alpha}_{1}{\alpha}_{2}}^{\left(6\right)}$, $\overline{s}{}_{\mu \nu {\alpha}_{1}{\alpha}_{2}{\alpha}_{3}{\alpha}_{4}}^{\left(8\right)}$,$\overline{s}{}_{\mu \nu {\alpha}_{1}{\alpha}_{2}{\alpha}_{3}{\alpha}_{4}{\alpha}_{5}{\alpha}_{6}}^{\left(10\right)*}$ | [14] | [14] [*] |

superconducting gravimeters | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$, $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [27,29] | [37] |

short-range gravity devices | ${\left({\overline{k}}_{\mathrm{eff}}\right)}_{jklm}$ ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$ | [22,27,29] | [38,39,40,41,42] |

gravitational-wave interferometers | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$, $\overline{s}{}_{\mu \nu {\alpha}_{1}{\alpha}_{2}}^{\left(6\right)}$, ${q}^{\left(5\right)}{}^{\mu \rho \alpha \nu \beta \sigma \gamma}$, ${k}^{\left(6\right)}{}^{\mu \alpha \nu \beta \rho \gamma \sigma \delta}$ | [23] | [23,32] |

lunar laser ranging | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [29] | [43,44] |

binary-pulsar observations | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [29] | [45,46,47] |

planetary ephemerides | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [48] | [48] |

gravity probe B | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [29] | [49] |

bound kinetic energy WEP | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$ | [27,50] | [50] |

atom interferometers | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$, $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [27,29] | [51,52,53] |

comagnetometry | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [16] | [16] |

perihelion precession | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$ | [27] | [27] |

equivalence-principle pendulum | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$ | [28] | [28] |

Solar-spin precession | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [29] | [29] |

System | Coefficients | Proposal | Experiments |
---|---|---|---|

atom interferometer | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$, $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [27] | [54] |

ring-laser gyroscopes | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [55,56] | [57] |

torsion pendula | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$ | [27] | [58] |

binary pulsars | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$, $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ ${K}_{jk}$ | [24,59] | [60] |

short-range gravity | ${K}_{jk}$ | [24] | [38,39,42] |

gravitational-wave detectors | ${s}^{\left(d\right)}{}^{\mu \rho \alpha \nu \sigma \beta \dots}$, ${k}^{\left(d\right)}{}^{\mu \alpha \nu \beta \rho \gamma \sigma \delta \dots}$, ${q}^{\left(d\right)}{}^{\mu \rho \alpha \nu \beta \sigma \gamma \dots}$ | [23] | [31,33] |

space-based WEP tests | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$ | [29,61] | [61,62,63,64] |

antimatter gravity | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$ | [27,65] | [66,67,68,69] |

charged matter WEP | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$ | [27] | [70] |

muonium free fall | ${\left(\overline{a}{}_{\mathrm{eff}}\right)}_{\mu}$, $\overline{c}{}_{\mu \nu}$ | [27] | [71] |

light bending | [72] | [73] | |

time-delay & Doppler tests | $\overline{s}{}_{\mu \nu}^{\left(4\right)}$ | [27,74] | [75,76,77] |

**Table 3.**Conservative constraints on coefficients ${\overline{s}}_{jm}^{\left(10\right)}$ in GeV${}^{-6}$.

j | Lower Bound | Coeff. | Upper Bound | j | Lower Bound | Coeff. | Upper Bound |
---|---|---|---|---|---|---|---|

0 | ${\overline{s}}_{00}^{\left(10\right)}$ | $<2\times {10}^{-66}$ | 6 | $-2\times {10}^{-61}<$ | ${\overline{s}}_{60}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | |

1 | $-1\times {10}^{-61}<$ | ${\overline{s}}_{10}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{61}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{11}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{61}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{11}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-8\times {10}^{-62}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{62}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | ||

2 | $-2\times {10}^{-61}<$ | ${\overline{s}}_{20}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{62}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{21}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{63}^{\left(10\right)}$ | $<9\times {10}^{-62}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{21}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{63}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{22}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{64}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{22}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{64}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

3 | $-2\times {10}^{-61}<$ | ${\overline{s}}_{30}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{65}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{31}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{65}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{31}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{66}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{32}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{66}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{32}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | 7 | $-2\times {10}^{-61}<$ | ${\overline{s}}_{70}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | |

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{33}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-9\times {10}^{-62}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{71}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{33}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{71}^{\left(10\right)}$ | $<9\times {10}^{-62}$ | ||

4 | $-2\times {10}^{-61}<$ | ${\overline{s}}_{40}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{72}^{\left(10\right)}$ | $<9\times {10}^{-62}$ | |

$-2\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{41}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{72}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{41}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{73}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{42}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{73}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{42}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{74}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{43}^{\left(10\right)}$ | $<9\times {10}^{-62}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{74}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{43}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{75}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{44}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{75}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{44}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{76}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

5 | $-1\times {10}^{-61}<$ | ${\overline{s}}_{50}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{76}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{51}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{77}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{51}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-2\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{77}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{52}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | 8 | $-2\times {10}^{-61}<$ | ${\overline{s}}_{80}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | |

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{52}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{81}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{53}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{81}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{53}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{82}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{54}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{82}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{54}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{83}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{55}^{\left(10\right)}$ | $<9\times {10}^{-62}$ | $-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{83}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{55}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | $-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{84}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | ||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{84}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{85}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{85}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{86}^{\left(10\right)}$ | $<2\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{86}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{87}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{87}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Re}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{88}^{\left(10\right)}$ | $<1\times {10}^{-61}$ | |||||

$-1\times {10}^{-61}<$ | $\mathrm{Im}\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}_{88}^{\left(10\right)}$ | $<1\times {10}^{-61}$ |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tasson, J.D.
The Standard-Model Extension and Gravitational Tests. *Symmetry* **2016**, *8*, 111.
https://doi.org/10.3390/sym8110111

**AMA Style**

Tasson JD.
The Standard-Model Extension and Gravitational Tests. *Symmetry*. 2016; 8(11):111.
https://doi.org/10.3390/sym8110111

**Chicago/Turabian Style**

Tasson, Jay D.
2016. "The Standard-Model Extension and Gravitational Tests" *Symmetry* 8, no. 11: 111.
https://doi.org/10.3390/sym8110111