# Modifications to Plane Gravitational Waves from Minimal Lorentz Violation

## Abstract

**:**

## 1. Introduction

## 2. Plane Waves with Minimal Lorentz Violation

## 3. The Perturbation Solution

#### 3.1. $\{{h}_{11}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{h}_{12}^{\left(1\right)}\}$

#### 3.2. $\{{h}_{00}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{h}_{33}^{\left(1\right)}\}$

#### 3.3. $\{{h}_{01}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{h}_{02}^{\left(1\right)}\}$

## 4. Geodesic Deviation

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The First-Order Riemann Tensor

## References

- Debono, I.; Smoot, G.F. General Relativity and Cosmology: Unsolved Questions and Future Directions. Universe
**2016**, 2, 23. [Google Scholar] [CrossRef] - Vishwakarma, R.G. Einstein and Beyond: A Critical Perspective on General Relativity. Universe
**2016**, 2, 11. [Google Scholar] [CrossRef] - Will, C.M. The confrontation between general relativity and experiment. Living Rev. Relativ.
**2014**, 14, 4. [Google Scholar] [CrossRef] [PubMed] - Berti, E.; Barausse, E.; Cardoso, V.; Gualtieri, L.; Pani, P.; Sperhake, U.; Stein, L.C.; Wex, N.; Yagi, K.; Baker, T.; et al. Testing General Relativity with Present and Future Astrophysical Observations. Class. Quantum Grav.
**2015**, 32, 243001. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Russell, N. Data tables for Lorentz and CPT violation. Rev. Mod. Phys.
**2011**, 83, 11. [Google Scholar] [CrossRef] - Battat, J.B.R.; Chandler, J.F.; Stubbs, C.W. Testing for Lorentz Violation: Constraints on Standard-Model Extension Parameters via Lunar Laser Ranging. Phys. Rev. Lett.
**2007**, 99, 241103. [Google Scholar] [CrossRef] - Muller, H.; Chiow, S.-W.; Herrmann, S.; Chu, S.; Chung, K.-Y. Atom Interferometry tests of the isotropy of post-Newtonian gravity. Phys. Rev. Lett.
**2008**, 100, 031101. [Google Scholar] [CrossRef] - Shao, L. Tests of local Lorentz invariance violation of gravity in the standard model extension with pulsars. Phys. Rev. Lett.
**2014**, 112, 111103. [Google Scholar] [CrossRef] - Shao, C.-G.; Tan, Y.-J.; Tan, W.-H.; Yang, S.-Q.; Luo, J.; Tobar, M.E.; Bailey, Q.G.; Long, J.C.; Weisman, E.; Xu, R.; et al. Combined search for Lorentz violation in short-range gravity. Phys. Rev. Lett.
**2016**, 117, 071102. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Testing local Lorentz invariance with gravitational waves. Phys. Lett. B
**2016**, 757, 510. [Google Scholar] [CrossRef] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, R.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Gravitational waves and gamma-rays from a binary Neutron star merger: GW170817 and GRB 170817A. Astrophys. J. Lett.
**2017**, 848, L13. [Google Scholar] [CrossRef] - Colladay, D.; Kostelecký, V.A. Lorentz-violating extension of the Standard Model. Phys. Rev. D
**1998**, 58, 116002. [Google Scholar] [CrossRef] - Kostelecký, V.A. Gravity, Lorentz violation, and the Standard Model. Phys. Rev. D
**2004**, 69, 105009. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Electrodynamics with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2009**, 80, 015020. [Google Scholar] [CrossRef] [Green Version] - Kostelecký, V.A.; Tasson, J.D. Matter-gravity couplings and Lorentz violation. Phys. Rev. D
**2011**, 83, 016013. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Neutrinos with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2012**, 85, 096005. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Fermions with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2013**, 88, 096006. [Google Scholar] [CrossRef] - Tasson, J.D. What do we know about Lorentz invariance? Rep. Prog. Phys.
**2014**, 77, 062901. [Google Scholar] [CrossRef] - Bailey, Q.G.; Kostelecký, V.A. Signals for Lorentz violation in post-newtonian gravity. Phys. Rev. D
**2006**, 74, 045001. [Google Scholar] [CrossRef] - Bailey, Q.G.; Kostelecký, V.A.; Xu, R. Short-range gravity and Lorentz violation. Phys. Rev. D
**2015**, 91, 022006. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Lorentz and Diffeomorphism Violations in Linearized Gravity. Phys. Lett. B
**2018**, 779, 136. [Google Scholar] [CrossRef] - Cervantes-Cota, J.L.; Galindo-Uribarri, S.; Smoot, G.F. A Brief History of Gravitational Waves. Universe
**2016**, 2, 22. [Google Scholar] [CrossRef] - Akutsu, T.; Ando, M.; Arai, K.; Arai, Y.; Araki, S.; Araya, A.; Aritomi, N.; Asada, H.; Aso, Y.; Atsuta, S.; et al. KAGRA: 2.5 Generation Interferometric Gravitational Wave Detector. arXiv
**2018**, arXiv:1811.08079. [Google Scholar] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. LIGO scientific collaboration and virgo collaboration. Gravitational wave astronomy with LIGO and similar detectors in the next decade. arXiv
**2019**, arXiv:1904.03187. [Google Scholar] - Mewes, M. Signals for Lorentz violation in gravitational waves. Phys. Rev. D
**2019**, 99, 104062. [Google Scholar] [CrossRef] [Green Version] - Sotiriou, T.P. Detecting Lorentz Violations with Gravitational Waves from Black Hole Binaries. Phys. Rev. Lett.
**2018**, 120, 041104. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Tasson, J.D. Constraints on Lorentz violation from gravitational Čerenkov radiation. Phys. Lett. B
**2015**, 749, 551. [Google Scholar] [CrossRef] - Poisson, E.; Will, C.M. Gravity; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]

© 2019 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**

Xu, R.
Modifications to Plane Gravitational Waves from Minimal Lorentz Violation. *Symmetry* **2019**, *11*, 1318.
https://doi.org/10.3390/sym11101318

**AMA Style**

Xu R.
Modifications to Plane Gravitational Waves from Minimal Lorentz Violation. *Symmetry*. 2019; 11(10):1318.
https://doi.org/10.3390/sym11101318

**Chicago/Turabian Style**

Xu, Rui.
2019. "Modifications to Plane Gravitational Waves from Minimal Lorentz Violation" *Symmetry* 11, no. 10: 1318.
https://doi.org/10.3390/sym11101318