Covariant Kinematics

Abstract

In this chapter, we examine the world line as a parameterized geometric object, with the purpose of introducing the dynamical time of the system in the next chapter. In Section 3.1, covariant vectors are defined. In Section 3.2, the Lorentz-invariant scalar products of these vectors are defined. In Section 3.3, regular parametric representations of the world line in terms of invariant parameters are defined. In Section 3.4, the covariant derivatives are discussed in both natural and arbitrary representations. In Section 3.5, an orthonormal tetrad of co-moving basis vectors as a function of arc-length derivatives is defined along the world line, and the Lorentz-invariant intrinsic curvature coordinates of the world line are given as a function of the frame-dependent kinematical variables. In Section 3.6, correlated representations of the n world lines of the many-body system are defined and discussed.