Cosmological principle

Definition

Let $(M, g)$ be a spacetime, that is, a time-oriented four-dimensional Lorentzian manifold. Let $u$ be a unit timelike vector field on $M$, the velocity of a family of observers. Let $(\Sigma_t)_{t \in I}$ be a foliation of $M$ by spacelike hypersurfaces orthogonal to $u$. The spacetime satisfies the cosmological principle with respect to $u$ and $(\Sigma_t)$ when it is spatially homogeneous and isotropic in the following sense.

1. Homogeneity: for each $t \in I$ and all points $p, q \in \Sigma_t$, there is an isometry of $(M, g)$ that maps $\Sigma_t$ onto itself and maps $p$ to $q$.
2. Isotropy: let $p \in M$, and let $v, w \in T_p M$ be unit vectors orthogonal to $u_p$. Then there is an isometry of $(M, g)$ that fixes $p$, maps $u$ to $u$, and has differential sending $v$ to $w$.