# Tests of Lorentz Symmetry in the Gravitational Sector

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Standard-Model Extension in the Gravitational Sector

## 3. Postfit Analysis Versus Full Modeling

## 4. Data Analysis

#### 4.1. Atomic Gravimetry

#### 4.2. Very Long Baseline Interferometry

#### 4.3. Lunar Laser Ranging

#### 4.4. Planetary Ephemerides

#### 4.5. Gravity Probe B

#### 4.6. Binary Pulsars

#### 4.7. Čerenkov Radiation

#### 4.8. Summary and Combined Analysis

## 5. The future

#### 5.1. The Gaia Mission

#### 5.2. Analysis of Cassini Conjunction Data

#### 5.3. Satellite Laser Ranging (LAGEOS/LARES)

#### 5.4. Gravity-Matter Coefficients and Breaking of the Einstein Equivalence Principle

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CMB | Cosmic Microwave Background |

ELPN | Éphéméride Lunaire Parisienne Numérique |

GPB | Gravity Probe B |

GR | General Relativity |

GRAIL | Gravity Recovery And Interior Laboratory |

INPOP | Intégrateur Numérique Planétaire de l’Observatoire de Paris |

IVS | International VLBI Service for Geodesy and Astrometry |

LARASE | LAser RAnged Satellites Experiment |

LLR | Lunar Laser Ranging |

LIV | Lorentz Invariance Violation |

mas | milliarcsecond |

Mpc | Megaparsec |

NFT | Numerical Fourier Transform |

PPN | Parametrized Post-Newtonian |

SME | Standard-Model Extension |

SSO | Solar System Object |

TAI | International Atomic Time |

TDB | Barycentric Dynamical Time |

TT | Terrestrial Time |

UTC | Universal Time Coordinate |

VLBI | Very Long Baseline Interferometry |

yr | year |

## References

- Iorio, L. Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein’s Theory of Gravitation in Its Centennial Year. Universe
**2015**, 1, 38–81. [Google Scholar] [CrossRef] - Will, C.M. The Confrontation between General Relativity and Experiment. Living Rev. Relativ.
**2014**, 17, 4. [Google Scholar] [CrossRef] - Turyshev, S.G. REVIEWS OF TOPICAL PROBLEMS: Experimental tests of general relativity: Recent progress and future directions. Phys. Uspekhi
**2009**, 52, 1–27. [Google Scholar] [CrossRef] - 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. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cervantes-Cota, J.; Galindo-Uribarri, S.; Smoot, G. A Brief History of Gravitational Waves. Universe
**2016**, 2, 22. [Google Scholar] [CrossRef] - Debono, I.; Smoot, G.F. General Relativity and Cosmology: Unsolved Questions and Future Directions. Universe
**2016**, 2, 23. [Google Scholar] [CrossRef] - 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 Gravity
**2015**, 32, 243001. [Google Scholar] [CrossRef] - Colladay, D.; Kostelecký, V.A. CPT violation and the standard model. Phys. Rev. D
**1997**, 55, 6760–6774. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Samuel, S. Gravitational phenomenology in higher-dimensional theories and strings. Phys. Rev. D
**1989**, 40, 1886–1903. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Samuel, S. Spontaneous breaking of Lorentz symmetry in string theory. Phys. Rev. D
**1989**, 39, 683–685. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Potting, R. CPT and strings. Nucl. Phys. B
**1991**, 359, 545. [Google Scholar] [CrossRef] - Gambini, R.; Pullin, J. Nonstandard optics from quantum space-time. Phys. Rev. D
**1999**, 59, 124021. [Google Scholar] [CrossRef] - Amelino-Camelia, G. Quantum-Spacetime Phenomenology. Living Rev. Relativ.
**2013**, 16, 5. [Google Scholar] [CrossRef] - Mavromatos, N.E. CPT Violation and Decoherence in Quantum Gravity; Lecture Notes in Physics; Kowalski-Glikman, J., Amelino-Camelia, G., Eds.; Springer: Berlin, Germany, 2005. [Google Scholar]
- Myers, R.C.; Pospelov, M. Ultraviolet Modifications of Dispersion Relations in Effective Field Theory. Phys. Rev. Lett.
**2003**, 90, 211601. [Google Scholar] [CrossRef] [PubMed] - Hayakawa, M. Perturbative analysis on infrared aspects of noncommutative QED on R
^{4}. Phys. Lett. B**2000**, 478, 394–400. [Google Scholar] [CrossRef] - Carroll, S.M.; Harvey, J.A.; Kostelecký, V.A.; Lane, C.D.; Okamoto, T. Noncommutative Field Theory and Lorentz Violation. Phys. Rev. Lett.
**2001**, 87, 141601. [Google Scholar] [CrossRef] [PubMed] - Bjorken, J.D. Cosmology and the standard model. Phys. Rev. D
**2003**, 67, 043508. [Google Scholar] [CrossRef] - Burgess, C.P.; Martineau, P.; Quevedo, F.; Rajesh, G.; Zhang, R.J. Brane-antibrane inflation in orbifold and orientifold models. J. High Energy Phys.
**2002**, 3, 052. [Google Scholar] [CrossRef] - Frey, A.R. String theoretic bounds on Lorentz-violating warped compactification. J. High Energy Phys.
**2003**, 4, 12. [Google Scholar] [CrossRef] - Cline, J.M.; Valcárcel, L. Asymmetrically warped compactifications and gravitational Lorentz violation. J. High Energy Phys.
**2004**, 3, 032. [Google Scholar] [CrossRef] - Tasson, J.D. What do we know about Lorentz invariance? Rep. Prog. Phys.
**2014**, 77, 062901. [Google Scholar] [CrossRef] [PubMed] - Mattingly, D. Modern Tests of Lorentz Invariance. Living Rev. Relativ.
**2005**, 8, 5. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Russell, N. Data tables for Lorentz and CPT violation. Rev. Mod. Phys.
**2011**, 83, 11–32. [Google Scholar] [CrossRef] - Colladay, D.; Kostelecký, V.A. Lorentz-violating extension of the standard model. Phys. Rev. D
**1998**, 58, 116002. [Google Scholar] [CrossRef] - Thorne, K.S.; Will, C.M. Theoretical Frameworks for Testing Relativistic Gravity. I. Foundations. Astrophys. J.
**1971**, 163, 595. [Google Scholar] [CrossRef] - Will, C.M. Theory and Experiment in Gravitational Physics; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Thorne, K.S.; Lee, D.L.; Lightman, A.P. Foundations for a Theory of Gravitation Theories. Phys. Rev. D
**1973**, 7, 3563–3578. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Tasson, J.D. Matter-gravity couplings and Lorentz violation. Phys. Rev. D
**2011**, 83, 016013. [Google Scholar] [CrossRef] - Tasson, J.D. The Standard-Model Extension and Gravitational Tests. Symmetry
**2016**, 8, 111. [Google Scholar] [CrossRef] - Fischbach, E.; Sudarsky, D.; Szafer, A.; Talmadge, C.; Aronson, S.H. Reanalysis of the Eotvos experiment. Phys. Rev. Lett.
**1986**, 56, 3–6. [Google Scholar] [CrossRef] [PubMed] - Talmadge, C.; Berthias, J.P.; Hellings, R.W.; Standish, E.M. Model-independent constraints on possible modifications of Newtonian gravity. Phys. Rev. Lett.
**1988**, 61, 1159–1162. [Google Scholar] [CrossRef] [PubMed] - Fischbach, E.; Talmadge, C.L. The Search for Non-Newtonian Gravity; Aip-Press Series; Springer: New York, NY, USA, 1999. [Google Scholar]
- Adelberger, E.G.; Heckel, B.R.; Nelson, A.E. Tests of the Gravitational Inverse-Square Law. Annu. Rev. Nucl. Part. Sci.
**2003**, 53, 77–121. [Google Scholar] [CrossRef] - Reynaud, S.; Jaekel, M.T. Testing the Newton Law at Long Distances. Int. J. Mod. Phys. A
**2005**, 20, 2294–2303. [Google Scholar] [CrossRef] - Bailey, Q.G. Gravity Sector of the SME. In Proceedings of the Seventh Meeting on CPT and Lorentz Symmetry, Bloomington, IN, USA, 20–24 June 2016.
- Kostelecký, V.A. Gravity, Lorentz violation, and the standard model. Phys. Rev. D
**2004**, 69, 105009. [Google Scholar] [CrossRef] - Bluhm, R.; Kostelecký, V.A. Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity. Phys. Rev. D
**2005**, 71, 065008. [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] - Bluhm, R. Nambu-Goldstone Modes in Gravitational Theories with Spontaneous Lorentz Breaking. Int. J. Mod. Phys. D
**2007**, 16, 2357–2363. [Google Scholar] [CrossRef] - Bluhm, R.; Fung, S.H.; Kostelecký, V.A. Spontaneous Lorentz and diffeomorphism violation, massive modes, and gravity. Phys. Rev. D
**2008**, 77, 065020. [Google Scholar] [CrossRef] - Bluhm, R. Explicit versus spontaneous diffeomorphism breaking in gravity. Phys. Rev. D
**2015**, 91, 065034. [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] - Bonder, Y. Lorentz violation in the gravity sector: The t puzzle. Phys. Rev. D
**2015**, 91, 125002. [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] [PubMed] - Bourgoin, A.; Hees, A.; Bouquillon, S.; Le Poncin-Lafitte, C.; Francou, G.; Angonin, M.C. Testing Lorentz symmetry with Lunar Laser Ranging. arXiv
**2016**. [Google Scholar] - Müller, 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] [PubMed] - Chung, K.Y.; Chiow, S.W.; Herrmann, S.; Chu, S.; Müller, H. Atom interferometry tests of local Lorentz invariance in gravity and electrodynamics. Phys. Rev. D
**2009**, 80, 016002. [Google Scholar] [CrossRef] - Iorio, L. Orbital effects of Lorentz-violating standard model extension gravitomagnetism around a static body: A sensitivity analysis. Class. Quantum Gravity
**2012**, 29, 175007. [Google Scholar] [CrossRef] - Hees, A.; Bailey, Q.G.; Le Poncin-Lafitte, C.; Bourgoin, A.; Rivoldini, A.; Lamine, B.; Meynadier, F.; Guerlin, C.; Wolf, P. Testing Lorentz symmetry with planetary orbital dynamics. Phys. Rev. D
**2015**, 92, 064049. [Google Scholar] [CrossRef] - Bennett, D.; Skavysh, V.; Long, J. Search for Lorentz Violation in a Short-Range Gravity Experiment. In Proceedings of the Fifth Meeting on CPT and Lorentz Symmetry, Bloomington, IN, USA, 28 June–2 July 2010.
- Bailey, Q.G.; Everett, R.D.; Overduin, J.M. Limits on violations of Lorentz symmetry from Gravity Probe B. Phys. Rev. D
**2013**, 88, 102001. [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] [PubMed] - Shao, L. New pulsar limit on local Lorentz invariance violation of gravity in the standard-model extension. Phys. Rev. D
**2014**, 90, 122009. [Google Scholar] [CrossRef] - Le Poncin-Lafitte, C.; Hees, A.; lambert, S. Lorentz symmetry and Very Long Baseline Interferometry. arXiv
**2016**. [Google Scholar] - Kostelecký, V.A.; Tasson, J.D. Constraints on Lorentz violation from gravitational Čerenkov radiation. Phys. Lett. B
**2015**, 749, 551–559. [Google Scholar] [CrossRef] - Shao, C.G.; Tan, Y.J.; Tan, W.H.; Yang, S.Q.; Luo, J.; Tobar, M.E. Search for Lorentz invariance violation through tests of the gravitational inverse square law at short ranges. Phys. Rev. D
**2015**, 91, 102007. [Google Scholar] [CrossRef] - Long, J.C.; Kostelecký, V.A. Search for Lorentz violation in short-range gravity. Phys. Rev. D
**2015**, 91, 092003. [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] [PubMed] - Kostelecký, V.A.; Mewes, M. Testing local Lorentz invariance with gravitational waves. Phys. Lett. B
**2016**, 757, 510–514. [Google Scholar] [CrossRef] - Yunes, N.; Yagi, K.; Pretorius, F. Theoretical physics implications of the binary black-hole mergers GW150914 and GW151226. Phys. Rev. D
**2016**, 94, 084002. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Russell, N.; Tasson, J.D. Constraints on Torsion from Bounds on Lorentz Violation. Phys. Rev. Lett.
**2008**, 100, 111102. [Google Scholar] [CrossRef] [PubMed] - Heckel, B.R.; Adelberger, E.G.; Cramer, C.E.; Cook, T.S.; Schlamminger, S.; Schmidt, U. Preferred-frame and CP-violation tests with polarized electrons. Phys. Rev. D
**2008**, 78, 092006. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Mewes, M. Signals for Lorentz violation in electrodynamics. Phys. Rev. D
**2002**, 66, 056005. [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] - Kostelecký, V.A. Riemann-Finsler geometry and Lorentz-violating kinematics. Phys. Lett. B
**2011**, 701, 137–143. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Russell, N. Classical kinematics for Lorentz violation. Phys. Lett. B
**2010**, 693, 443–447. [Google Scholar] [CrossRef] - Jacobson, T.; Mattingly, D. Gravity with a dynamical preferred frame. Phys. Rev. D
**2001**, 64, 024028. [Google Scholar] [CrossRef] - Jackiw, R.; Pi, S.Y. Chern-Simons modification of general relativity. Phys. Rev. D
**2003**, 68, 104012. [Google Scholar] [CrossRef] - Hernaski, C.A.; Belich, H. Lorentz violation and higher derivative gravity. Phys. Rev. D
**2014**, 89, 104027. [Google Scholar] [CrossRef] - Balakin, A.B.; Lemos, J.P.S. Einstein-aether theory with a Maxwell field: General formalism. Ann. Phys.
**2014**, 350, 454–484. [Google Scholar] [CrossRef] - Yagi, K.; Blas, D.; Yunes, N.; Barausse, E. Strong Binary Pulsar Constraints on Lorentz Violation in Gravity. Phys. Rev. Lett.
**2014**, 112, 161101. [Google Scholar] [CrossRef] [PubMed] - Yagi, K.; Blas, D.; Barausse, E.; Yunes, N. Constraints on Einstein-AEther theory and Hor̆ava gravity from binary pulsar observations. Phys. Rev. D
**2014**, 89, 084067. [Google Scholar] [CrossRef] - Hernaski, C.A. Quantization and stability of bumblebee electrodynamics. Phys. Rev. D
**2014**, 90, 124036. [Google Scholar] [CrossRef] - Seifert, M.D. Vector models of gravitational Lorentz symmetry breaking. Phys. Rev. D
**2009**, 79, 124012. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Potting, R. Gravity from local Lorentz violation. Gen. Relativ. Gravit.
**2005**, 37, 1675–1679. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Potting, R. Gravity from spontaneous Lorentz violation. Phys. Rev. D
**2009**, 79, 065018. [Google Scholar] [CrossRef] - Altschul, B.; Bailey, Q.G.; Kostelecký, V.A. Lorentz violation with an antisymmetric tensor. Phys. Rev. D
**2010**, 81, 065028. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Vargas, A.J. Lorentz and C P T tests with hydrogen, antihydrogen, and related systems. Phys. Rev. D
**2015**, 92, 056002. [Google Scholar] [CrossRef] - Bailey, Q.G. Time delay and Doppler tests of the Lorentz symmetry of gravity. Phys. Rev. D
**2009**, 80, 044004. [Google Scholar] [CrossRef] - Bailey, Q.G. Lorentz-violating gravitoelectromagnetism. Phys. Rev. D
**2010**, 82, 065012. [Google Scholar] [CrossRef] - Tso, R.; Bailey, Q.G. Light-bending tests of Lorentz invariance. Phys. Rev. D
**2011**, 84, 085025. [Google Scholar] [CrossRef] - Touboul, P.; Rodrigues, M. The MICROSCOPE space mission. Class. Quantum Gravity
**2001**, 18, 2487–2498. [Google Scholar] [CrossRef] - Touboul, P.; Métris, G.; Lebat, V.; Robert, A. The MICROSCOPE experiment, ready for the in-orbit test of the equivalence principle. Class. Quantum Gravity
**2012**, 29, 184010. [Google Scholar] [CrossRef] - Kostelecký, V.A.; Tasson, J.D. Prospects for Large Relativity Violations in Matter-Gravity Couplings. Phys. Rev. Lett.
**2009**, 102, 010402. [Google Scholar] [CrossRef] [PubMed] - Kostelecký, V.A.; Lehnert, R. Stability, causality, and Lorentz and CPT violation. Phys. Rev. D
**2001**, 63, 065008. [Google Scholar] [CrossRef] - Nordtvedt, K., Jr.; Will, C.M. Conservation Laws and Preferred Frames in Relativistic Gravity. II. Experimental Evidence to Rule Out Preferred-Frame Theories of Gravity. Astrophys. J.
**1972**, 177, 775. [Google Scholar] [CrossRef] - Warburton, R.J.; Goodkind, J.M. Search for evidence of a preferred reference frame. Astrophys. J.
**1976**, 208, 881–886. [Google Scholar] [CrossRef] - Nordtvedt, K., Jr. Anisotropic parametrized post-Newtonian gravitational metric field. Phys. Rev. D
**1976**, 14, 1511–1517. [Google Scholar] [CrossRef] - Bordeé, C.J. Atomic interferometry with internal state labelling. Phys. Lett. A
**1989**, 140, 10–12. [Google Scholar] [CrossRef] - Farah, T.; Guerlin, C.; Landragin, A.; Bouyer, P.; Gaffet, S.; Pereira Dos Santos, F.; Merlet, S. Underground operation at best sensitivity of the mobile LNE-SYRTE cold atom gravimeter. Gyroscopy Navig.
**2014**, 5, 266–274. [Google Scholar] [CrossRef] - Hauth, M.; Freier, C.; Schkolnik, V.; Senger, A.; Schmidt, M.; Peters, A. First gravity measurements using the mobile atom interferometer GAIN. Appl. Phys. B Lasers Opt.
**2013**, 113, 49–55. [Google Scholar] [CrossRef] - Hu, Z.K.; Sun, B.L.; Duan, X.C.; Zhou, M.K.; Chen, L.L.; Zhan, S.; Zhang, Q.Z.; Luo, J. Demonstration of an ultrahigh-sensitivity atom-interferometry absolute gravimeter. Phys. Rev. A
**2013**, 88, 043610. [Google Scholar] [CrossRef] - Tamura, Y. Bulletin d’Information Marées Terrestres; Royal Observatory of Belgium: Uccle, Belgium, 1987; Volume 99, p. 6813. [Google Scholar]
- Dehant, V.; Defraigne, P.; Wahr, J.M. Tides for a convective Earth. J. Geophys. Res.
**1999**, 104, 1035–1058. [Google Scholar] [CrossRef] - Egbert, G.D.; Bennett, A.F.; Foreman, M.G.G. TOPEX/POSEIDON tides estimated using a global inverse model. J. Geophys. Res.
**1994**, 99, 24821. [Google Scholar] [CrossRef] - Peters, A.; Chung, K.Y.; Chu, S. High-precision gravity measurements using atom interferometry. Metrologia
**2001**, 38, 25–61. [Google Scholar] [CrossRef] - Merlet, S.; Kopaev, A.; Diament, M.; Geneves, G.; Landragin, A.; Pereira Dos Santos, F. Micro-gravity investigations for the LNE watt balance project. Metrologia
**2008**, 45, 265–274. [Google Scholar] [CrossRef] - Fey, A.L.; Gordon, D.; Jacobs, C.S.; Ma, C.; Gaume, R.A.; Arias, E.F.; Bianco, G.; Boboltz, D.A.; Böckmann, S.; Bolotin, S.; et al. The Second Realization of the International Celestial Reference Frame by Very Long Baseline Interferometry. Astron. J.
**2015**, 150, 58. [Google Scholar] [CrossRef] - Soffel, M.; Klioner, S.A.; Petit, G.; Wolf, P.; Kopeikin, S.M.; Bretagnon, P.; Brumberg, V.A.; Capitaine, N.; Damour, T.; Fukushima, T.; et al. The IAU 2000 Resolutions for Astrometry, Celestial Mechanics, and Metrology in the Relativistic Framework: Explanatory Supplement. Astron. J.
**2003**, 126, 2687–2706. [Google Scholar] [CrossRef] - Lambert, S.B.; Le Poncin-Lafitte, C. Determining the relativistic parameter γ using very long baseline interferometry. Astron. Astrophys.
**2009**, 499, 331–335. [Google Scholar] [CrossRef] - Lambert, S.B.; Le Poncin-Lafitte, C. Improved determination of γ by VLBI. Astron. Astrophys.
**2011**, 529, A70. [Google Scholar] [CrossRef] - Le Poncin-Lafitte, C.; Linet, B.; Teyssandier, P. World function and time transfer: General post-Minkowskian expansions. Class. Quantum Gravity
**2004**, 21, 4463–4483. [Google Scholar] [CrossRef] - Teyssandier, P.; Le Poncin-Lafitte, C. General post-Minkowskian expansion of time transfer functions. Class. Quantum Gravity
**2008**, 25, 145020. [Google Scholar] [CrossRef] - Le Poncin-Lafitte, C.; Teyssandier, P. Influence of mass multipole moments on the deflection of a light ray by an isolated axisymmetric body. Phys. Rev. D
**2008**, 77, 044029. [Google Scholar] [CrossRef] - Hees, A.; Bertone, S.; Le Poncin-Lafitte, C. Relativistic formulation of coordinate light time, Doppler, and astrometric observables up to the second post-Minkowskian order. Phys. Rev. D
**2014**, 89, 064045. [Google Scholar] [CrossRef] - Hees, A.; Bertone, S.; Le Poncin-Lafitte, C. Light propagation in the field of a moving axisymmetric body: Theory and applications to the Juno mission. Phys. Rev. D
**2014**, 90, 084020. [Google Scholar] [CrossRef] - Finkelstein, A.M.; Kreinovich, V.I.; Pandey, S.N. Relativistic reductions for radiointerferometric observables. Astrophys. Space Sci.
**1983**, 94, 233–247. [Google Scholar] [CrossRef] - Petit, G.; Luzum, B. IERS Conventions (2010); Bundesamt für Kartographie und Geodäsie: Frankfurt am Main, Germany, 2010. [Google Scholar]
- Chapront, J.; Chapront-Touzé, M.; Francou, G. Determination of the lunar orbital and rotational parameters and of the ecliptic reference system orientation from LLR measurements and IERS data. Astron. Astrophys.
**1999**, 343, 624–633. [Google Scholar] - Dickey, J.O.; Bender, P.L.; Faller, J.E.; Newhall, X.X.; Ricklefs, R.L.; Ries, J.G.; Shelus, P.J.; Veillet, C.; Whipple, A.L.; Wiant, J.R.; et al. Lunar Laser Ranging: A Continuing Legacy of the Apollo Program. Science
**1994**, 265, 482–490. [Google Scholar] [CrossRef] [PubMed] - Nordtvedt, K. Equivalence Principle for Massive Bodies. I. Phenomenology. Phys. Rev.
**1968**, 169, 1014–1016. [Google Scholar] [CrossRef] - Nordtvedt, K. Equivalence Principle for Massive Bodies. II. Theory. Phys. Rev.
**1968**, 169, 1017–1025. [Google Scholar] [CrossRef] - Nordtvedt, K. Testing Relativity with Laser Ranging to the Moon. Phys. Rev.
**1968**, 170, 1186–1187. [Google Scholar] [CrossRef] - Williams, J.G.; Turyshev, S.G.; Boggs, D.H. Progress in Lunar Laser Ranging Tests of Relativistic Gravity. Phys. Rev. Lett.
**2004**, 93, 261101. [Google Scholar] [CrossRef] [PubMed] - Williams, J.G.; Turyshev, S.G.; Boggs, D.H. Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon. Int. J. Mod. Phys. D
**2009**, 18, 1129–1175. [Google Scholar] [CrossRef] - Merkowitz, S.M. Tests of Gravity Using Lunar Laser Ranging. Living Rev. Relativ.
**2010**, 13, 7. [Google Scholar] [CrossRef] - Bertotti, B.; Iess, L.; Tortora, P. A test of general relativity using radio links with the Cassini spacecraft. Nature
**2003**, 425, 374–376. [Google Scholar] [CrossRef] [PubMed] - Müller, J.; Williams, J.G.; Turyshev, S.G. Lunar Laser Ranging Contributions to Relativity and Geodesy. In Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space; Astrophysics and Space Science Library; Dittus, H., Lammerzahl, C., Turyshev, S.G., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; Volume 349, pp. 457–472. [Google Scholar]
- Folkner, W.M.; Williams, J.G.; Boggs, D.H.; Park, R.; Kuchynka, P. The Planetary and Lunar Ephemeris DE 430 and DE431. IPN Prog. Report.
**2014**, 42, 196. [Google Scholar] - Lupton, R. Statistics in theory and practice. Econ. J.
**1993**, 43, 688–690. [Google Scholar] - Gottlieb, A.D. Asymptotic equivalence of the jackknife and infinitesimal jackknife variance estimators for some smooth statistics. Ann. Inst. Stat. Math.
**2003**, 55, 555–561. [Google Scholar] [CrossRef] - Konopliv, A.S.; Park, R.S.; Yuan, D.N.; Asmar, S.W.; Watkins, M.M.; Williams, J.G.; Fahnestock, E.; Kruizinga, G.; Paik, M.; Strekalov, D.; et al. High-resolution lunar gravity fields from the GRAIL Primary and Extended Missions. Gepphys. Res. Lett.
**2014**, 41, 1452–1458. [Google Scholar] [CrossRef] - Lemoine, F.G.; Goossens, S.; Sabaka, T.J.; Nicholas, J.B.; Mazarico, E.; Rowlands, D.D.; Loomis, B.D.; Chinn, D.S.; Neumann, G.A.; Smith, D.E.; et al. GRGM900C: A degree 900 lunar gravity model from GRAIL primary and extended mission data. Gepphys. Res. Lett.
**2014**, 41, 3382–3389. [Google Scholar] [CrossRef] [PubMed] - Arnold, D.; Bertone, S.; Jäggi, A.; Beutler, G.; Mervart, L. GRAIL gravity field determination using the Celestial Mechanics Approach. Icarus
**2015**, 261, 182–192. [Google Scholar] [CrossRef] - Standish, E.M. The JPL planetary ephemerides. Celest. Mech.
**1982**, 26, 181–186. [Google Scholar] [CrossRef] - Newhall, X.X.; Standish, E.M.; Williams, J.G. DE 102—A numerically integrated ephemeris of the moon and planets spanning forty-four centuries. Astron. Astrophys.
**1983**, 125, 150–167. [Google Scholar] - Standish, E.M., Jr. The observational basis for JPL’s DE 200, the planetary ephemerides of the Astronomical Almanac. Astron. Astrophys.
**1990**, 233, 252–271. [Google Scholar] - Standish, E.M. Testing alternate gravitational theories. IAU Symp.
**2010**, 261, 179–182. [Google Scholar] [CrossRef] - Standish, E.M.; Williams, J.G. Orbital Ephemerides of the Sun, Moon, and Planets. In Explanatory Supplement to the Astronomical Almanac, 3rd ed.; Urban, S.E., Seidelmann, P.K., Eds.; Univeristy Science Books: Herndon, VA, USA, 2012; Chapter 8; pp. 305–346. [Google Scholar]
- Hees, A.; Folkner, W.M.; Jacobson, R.A.; Park, R.S. Constraints on modified Newtonian dynamics theories from radio tracking data of the Cassini spacecraft. Phys. Rev. D
**2014**, 89, 102002. [Google Scholar] [CrossRef] - Fienga, A.; Manche, H.; Laskar, J.; Gastineau, M. INPOP06: A new numerical planetary ephemeris. Astron. Astrophys.
**2008**, 477, 315–327. [Google Scholar] [CrossRef] - Fienga, A.; Laskar, J.; Morley, T.; Manche, H.; Kuchynka, P.; Le Poncin-Lafitte, C.; Budnik, F.; Gastineau, M.; Somenzi, L. INPOP08, a 4-D planetary ephemeris: From asteroid and time-scale computations to ESA Mars Express and Venus Express contributions. Astron. Astrophys.
**2009**, 507, 1675–1686. [Google Scholar] [CrossRef] - Fienga, A.; Laskar, J.; Kuchynka, P.; Le Poncin-Lafitte, C.; Manche, H.; Gastineau, M. Gravity tests with INPOP planetary ephemerides. IAU Symp.
**2010**, 261, 159–169. [Google Scholar] [CrossRef] - Fienga, A.; Laskar, J.; Kuchynka, P.; Manche, H.; Desvignes, G.; Gastineau, M.; Cognard, I.; Theureau, G. The INPOP10a planetary ephemeris and its applications in fundamental physics. Celest. Mech. Dyn. Astron.
**2011**, 111, 363–385. [Google Scholar] [CrossRef] [Green Version] - Verma, A.K.; Fienga, A.; Laskar, J.; Manche, H.; Gastineau, M. Use of MESSENGER radioscience data to improve planetary ephemeris and to test general relativity. Astron. Astrophys.
**2014**, 561, A115. [Google Scholar] [CrossRef] [Green Version] - Fienga, A.; Laskar, J.; Exertier, P.; Manche, H.; Gastineau, M. Numerical estimation of the sensitivity of INPOP planetary ephemerides to general relativity parameters. Celest. Mech. Dyn. Astron.
**2015**, 123, 325–349. [Google Scholar] [CrossRef] - Pitjeva, E.V. High-Precision Ephemerides of Planets EPM and Determination of Some Astronomical Constants. Sol. Syst. Res.
**2005**, 39, 176–186. [Google Scholar] [CrossRef] - Pitjeva, E.V. EPM ephemerides and relativity. IAU Symp.
**2010**, 261, 170–178. [Google Scholar] [CrossRef] - Pitjeva, E.V.; Pitjev, N.P. Relativistic effects and dark matter in the Solar system from observations of planets and spacecraft. Mon. Not. R. Astrono. Soc.
**2013**, 432, 3431–3437. [Google Scholar] [CrossRef] - Pitjeva, E.V. Updated IAA RAS planetary ephemerides-EPM2011 and their use in scientific research. Sol. Syst. Res.
**2013**, 47, 386–402. [Google Scholar] [CrossRef] - Pitjeva, E.V.; Pitjev, N.P. Development of planetary ephemerides EPM and their applications. Celest. Mech. Dyn. Astron.
**2014**, 119, 237–256. [Google Scholar] [CrossRef] - Konopliv, A.S.; Asmar, S.W.; Folkner, W.M.; Karatekin, O.; Nunes, D.C.; Smrekar, S.E.; Yoder, C.F.; Zuber, M.T. Mars high resolution gravity fields from MRO, Mars seasonal gravity, and other dynamical parameters. Icarus
**2011**, 211, 401–428. [Google Scholar] [CrossRef] - Pitjev, N.P.; Pitjeva, E.V. Constraints on dark matter in the solar system. Astron. Lett.
**2013**, 39, 141–149. [Google Scholar] [CrossRef] - Milgrom, M. MOND effects in the inner Solar system. Mon. Not. R. Astrono. Soc.
**2009**, 399, 474–486. [Google Scholar] [CrossRef] - Blanchet, L.; Novak, J. External field effect of modified Newtonian dynamics in the Solar system. Mon. Not. R. Astrono. Soc.
**2011**, 412, 2530–2542. [Google Scholar] [CrossRef] [Green Version] - Hees, A.; Famaey, B.; Angus, G.W.; Gentile, G. Combined Solar system and rotation curve constraints on MOND. Mon. Not. R. Astrono. Soc.
**2016**, 455, 449–461. [Google Scholar] [CrossRef] - Hees, A.; Lamine, B.; Reynaud, S.; Jaekel, M.T.; Le Poncin-Lafitte, C.; Lainey, V.; Füzfa, A.; Courty, J.M.; Dehant, V.; Wolf, P. Radioscience simulations in General Relativity and in alternative theories of gravity. Class. Quantum Gravity
**2012**, 29, 235027. [Google Scholar] [CrossRef] [Green Version] - de Sitter, W. Einstein’s theory of gravitation and its astronomical consequences. Mon. Not. R. Astrono. Soc.
**1916**, 76, 699–728. [Google Scholar] [CrossRef] - Lense, J.; Thirring, H. Über den Einfluß der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie. Phys. Z.
**1918**, 19, 156. [Google Scholar] - Schiff, L.I. Possible New Experimental Test of General Relativity Theory. Phys. Rev. Lett.
**1960**, 4, 215–217. [Google Scholar] [CrossRef] - Pugh, G.E. Proposal for a Satellite Test of the Coriolis Predicition of General Relativity. In Nonlinear Gravitodynamics: The Lense-Thirring Effect; Word Scientific Publishing: Singapore, 1959. [Google Scholar]
- Everitt, C.W.F.; Debra, D.B.; Parkinson, B.W.; Turneaure, J.P.; Conklin, J.W.; Heifetz, M.I.; Keiser, G.M.; Silbergleit, A.S.; Holmes, T.; Kolodziejczak, J.; et al. Gravity Probe B: Final Results of a Space Experiment to Test General Relativity. Phys. Rev. Lett.
**2011**, 106, 221101. [Google Scholar] [CrossRef] [PubMed] - Stella, L.; Vietri, M. kHz Quasiperiodic Oscillations in Low-Mass X-Ray Binaries as Probes of General Relativity in the Strong-Field Regime. Phys. Rev. Lett.
**1999**, 82, 17–20. [Google Scholar] [CrossRef] - Hulse, R.A.; Taylor, J.H. Discovery of a pulsar in a binary system. Astrophys. J.
**1975**, 195, L51–L53. [Google Scholar] [CrossRef] - Taylor, J.H.; Hulse, R.A.; Fowler, L.A.; Gullahorn, G.E.; Rankin, J.M. Further observations of the binary pulsar PSR 1913+16. Astrophys. J.
**1976**, 206, L53–L58. [Google Scholar] [CrossRef] - Taylor, J.H.; Fowler, L.A.; McCulloch, P.M. Measurements of general relativistic effects in the binary pulsar PSR 1913+16. Nature
**1979**, 277, 437–440. [Google Scholar] [CrossRef] - Damour, T.; Deruelle, N. General relativistic celestial mechanics of binary systems. II. The post-Newtonian timing formula. Ann. Inst. Henri Poincaré Phys. Théor.
**1986**, 44, 263–292. [Google Scholar] - Stairs, I.H. Testing General Relativity with Pulsar Timing. Living Rev. Relativ.
**2003**, 6, 5. [Google Scholar] [CrossRef] - Lorimer, D.R. Binary and Millisecond Pulsars. Living Rev. Relativ.
**2008**, 11, 8. [Google Scholar] [CrossRef] - Damour, T.; Deruelle, N. General relativistic celestial mechanics of binary systems. I. The post-Newtonian motion. Ann. Inst. Henri Poincaré Phys. Théor.
**1985**, 43, 107–132. [Google Scholar] - Wex, N. The second post-Newtonian motion of compact binary-star systems with spin. Class. Quantum Gravity
**1995**, 12, 983–1005. [Google Scholar] [CrossRef] - Edwards, R.T.; Hobbs, G.B.; Manchester, R.N. TEMPO2, a new pulsar timing package - II. The timing model and precision estimates. Mon. Not. R. Astrono. Soc.
**2006**, 372, 1549–1574. [Google Scholar] [CrossRef] - Wex, N. Testing Relativistic Gravity with Radio Pulsars. In Frontiers in Relativistic Celestial Mechanics; Applications and Experiments; Kopeikin, S., Ed.; De Gruyter: Berlin, Germany, 2014; Volume 2. [Google Scholar]
- Kramer, M. Pulsars as probes of gravity and fundamental physics. Int. J. Mod. Phys. D
**2016**, 25, 14. [Google Scholar] [CrossRef] - Kramer, M.; Stairs, I.H.; Manchester, R.N.; McLaughlin, M.A.; Lyne, A.G.; Ferdman, R.D.; Burgay, M.; Lorimer, D.R.; Possenti, A.; D’Amico, N.; et al. Tests of General Relativity from Timing the Double Pulsar. Science
**2006**, 314, 97–102. [Google Scholar] [CrossRef] [PubMed] - Damour, T.; Esposito-Farese, G. Tensor-multi-scalar theories of gravitation. Class. Quantum Gravity
**1992**, 9, 2093–2176. [Google Scholar] [CrossRef] - Damour, T.; Taylor, J.H. Strong-field tests of relativistic gravity and binary pulsars. Phys. Rev. D
**1992**, 45, 1840–1868. [Google Scholar] [CrossRef] - Damour, T.; Esposito-Farèse, G. Tensor-scalar gravity and binary-pulsar experiments. Phys. Rev. D
**1996**, 54, 1474–1491. [Google Scholar] [CrossRef] [Green Version] - Freire, P.C.C.; Wex, N.; Esposito-Farèse, G.; Verbiest, J.P.W.; Bailes, M.; Jacoby, B.A.; Kramer, M.; Stairs, I.H.; Antoniadis, J.; Janssen, G.H. The relativistic pulsar-white dwarf binary PSR J1738+0333 - II. The most stringent test of scalar-tensor gravity. Mon. Not. R. Astrono. Soc.
**2012**, 423, 3328–3343. [Google Scholar] [CrossRef] - Ransom, S.M.; Stairs, I.H.; Archibald, A.M.; Hessels, J.W.T.; Kaplan, D.L.; van Kerkwijk, M.H.; Boyles, J.; Deller, A.T.; Chatterjee, S.; Schechtman-Rook, A.; et al. A millisecond pulsar in a stellar triple system. Nature
**2014**, 505, 520–524. [Google Scholar] [CrossRef] [PubMed] - Damour, T.; Esposito-Farese, G. Nonperturbative strong-field effects in tensor-scalar theories of gravitation. Phys. Rev. Lett.
**1993**, 70, 2220–2223. [Google Scholar] [CrossRef] [PubMed] - Foster, B.Z. Strong field effects on binary systems in Einstein-aether theory. Phys. Rev. D
**2007**, 76, 084033. [Google Scholar] [CrossRef] - Wex, N.; Kramer, M. A characteristic observable signature of preferred-frame effects in relativistic binary pulsars. Mon. Not. R. Astrono. Soc.
**2007**, 380, 455–465. [Google Scholar] [CrossRef] - Shao, L.; Wex, N. New tests of local Lorentz invariance of gravity with small-eccentricity binary pulsars. Class. Quantum Gravity
**2012**, 29, 215018. [Google Scholar] [CrossRef] - Nordtvedt, K. Probing gravity to the second post-Newtonian order and to one part in 10 to the 7th using the spin axis of the sun. Astrophys. J.
**1987**, 320, 871–874. [Google Scholar] [CrossRef] - Shao, L.; Caballero, R.N.; Kramer, M.; Wex, N.; Champion, D.J.; Jessner, A. A new limit on local Lorentz invariance violation of gravity from solitary pulsars. Class. Quantum Gravity
**2013**, 30, 165019. [Google Scholar] [CrossRef] - Bell, J.F.; Damour, T. A new test of conservation laws and Lorentz invariance in relativistic gravity. Class. Quantum Gravity
**1996**, 13, 3121–3127. [Google Scholar] [CrossRef] - Gonzalez, M.E.; Stairs, I.H.; Ferdman, R.D.; Freire, P.C.C.; Nice, D.J.; Demorest, P.B.; Ransom, S.M.; Kramer, M.; Camilo, F.; Hobbs, G.; et al. High-precision Timing of Five Millisecond Pulsars: Space Velocities, Binary Evolution, and Equivalence Principles. Astrophys. J.
**2011**, 743, 102. [Google Scholar] [CrossRef] - Lorimer, D.R.; Kramer, M. Handbook of Pulsar Astronomy; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Jennings, R.J.; Tasson, J.D.; Yang, S. Matter-sector Lorentz violation in binary pulsars. Phys. Rev. D
**2015**, 92, 125028. [Google Scholar] [CrossRef] - Moore, G.D.; Nelson, A.E. Lower bound on the propagation speed of gravity from gravitational Cherenkov radiation. J. High Energy Phys.
**2001**, 9, 023. [Google Scholar] [CrossRef] - Kiyota, S.; Yamamoto, K. Constraint on modified dispersion relations for gravitational waves from gravitational Cherenkov radiation. Phys. Rev. D
**2015**, 92, 104036. [Google Scholar] [CrossRef] - Elliott, J.W.; Moore, G.D.; Stoica, H. Constraining the New Aether: Gravitational Cherenkov radiation. J. High Energy Phys.
**2005**, 8, 066. [Google Scholar] [CrossRef] - Kimura, R.; Yamamoto, K. Constraints on general second-order scalar-tensor models from gravitational Cherenkov radiation. J. Cosmol. Astropart. Phys.
**2012**, 7, 050. [Google Scholar] [CrossRef] - De Laurentis, M.; Capozziello, S.; Basini, G. Gravitational Cherenkov Radiation from Extended Theories of Gravity. Mod. Phys. Lett. A
**2012**, 27, 1250136. [Google Scholar] [CrossRef] - Kimura, R.; Tanaka, T.; Yamamoto, K.; Yamashita, Y. Constraint on ghost-free bigravity from gravitational Cherenkov radiation. Phys. Rev. D
**2016**, 94, 064059. [Google Scholar] [CrossRef] - Takeda, M.; Hayashida, N.; Honda, K.; Inoue, N.; Kadota, K.; Kakimoto, F.; Kamata, K.; Kawaguchi, S.; Kawasaki, Y.; Kawasumi, N.; et al. Small-Scale Anisotropy of Cosmic Rays above 10
^{19}eV Observed with the Akeno Giant Air Shower Array. Astrophys. J.**1999**, 522, 225–237. [Google Scholar] [CrossRef] - Bird, D.J.; Corbato, S.C.; Dai, H.Y.; Elbert, J.W.; Green, K.D.; Huang, M.A.; Kieda, D.B.; Ko, S.; Larsen, C.G.; Loh, E.C.; et al. Detection of a cosmic ray with measured energy well beyond the expected spectral cutoff due to cosmic microwave radiation. Astrophys. J.
**1995**, 441, 144–150. [Google Scholar] [CrossRef] - Wada, M. Catalogue of Highest Energy Cosmic Rays. Giant Extensive Air Showers. No._1. Volcano Ranch, Haverah Park; Institute of Physical and Chemical Research: Tokyo, Japan, 1980. [Google Scholar]
- High Resolution Fly’S Eye Collaboration.; Abbasi, R.U.; Abu-Zayyad, T.; Allen, M.; Amman, J.F.; Archbold, G.; Belov, K.; Belz, J.W.; BenZvi, S.Y.; Bergman, D.R.; et al. Search for correlations between HiRes stereo events and active galactic nuclei. Astropart. Phys.
**2008**, 30, 175–179. [Google Scholar] [CrossRef] - Aab, A.; Abreu, P.; Aglietta, M.; Ahn, E.J.; Al Samarai, I.; Albuquerque, I.F.M.; Allekotte, I.; Allen, J.; Allison, P.; Almela, A.; et al. Searches for Anisotropies in the Arrival Directions of the Highest Energy Cosmic Rays Detected by the Pierre Auger Observatory. Astrophys. J.
**2015**, 804, 15. [Google Scholar] [CrossRef] - Winn, M.M.; Ulrichs, J.; Peak, L.S.; McCusker, C.B.A.; Horton, L. The cosmic-ray energy spectrum above 10
^{17}eV. J. Phys. G Nucl. Phys.**1986**, 12, 653–674. [Google Scholar] [CrossRef] - Abbasi, R.U.; Abe, M.; Abu-Zayyad, T.; Allen, M.; Anderson, R.; Azuma, R.; Barcikowski, E.; Belz, J.W.; Bergman, D.R.; Blake, S.A.; et al. Indications of Intermediate-scale Anisotropy of Cosmic Rays with Energy Greater Than 57 EeV in the Northern Sky Measured with the Surface Detector of the Telescope Array Experiment. Astrophys. J.
**2014**, 790, L21. [Google Scholar] [CrossRef] - Pravdin, M.I.; Glushkov, A.V.; Ivanov, A.A.; Knurenko, S.P.; Kolosov, V.A.; Makarov, I.T.; Sabourov, A.V.; Sleptsov, I.Y.; Struchkov, G.G. Estimation of the giant shower energy at the Yakutsk EAS Array. Int. Cosm. Ray Conf.
**2005**, 7, 243. [Google Scholar] - De Bruijne, J.H.J. Science performance of Gaia, ESA’s space-astrometry mission. Astrophys. Space Sci.
**2012**, 341, 31–41. [Google Scholar] [CrossRef] - Mignard, F.; Klioner, S.A. Gaia: Relativistic modelling and testing. IAU Symp.
**2010**, 261, 306–314. [Google Scholar] [CrossRef] - Jaekel, M.T.; Reynaud, S. Gravity Tests in the Solar System and the Pioneer Anomaly. Mod. Phys. Lett. A
**2005**, 20, 1047–1055. [Google Scholar] [CrossRef] - Jaekel, M.T.; Reynaud, S. Post-Einsteinian tests of linearized gravitation. Class. Quantum Gravity
**2005**, 22, 2135–2157. [Google Scholar] [CrossRef] - Jaekel, M.T.; Reynaud, S. Post-Einsteinian tests of gravitation. Class. Quantum Gravity
**2006**, 23, 777–798. [Google Scholar] [CrossRef] - Reynaud, S.; Jaekel, M.T. Long Range Gravity Tests and the Pioneer Anomaly. Int. J. Mod. Phys. D
**2007**, 16, 2091–2105. [Google Scholar] [CrossRef] - Reynaud, S.; Jaekel, M.T. Tests of general relativity in the Solar System. Atom Opt. Space Phys.
**2009**, 168, 203–207. [Google Scholar] - Gai, M.; Vecchiato, A.; Ligori, S.; Sozzetti, A.; Lattanzi, M.G. Gravitation astrometric measurement experiment. Exp. Astron.
**2012**, 34, 165–180. [Google Scholar] [CrossRef] - Turyshev, S.G.; Shao, M. Laser Astrometric Test of Relativity: Science, Technology and Mission Design. Int. J. Mod. Phys. D
**2007**, 16, 2191–2203. [Google Scholar] [CrossRef] - Mouret, S. Tests of fundamental physics with the Gaia mission through the dynamics of minor planets. Phys. Rev. D
**2011**, 84, 122001. [Google Scholar] [CrossRef] - Hees, A.; Hestroffer, D.; Le Poncin-Lafitte, C.; David, P. Tests of gravitation with GAIA observations of Solar System Objects. In Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics (SF2A-2015), Toulouse, France, 2–5 June 2015; Martins, F., Boissier, S., Buat, V., Cambrésy, L., Petit, P., Eds.; pp. 125–131.
- Margot, J.L.; Giorgini, J.D. Probing general relativity with radar astrometry in the inner solar system. IAU Symp.
**2010**, 261, 183–188. [Google Scholar] [CrossRef] - Hees, A.; Lamine, B.; Poncin-Lafitte, C.L.; Wolf, P. How to Test the SME with Space Missions? In Proceedings of the Sixth Meeting CPT and Lorentz Symmetry, Bloomington, IN, USA, 17–21 June 2013; Kostelecky, A., Ed.; pp. 107–110.
- Hees, A.; Lamine, B.; Reynaud, S.; Jaekel, M.T.; Le Poncin-Lafitte, C.; Lainey, V.; Füzfa, A.; Courty, J.M.; Dehant, V.; Wolf, P. Simulations of Solar System Observations in Alternative Theories of Gravity. In Proceedings of the Thirteenth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories, Stockholm, Sweden, 1–7 July 2012; Rosquist, K., Ed.; pp. 2357–2359.
- Iess, L.; Asmar, S. Probing Space-Time in the Solar System: From Cassini to Bepicolombo. Int. J. Mod. Phys. D
**2007**, 16, 2117–2126. [Google Scholar] [CrossRef] - Kliore, A.J.; Anderson, J.D.; Armstrong, J.W.; Asmar, S.W.; Hamilton, C.L.; Rappaport, N.J.; Wahlquist, H.D.; Ambrosini, R.; Flasar, F.M.; French, R.G.; et al. Cassini Radio Science. Space Sci. Rev.
**2004**, 115, 1–70. [Google Scholar] [CrossRef] - Iorio, L.; Ciufolini, I.; Pavlis, E.C. Measuring the relativistic perigee advance with satellite laser ranging. Class. Quantum Gravity
**2002**, 19, 4301–4309. [Google Scholar] [CrossRef] - Lucchesi, D.M.; Peron, R. Accurate Measurement in the Field of the Earth of the General-Relativistic Precession of the LAGEOS II Pericenter and New Constraints on Non-Newtonian Gravity. Phys. Rev. Lett.
**2010**, 105, 231103. [Google Scholar] [CrossRef] [PubMed] - Lucchesi, D.M.; Peron, R. LAGEOS II pericenter general relativistic precession (1993–2005): Error budget and constraints in gravitational physics. Phys. Rev. D
**2014**, 89, 082002. [Google Scholar] [CrossRef] - Ciufolini, I.; Pavlis, E.C. A confirmation of the general relativistic prediction of the Lense-Thirring effect. Nature
**2004**, 431, 958–960. [Google Scholar] [CrossRef] [PubMed] - Ciufolini, I.; Paolozzi, A.; Pavlis, E.C.; Koenig, R.; Ries, J.; Gurzadyan, V.; Matzner, R.; Penrose, R.; Sindoni, G.; Paris, C.; et al. A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model. Measurement of Earth’s dragging of inertial frames. Eur. Phys. J. C
**2016**, 76, 120. [Google Scholar] [CrossRef] [PubMed] - Ciufolini, I.; Paolozzi, A.; Pavlis, E.C.; Ries, J.C.; Koenig, R.; Matzner, R.A.; Sindoni, G.; Neumayer, H. Towards a One Percent Measurement of Frame Dragging by Spin with Satellite Laser Ranging to LAGEOS, LAGEOS 2 and LARES and GRACE Gravity Models. Space Sci. Rev.
**2009**, 148, 71–104. [Google Scholar] [CrossRef] - Ciufolini, I.; Paolozzi, A.; Pavlis, E.; Ries, J.; Gurzadyan, V.; Koenig, R.; Matzner, R.; Penrose, R.; Sindoni, G. Testing General Relativity and gravitational physics using the LARES satellite. Eur. Phys. J. Plus
**2012**, 127, 133. [Google Scholar] [CrossRef] - Ciufolini, I.; Pavlis, E.C.; Paolozzi, A.; Ries, J.; Koenig, R.; Matzner, R.; Sindoni, G.; Neumayer, K.H. Phenomenology of the Lense-Thirring effect in the Solar System: Measurement of frame-dragging with laser ranged satellites. New Astron.
**2012**, 17, 341–346. [Google Scholar] [CrossRef] - Paolozzi, A.; Ciufolini, I.; Vendittozzi, C. Engineering and scientific aspects of LARES satellite. Acta Astronaut.
**2011**, 69, 127–134. [Google Scholar] [CrossRef] - Iorio, L. Towards a 1% measurement of the Lense-Thirring effect with LARES? Adv. Space Res.
**2009**, 43, 1148–1157. [Google Scholar] [CrossRef] - Iorio, L. Will the recently approved LARES mission be able to measure the Lense-Thirring effect at 1%? Gen. Relativ. Gravit.
**2009**, 41, 1717–1724. [Google Scholar] [CrossRef] - Iorio, L. An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense-Thirring Effect with Satellite Laser Ranging. Space Sci. Rev.
**2009**, 148, 363–381. [Google Scholar] [CrossRef] - Iorio, L.; Lichtenegger, H.I.M.; Ruggiero, M.L.; Corda, C. Phenomenology of the Lense-Thirring effect in the solar system. Astrophys. Space Sci.
**2011**, 331, 351–395. [Google Scholar] [CrossRef] - Renzetti, G. Are higher degree even zonals really harmful for the LARES/LAGEOS frame-dragging experiment? Can. J. Phys.
**2012**, 90, 883–888. [Google Scholar] [CrossRef] - Renzetti, G. First results from LARES: An analysis. New Astron.
**2013**, 23, 63–66. [Google Scholar] [CrossRef] - Lucchesi, D.M.; Anselmo, L.; Bassan, M.; Pardini, C.; Peron, R.; Pucacco, G.; Visco, M. Testing the gravitational interaction in the field of the Earth via satellite laser ranging and the Laser Ranged Satellites Experiment (LARASE). Class. Quantum Gravity
**2015**, 32, 155012. [Google Scholar] [CrossRef] - Lämmerzahl, C.; Ciufolini, I.; Dittus, H.; Iorio, L.; Müller, H.; Peters, A.; Samain, E.; Scheithauer, S.; Schiller, S. OPTIS–An Einstein Mission for Improved Tests of Special and General Relativity. Gen. Relativ. Gravit.
**2004**, 36, 2373–2416. [Google Scholar] [CrossRef] - Hohensee, M.A.; Chu, S.; Peters, A.; Müller, H. Equivalence Principle and Gravitational Redshift. Phys. Rev. Lett.
**2011**, 106, 151102. [Google Scholar] [CrossRef] [PubMed] - Hohensee, M.A.; Leefer, N.; Budker, D.; Harabati, C.; Dzuba, V.A.; Flambaum, V.V. Limits on Violations of Lorentz Symmetry and the Einstein Equivalence Principle using Radio-Frequency Spectroscopy of Atomic Dysprosium. Phys. Rev. Lett.
**2013**, 111, 050401. [Google Scholar] [CrossRef] [PubMed] - Hohensee, M.A.; Müller, H.; Wiringa, R.B. Equivalence Principle and Bound Kinetic Energy. Phys. Rev. Lett.
**2013**, 111, 151102. [Google Scholar] [CrossRef] [PubMed] - Delva, P.; Hees, A.; Bertone, S.; Richard, E.; Wolf, P. Test of the gravitational redshift with stable clocks in eccentric orbits: Application to Galileo satellites 5 and 6. Class. Quantum Gravity
**2015**, 32, 232003. [Google Scholar] [CrossRef] - Cacciapuoti, L.; Salomon, C. Atomic clock ensemble in space. J. Phys. Conf. Ser.
**2011**, 327, 012049. [Google Scholar] [CrossRef]

**Figure 1.**This figure represents the distribution of the orbital parameters for the Solar System Objects (SSOs) expected to be observed by the Gaia satellite. The red stars represent the innermost planets of the Solar System.

**Figure 2.**Doppler signature produced by ${\overline{s}}^{TT}=2\times {10}^{-5}$ on the 2-way Doppler link Earth-Cassini-Earth during the 2002 Solar conjunction.

Coefficient | ||||||

${\overline{s}}^{TX}$ | $\left(-3.1\pm 5.1\right)\times {10}^{-5}$ | |||||

${\overline{s}}^{TY}$ | $\left(0.1\pm 5.4\right)\times {10}^{-5}$ | |||||

${\overline{s}}^{TZ}$ | $\left(1.4\pm 6.6\right)\times {10}^{-5}$ | |||||

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $\left(4.4\pm 11\right)\times {10}^{-9}$ | |||||

${\overline{s}}^{XY}$ | $\left(0.2\pm 3.9\right)\times {10}^{-9}$ | |||||

${\overline{s}}^{XZ}$ | $\left(-2.6\pm 4.4\right)\times {10}^{-9}$ | |||||

${\overline{s}}^{YZ}$ | $\left(-0.3\pm 4.5\right)\times {10}^{-9}$ | |||||

Correlation Coefficients | ||||||

1 | ||||||

0.05 | 1 | |||||

0.11 | −0.16 | 1 | ||||

−0.82 | 0.34 | −0.16 | 1 | |||

−0.38 | −0.86 | 0.10 | −0.01 | 1 | ||

−0.41 | 0.13 | −0.89 | 0.38 | 0.02 | 1 | |

−0.12 | −0.19 | −0.89 | 0.04 | 0.20 | 0.80 | 1 |

**Table 2.**Estimation of Standard-Model Extension (SME) coefficients from Lunar Laser Ranging (LLR) postfit data analysis from [45]. No correlations coefficients have been derived in this analysis. The coefficients ${\overline{s}}^{ij}$ are projections of the ${\overline{s}}^{IJ}$ into the lunar orbital plane (see Equation (107) from [39]) while the linear combinations ${\overline{s}}_{{\mathsf{\Omega}}_{\oplus}c}$ and ${\overline{s}}_{{\mathsf{\Omega}}_{\oplus}s}$ are given by Equation (108) from [39].

Coefficient | |
---|---|

${\overline{s}}^{11}-{\overline{s}}^{22}$ | $(1.3\pm 0.9)\times {10}^{-10}$ |

${\overline{s}}^{12}$ | $(6.9\pm 4.5)\times {10}^{-11}$ |

${\overline{s}}^{01}$ | $(-0.8\pm 1.1)\times {10}^{-6\phantom{\rule{4pt}{0ex}}}$ |

${\overline{s}}^{02}$ | $(-5.2\pm 4.8)\times {10}^{-7\phantom{\rule{4pt}{0ex}}}$ |

${\overline{s}}_{{\mathsf{\Omega}}_{\oplus}c}$ | $(0.2\pm 3.9)\times {10}^{-7\phantom{\rule{4pt}{0ex}}}$ |

${\overline{s}}_{{\mathsf{\Omega}}_{\oplus}s}$ | $(-1.3\pm 4.1)\times {10}^{-7\phantom{\rule{4pt}{0ex}}}$ |

**Table 3.**Estimation of SME coefficients from a full LLR data analysis from [46] and associated correlation coefficients.

Coefficient | Estimates | ||||
---|---|---|---|---|---|

${\overline{s}}^{TX}$ | $\left(-0.9\pm 1.0\right)\times {10}^{-8\phantom{0}}$ | ||||

${\overline{s}}^{XY}$ | $\left(-5.7\pm 7.7\right)\times {10}^{-12}$ | ||||

${\overline{s}}^{XZ}$ | $\left(-2.2\pm 5.9\right)\times {10}^{-12}$ | ||||

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $\left(0.6\pm 4.2\right)\times {10}^{-11}$ | ||||

${\overline{s}}^{TY}+0.43\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}^{TZ}$ | $\left(6.2\pm 7.9\right)\times {10}^{-9\phantom{0}}$ | ||||

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}-4.5\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}^{YZ}$ | $\left(2.3\pm 4.5\right)\times {10}^{-11}$ | ||||

Correlation Coefficients | |||||

1 | |||||

−0.06 | 1 | ||||

−0.04 | 0.29 | 1 | |||

0.58 | −0.12 | −0.16 | 1 | ||

0.16 | −0.01 | −0.09 | 0.25 | 1 | |

0.07 | −0.10 | −0.13 | −0.10 | 0.03 | 1 |

**Table 4.**Estimations of the SME coefficients from a postfit data analysis based on planetary ephemerides from [50]. The uncertainties correspond to the 68% Bayesian confidence levels of the marginal posterior probability distribution function. The associated correlation coefficients can be found in Table III from [50].

Coefficient | |||||||
---|---|---|---|---|---|---|---|

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $(-0.8\pm 2.0)\times {10}^{-10}$ | ||||||

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}^{ZZ}$ | $(-0.8\pm 2.7)\times {10}^{-10}$ | ||||||

${\overline{s}}^{XY}$ | $(-0.3\pm 1.1)\times {10}^{-10}$ | ||||||

${\overline{s}}^{XZ}$ | $(-1.0\pm 3.5)\times {10}^{-11}$ | ||||||

${\overline{s}}^{YZ}$ | $(5.5\pm 5.2)\times {10}^{-12}$ | ||||||

${\overline{s}}^{TX}$ | $(-2.9\pm 8.3)\times {10}^{-9\phantom{1}}$ | ||||||

${\overline{s}}^{TY}$ | $(0.3\pm 1.4)\times {10}^{-8\phantom{1}}$ | ||||||

${\overline{s}}^{TZ}$ | $(-0.2\pm 5.0)\times {10}^{-8\phantom{1}}$ | ||||||

Correlation coefficients | |||||||

1 | |||||||

0.99 | 1 | ||||||

0.99 | 0.99 | 1 | |||||

0.98 | 0.98 | 0.99 | 1 | ||||

−0.32 | −0.24 | −0.26 | −0.26 | 1 | |||

0.99 | 0.98 | 0.98 | 0.98 | −0.32 | 1 | ||

0.62 | 0.67 | 0.62 | 0.59 | 0.36 | 0.60 | 1 | |

−0.83 | −0.86 | −0.83 | −0.81 | −0.14 | −0.82 | −0.95 | 1 |

**Table 5.**Estimations of the SME coefficients from a postfit data analysis based on Gravity Probe B (GPB) [52].

Coefficient | ||
---|---|---|

${\overline{s}}_{\mathrm{GPB}}^{\left(1\right)}$ | = ${\overline{s}}^{TT}+970\left({\overline{s}}^{XX}-{\overline{s}}^{YY}\right)-0.05\left({\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}\right)$ $+2895\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XY}-3235\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XZ}-11\phantom{\rule{0.166667em}{0ex}}240\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{YZ}$ | $(0.7\pm 3.1)\times {10}^{-3}$ |

${\overline{s}}_{\mathrm{GPB}}^{\left(2\right)}$ | = ${\overline{s}}^{XX}-{\overline{s}}^{YY}+3.02\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XY}+0.32\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XZ}+1.05\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{YZ}$ | $(-1.1\pm 3.8)\times {10}^{-7}$ |

**Table 6.**Estimation of SME coefficients from binary pulsars data analysis from [53,54]. No correlations coefficients have been derived in this analysis. These estimates should be considered as estimates on the strong field version of the SME coefficients that may include non perturbative strong field effects due to the gravitational binding energy.

Coefficient | |
---|---|

$\left|{\overline{s}}^{TT}\right|$ | $<2.8\times {10}^{-4\phantom{1}}$ |

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $(0.2\pm 9.9)\times {10}^{-11}$ |

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}$ | $(-0.05\pm 12.25)\times {10}^{-11}$ |

${\overline{s}}^{XY}$ | $(0.05\pm 3.55)\times {10}^{-11}$ |

${\overline{s}}^{XZ}$ | $(0.0\pm 2.0)\times {10}^{-11}$ |

${\overline{s}}^{YZ}$ | $(0.0\pm 3.3)\times {10}^{-11}$ |

${\overline{s}}^{TX}$ | $(0.05\pm 5.25)\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{TY}$ | $(0.5\pm 8.0)\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{TZ}$ | $(-0.05\pm 5.85)\times {10}^{-9\phantom{1}}$ |

**Table 7.**Lower and upper limits on the SME coefficients decomposed in spherical harmonics derived from Čerenkov radiation [56].

Coefficient | Lower Bound | Upper Bound |
---|---|---|

${\overline{s}}_{00}^{\left(SH\right)}$ | $-3\times {10}^{-14}$ | |

${\overline{s}}_{10}^{\left(SH\right)}$ | $-1\times {10}^{-13}$ | $7\times {10}^{-14}$ |

Re ${\overline{s}}_{11}^{\left(SH\right)}$ | $-8\times {10}^{-14}$ | $8\times {10}^{-14}$ |

Im ${\overline{s}}_{11}^{\left(SH\right)}$ | $-7\times {10}^{-14}$ | $9\times {10}^{-14}$ |

${\overline{s}}_{20}^{\left(SH\right)}$ | $-7\times {10}^{-14}$ | $1\times {10}^{-13}$ |

Re ${\overline{s}}_{21}^{\left(SH\right)}$ | $-7\times {10}^{-14}$ | $7\times {10}^{-14}$ |

Im ${\overline{s}}_{21}^{\left(SH\right)}$ | $-5\times {10}^{-14}$ | $8\times {10}^{-14}$ |

Re ${\overline{s}}_{22}^{\left(SH\right)}$ | $-6\times {10}^{-14}$ | $8\times {10}^{-14}$ |

Im ${\overline{s}}_{22}^{\left(SH\right)}$ | $-7\times {10}^{-14}$ | $7\times {10}^{-14}$ |

Atomic Grav. [48] | LLR [46] | Planetary Eph. [50] | Pulsars [53,54] | Čerenkov rad. [56] | ||
---|---|---|---|---|---|---|

Lower Bound | Upper Bound | |||||

${\overline{s}}^{TT}$ | $<2.8\times {10}^{-4\phantom{1}}$ | $-6\times {10}^{-15}<$ | ||||

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $\left(4.4\pm 11\right)\times {10}^{-9}$ | $\left(0.6\pm 4.2\right)\times {10}^{-11}$ | $(-0.8\pm 2.0)\times {10}^{-10}$ | $(0.2\pm 9.9)\times {10}^{-11}$ | $-9\times {10}^{-14}<$ | $<1.2\times {10}^{-13}$ |

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}^{ZZ}$ | $(-0.8\pm 2.7)\times {10}^{-10}$ | $(-0.05\pm 12.25)\times {10}^{-11}$ | $-1.9\times {10}^{-13}<$ | $<1.3\times {10}^{-13}$ | ||

${\overline{s}}^{XY}$ | $\left(0.2\pm 3.9\right)\times {10}^{-9}$ | $\left(-5.7\pm 7.7\right)\times {10}^{-12}$ | $(-0.3\pm 1.1)\times {10}^{-10}$ | $(0.05\pm 3.55)\times {10}^{-11}$ | $-3.9\times {10}^{-14}<$ | $<6.2\times {10}^{-14}$ |

${\overline{s}}^{XZ}$ | $\left(-2.6\pm 4.4\right)\times {10}^{-9}$ | $\left(-2.2\pm 5.9\right)\times {10}^{-12}$ | $(-1.0\pm 3.5)\times {10}^{-11}$ | $(0.0\pm 2.0)\times {10}^{-11}$ | $-5.4\times {10}^{-14}<$ | $<5.4\times {10}^{-14}$ |

${\overline{s}}^{YZ}$ | $\left(-0.3\pm 4.5\right)\times {10}^{-9}$ | $(5.5\pm 5.2)\times {10}^{-12}$ | $(0.0\pm 3.3)\times {10}^{-11}$ | $-3.9\times {10}^{-14}<$ | $<6.2\times {10}^{-14}$ | |

${\overline{s}}^{TX}$ | $\left(-3.1\pm 5.1\right)\times {10}^{-5}$ | $\left(-0.9\pm 1.0\right)\times {10}^{-8\phantom{0}}$ | $(-2.9\pm 8.3)\times {10}^{-9\phantom{1}}$ | $(0.05\pm 5.25)\times {10}^{-9\phantom{1}}$ | $2.8\times {10}^{-14}<$ | $<2.8\times {10}^{-14}$ |

${\overline{s}}^{TY}$ | $\left(0.1\pm 5.4\right)\times {10}^{-5}$ | $(0.3\pm 1.4)\times {10}^{-8\phantom{1}}$ | $(0.5\pm 8.0)\times {10}^{-9\phantom{1}}$ | $3.1\times {10}^{-14}<$ | $<2.4\times {10}^{-14}$ | |

${\overline{s}}^{TZ}$ | $\left(1.4\pm 6.6\right)\times {10}^{-5}$ | $(-0.2\pm 5.0)\times {10}^{-8\phantom{1}}$ | $(-0.05\pm 5.85)\times {10}^{-9\phantom{1}}$ | $1.7\times {10}^{-14}<$ | $<2.4\times {10}^{-14}$ | |

${\overline{s}}^{TY}+0.43\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}^{TZ}$ | $\left(6.2\pm 7.9\right)\times {10}^{-9\phantom{0}}$ | |||||

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}-4.5\phantom{\rule{3.33333pt}{0ex}}{\overline{s}}^{YZ}$ | $\left(2.3\pm 4.5\right)\times {10}^{-11}$ | |||||

VLBI [55] | GPB [52] | |||||

${\overline{s}}^{TT}$ | $(-5\pm 8)\times {10}^{-5}$ | |||||

${\overline{s}}^{TT}+970\left({\overline{s}}^{XX}-{\overline{s}}^{YY}\right)-0.05\left({\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}\right)+2895\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XY}-3235\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XZ}-11\phantom{\rule{0.166667em}{0ex}}240\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{YZ}$ | $(0.7\pm 3.1)\times {10}^{-3}$ | |||||

${\overline{s}}^{XX}-{\overline{s}}^{YY}+3.02\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XY}+0.32\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{XZ}+1.05\phantom{\rule{0.166667em}{0ex}}{\overline{s}}^{YZ}$ | $(-1.1\pm 3.8)\times {10}^{-7}$ |

**Table 9.**Estimation of SME coefficients resulting from a fit combining results from: atomic gravimetry (see Table 1), VLBI (see Equation (13)), LLR (see Table 3), planetary ephemerides (see Table 4), Gravity Probe B (see Table 5). The correlation matrices from all these analyses have been used in the combined fit. The right column includes the pulsars results from Table 6 as well. The three estimates on ${\overline{s}}^{JJ}$ are obtained by using the traceless condition ${\overline{s}}^{TT}={\overline{s}}^{XX}+{\overline{s}}^{YY}+{\overline{s}}^{ZZ}$.

Coefficient | Without Pulsars | With Pulsars |
---|---|---|

${\overline{s}}^{TT}$ | $(-5.\pm 8.)\times {10}^{-5\phantom{1}}$ | $(-4.6\pm 7.7)\times {10}^{-5\phantom{1}}$ |

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $(-0.5\pm 1.9)\times {10}^{-11}$ | $(-0.5\pm 1.9)\times {10}^{-11}$ |

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}$ | $(1.6\pm 3.1)\times {10}^{-11}$ | $(0.8\pm 2.5)\times {10}^{-11}$ |

${\overline{s}}^{XY}$ | $(-1.5\pm 6.8)\times {10}^{-12}$ | $(-1.6\pm 6.6)\times {10}^{-12}$ |

${\overline{s}}^{XZ}$ | $(-1.0\pm 4.1)\times {10}^{-12}$ | $(-0.8\pm 3.9)\times {10}^{-12}$ |

${\overline{s}}^{YZ}$ | $(2.6\pm 4.7)\times {10}^{-12}$ | $(1.1\pm 3.2)\times {10}^{-12}$ |

${\overline{s}}^{TX}$ | $(-0.1\pm 1.3)\times {10}^{-9\phantom{1}}$ | $(-0.1\pm 1.3)\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{TY}$ | $(0.5\pm 1.1)\times {10}^{-8\phantom{1}}$ | $(0.4\pm 2.3)\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{TZ}$ | $(-1.2\pm 2.7)\times {10}^{-8\phantom{1}}$ | $(-0.6\pm 5.5)\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{XX}$ | $(-1.7\pm 2.7)\times {10}^{-5\phantom{1}}$ | $(-1.5\pm 2.6)\times {10}^{-5\phantom{1}}$ |

${\overline{s}}^{YY}$ | $(-1.7\pm 2.7)\times {10}^{-5\phantom{1}}$ | $(-1.5\pm 2.6)\times {10}^{-5\phantom{1}}$ |

${\overline{s}}^{ZZ}$ | $(-1.7\pm 2.7)\times {10}^{-5\phantom{1}}$ | $(-1.5\pm 2.6)\times {10}^{-5\phantom{1}}$ |

**Table 10.**Sensitivity of the SME coefficients to the measurement of the light deflection by several space missions or proposals (these estimates are based on Table I from [82]).

Mission | ${\overline{\mathit{s}}}^{\mathit{TT}}$ | ${\overline{\mathit{s}}}^{\mathit{TJ}}$ | ${\overline{\mathit{s}}}^{\mathit{IJ}}$ |
---|---|---|---|

Gaia [196] | ${10}^{-6}$ | ${10}^{-6}$ | ${10}^{-5}$ |

AGP [203] | ${10}^{-7}$ | ${10}^{-7}$ | ${10}^{-6}$ |

LATOR [204] | ${10}^{-8}$ | ${10}^{-8}$ | ${10}^{-7}$ |

**Table 11.**Sensitivity of the SME coefficients to the observations of 360,000 asteroids by the Gaia satellite during a period of 5 years.

SME Coefficients | Sensitivity ($\mathbf{1}\mathbf{-}\mathit{\sigma}$) |
---|---|

${\overline{s}}^{XX}-{\overline{s}}^{YY}$ | $3.7\times {10}^{-12}$ |

${\overline{s}}^{XX}+{\overline{s}}^{YY}-2{\overline{s}}^{ZZ}$ | $6.4\times {10}^{-12}$ |

${\overline{s}}^{XY}$ | $1.6\times {10}^{-12}$ |

${\overline{s}}^{XZ}$ | $9.2\times {10}^{-13}$ |

${\overline{s}}^{YZ}$ | $1.7\times {10}^{-12}$ |

${\overline{s}}^{TX}$ | $5.6\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{TY}$ | $8.8\times {10}^{-9\phantom{1}}$ |

${\overline{s}}^{TZ}$ | $1.6\times {10}^{-8\phantom{1}}$ |

© 2016 by the authors; 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**

Hees, A.; Bailey, Q.G.; Bourgoin, A.; Pihan-Le Bars, H.; Guerlin, C.; Le Poncin-Lafitte, C.
Tests of Lorentz Symmetry in the Gravitational Sector. *Universe* **2016**, *2*, 30.
https://doi.org/10.3390/universe2040030

**AMA Style**

Hees A, Bailey QG, Bourgoin A, Pihan-Le Bars H, Guerlin C, Le Poncin-Lafitte C.
Tests of Lorentz Symmetry in the Gravitational Sector. *Universe*. 2016; 2(4):30.
https://doi.org/10.3390/universe2040030

**Chicago/Turabian Style**

Hees, Aurélien, Quentin G. Bailey, Adrien Bourgoin, Hélène Pihan-Le Bars, Christine Guerlin, and Christophe Le Poncin-Lafitte.
2016. "Tests of Lorentz Symmetry in the Gravitational Sector" *Universe* 2, no. 4: 30.
https://doi.org/10.3390/universe2040030