Basic Notions

Abstract

A general spacetime is a 4-dimensional differentiable manifold whose tangent space is, at each point, a Minkowski spacetime. Linear frames and tetrad fields are constitutive parts of its structure as a manifold, and instrumental in relativistic physics and gravitation. They are defined up to point-dependent Lorentz transformations, under which usual derivatives exhibit a non-covariance that can be just compensated by the non-covariance of connections, objects thereby essential to produce meaningful, covariant derivatives. Each connection defines a covariant derivative, from which two basic covariant objects result: curvature and torsion. These quantities satisfy two mandatory relations, the Bianchi identities.