The pure-gravity sector of the minimal standard-model extension is studied in the limit of Riemann spacetime. A method is developed to extract the modified Einstein field equations in the limit of small metric fluctuations about the Minkowski vacuum, while allowing for the dynamics of the 20 independent coefficients for Lorentz violation. The linearized effective equations are solved to obtain the post-Newtonian metric. The corresponding post-Newtonian behavior of a perfect fluid is studied and applied to the gravitating many-body system. Illustrative examples of the methodology are provided using bumblebee models. The implications of the general theoretical results are studied for a variety of existing and proposed gravitational experiments, including lunar and satellite laser-ranging, laboratory experiments with gravimeters and torsion pendula, measurements of the spin precession of orbiting gyroscopes, timing studies of signals from binary pulsars, and the classic tests involving the perihelion precession and the time delay of light. For each type of experiment considered, estimates of the attainable sensitivities are provided. Numerous effects of local Lorentz violation can be studied in existing or near-future experiments at sensitivities ranging from parts in ${10}^4$ down to parts in ${10}^{15}$.

Field theories with spontaneous Lorentz violation involving an antisymmetric 2-tensor are studied. A general action including nonminimal gravitational couplings is constructed, and features of the Nambu-Goldstone and massive modes are discussed. Minimal models in Minkowski spacetime exhibit dualities with Lorentz-violating vector and scalar theories. The post-Newtonian expansion for nonminimal models in Riemann spacetime involves qualitatively new features, including the absence of an isotropic limit. Certain interactions producing stable Lorentz-violating theories in Minkowski spacetime solve the renormalization-group equations in the tadpole approximation.

We present in detail the scientific objectives in fundamental physics of the Space–Time Explorer and QUantum Equivalence Space\ud Test (STE–QUEST) space mission. STE–QUEST was pre-selected by the European Space Agency together with four other missions\ud for the cosmic vision M3 launch opportunity planned around 2024. It carries out tests of different aspects of the Einstein Equivalence\ud Principle using atomic clocks, matter wave interferometry and long distance time/frequency links, providing fascinating science at the\ud interface between quantum mechanics and gravitation that cannot be achieved, at that level of precision, in ground experiments. We\ud especially emphasize the specific strong interest of performing Equivalence Principle tests in the quantum regime, i.e. using quantum\ud atomic wave interferometry. Although STE–QUEST was finally not selected in early 2014 because of budgetary and technological reasons, its science case was very highly rated. Our aim is to expose that science to a large audience in order to allow future projects and\ud proposals to take advantage of the STE–QUEST experience

The static limit of Lorentz-violating electrodynamics in vacuum and in media is investigated. Features of the general solutions include the need for unconventional boundary conditions and the mixing of electrostatic and magnetostatic effects. Explicit solutions are provided for some simple cases. Electromagnetostatics experiments show promise for improving existing sensitivities to parity-odd coefficients for Lorentz violation in the photon sector.

Comparatively few searches have been performed for violations of local Lorentz invariance in the pure-gravity sector. We show that tests of short-range gravity are sensitive to a broad class of unconstrained and novel signals that depend on the experimental geometry and on sidereal time.

Lorentz symmetry is one of the pillars of both General Relativity and the Standard Model of particle physics. Motivated by ideas about quantum gravity, unification theories and violations of CPT symmetry, a significant effort has been put the last decades into testing Lorentz symmetry. This review focuses on Lorentz symmetry tests performed in the gravitational sector. We briefly review the basics of the pure gravitational sector of the Standard-Model Extension (SME) framework, a formalism developed in order to systematically parametrize hypothetical violations of the Lorentz invariance. Furthermore, we discuss the latest constraints obtained within this formalism including analyses of the following measurements: atomic gravimetry, Lunar Laser Ranging, Very Long Baseline Interferometry, planetary ephemerides, Gravity Probe B, binary pulsars, high energy cosmic rays, … In addition, we propose a combined analysis of all these results. We also discuss possible improvements on current analyses and present some sensitivity analyses for future observations.