4 - Kinematics of cosmological models

Summary

In cosmology, the matter components allow us to make a physically motivated choice of preferred motion. For example, we could choose the CMB frame, in which the radiation dipole vanishes, or the frame in which the total momentum density of all components vanishes. Such a choice corresponds to a preferred 4-velocity field u^a that generates a family of preferred world lines. We can then make a 1+3 split relative to u^a, in order to relate the physics and geometry to the observations. In this chapter we discuss how to do this for the kinematics of cosmological models; the following chapter will consider the dynamics.

The (real or fictitious) observers are comoving with the matter-defined 4-velocity u^a, and we can call the observers and the 4-velocity ‘fundamental’. If we change our choice of fundamental 4-velocity, the kinematics and dynamics transform in a well-defined way, as discussed in the following chapter.

Comoving coordinates

To describe the spacetime geometry it is convenient to use comoving (Lagrangian) coordinates, adapted to the fundamental world lines. These are locally defined as follows.

(1) Choose a surface S that intersects each world line once only (note that no unique choice is available in general). Label each world line where it intersects this surface; as the surface is three-dimensional, three labels yⁱ, (i =1,2,3), are required to label all the world lines.

(2) Extend this labelling off the surface S by maintaining the same labelling for the world lines at later and earlier times.