  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}.
