The paper by V. Alan Kostelecký explores the role of the gravitational sector in the Lorentz- and CPT-violating Standard-Model Extension (SME) within the framework of Riemann-Cartan spacetimes. Riemann-Cartan spacetimes, which include both Riemannian and Minkowski geometries as limiting cases, are used to address Lorentz violation in a more general context. The methodology is first illustrated using the QED extension in a Riemann-Cartan background, and then applied to the full SME, constructing leading-order terms involving operators of mass dimensions three and four. The paper discusses the incorporation of arbitrary Lorentz and CPT violation into general relativity and other gravity theories based on Riemann-Cartan geometries. It finds that explicit Lorentz symmetry breaking is incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking can evade this difficulty. The paper also provides a detailed analysis of the QED extension in Riemann-Cartan spacetime, including the fermion and photon sectors, and discusses the observability of various types of Lorentz violation.The paper by V. Alan Kostelecký explores the role of the gravitational sector in the Lorentz- and CPT-violating Standard-Model Extension (SME) within the framework of Riemann-Cartan spacetimes. Riemann-Cartan spacetimes, which include both Riemannian and Minkowski geometries as limiting cases, are used to address Lorentz violation in a more general context. The methodology is first illustrated using the QED extension in a Riemann-Cartan background, and then applied to the full SME, constructing leading-order terms involving operators of mass dimensions three and four. The paper discusses the incorporation of arbitrary Lorentz and CPT violation into general relativity and other gravity theories based on Riemann-Cartan geometries. It finds that explicit Lorentz symmetry breaking is incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking can evade this difficulty. The paper also provides a detailed analysis of the QED extension in Riemann-Cartan spacetime, including the fermion and photon sectors, and discusses the observability of various types of Lorentz violation.