16 - Relativistic Cosmology

Summary

Consider a spacetime M that, in addition to its metric gab and its various matter fields, contains a preferred function t (unique up to an additive constant) and a preferred four-velocity vector field v^a where v^a∇_at = 1. The function t determines a family of hypersurfaces Σ_t on which t is constant, and v^a determines a timelike congruence with t as a parameter function. Our intention is to take M as model of the universe as a whole with the hypersurfaces Σ_t representing different eras, t representing universal time, and the curves of the congruence representing the world lines of comoving particles.

We say that a tensor field is preferred if it can be constructed from nothing more than the available structure on M, that is g_ab, t, v^a, and the various matter fields. For example, if a Maxwell field F_ab is one of the matter fields, then E_a = F_abv^b will be a preferred vector field. Using this notion, we impose the cosmological principle by demanding that the Σ_t surfaces be isotropic in that they contain no intrinsic, preferred vector fields and hence no preferred directions. This is a very strong condition and implies the following results:

1. (i) v^a is orthogonal to Σ_t. If not, then its projection in Σ_t would give a preferred vector field in contradiction to isotropy.

2. (ii) v^a = ∇_at. If v_a - ∇_at ≠ 0, it would be orthogonal to v^a and hence a preferred vector field in Σ_t.

3. (iii) Dv^a = 0, that is, the world lines are geodesics. If Dv^a ≠ 0, it would be orthogonal to v^a and therefore a preferred vector field in Σ_t.

4. […]