  The role of the gravitational sector in the Lorentz- and CPT-violating
Standard-Model Extension (SME) is studied. A framework is developed for
addressing this topic in the context of Riemann-Cartan spacetimes, which
include as limiting cases the usual Riemann and Minkowski geometries. The
methodology is first illustrated in the context of the QED extension in a
Riemann-Cartan background. The full SME in this background is then considered,
and the leading-order terms in the SME action involving operators of mass
dimension three and four are constructed. The incorporation of arbitrary
Lorentz and CPT violation into general relativity and other theories of gravity
based on Riemann-Cartan geometries is discussed. The dominant terms in the
effective low-energy action for the gravitational sector are provided, thereby
completing the formulation of the leading-order terms in the SME with gravity.
Explicit Lorentz symmetry breaking is found to be incompatible with generic
Riemann-Cartan geometries, but spontaneous Lorentz breaking evades this
difficulty.
