So the FLRW metric takes the following form in reduced-circumference polar coordinates.
$$\mathrm ds^2 = -c^2 \mathrm dt^2 + a^2(t) \left(\frac{\mathrm dr^2}{1 – k\, r^2} + r^2 (\mathrm d\theta^2+\sin^2\theta\,\mathrm d\phi^2)\right)$$
It's clear to me how this is derived from the restrictions of the cosmological principle applied to the most general possible metric but what is not clear to me is the reason behind assuming a constant curvature term $k$. It cannot be position dependent and be compatible with the cosmological principle but it seems like it should have the freedom to be time dependent much like the scale factor as long as it varies everywhere the same. Is there some coordinate redefinition possible such that the time dependence can be removed or am I missing something more?
Best Answer
Definition 1. A spacetime is said to be spatially homogeneous if there is a one-parameter family of spacelike hypersurfaces $\Sigma_t$ foliating the spacetime such that for each $t$ and for any points $p,q\in\Sigma_t$ there is an isometry of the spacetime metric $g$ which takes $p$ to $q$.
Definition 2. A spacetime is said to be isotropic if at each point there is a congruence of timelike curves, with tangents denoted $u$, satisfying: Given any point p and two unit spacelike vectors in $T_pM$, there is an isometry of $g$ which leaves $p$ and $u$ fixed but rotates one of these spacelike vectors into the other.
Restrict $g$ to a Riemannian metric $h$ on $\Sigma_t$. The geometry of each "leaf" of the foliation must inherit homogeneity and isotropy.
Let ${}^{(3)}\operatorname{Riem}$ be the Riemann tensor on $\Sigma_t$, $R_\Sigma$ be the scalar curvature and $T$ be the tensor field $$T(X,Y)Z=6\left[h(Z,Y)X-h(Z,X)Y\right]$$ for vector fields $X,Y,Z$.
Theorem. Homogeneity and isotropy of $\Sigma_t$ $\Leftrightarrow$ ${}^{(3)}\operatorname{Riem}=R_\Sigma T$, $R_\Sigma=\text{const.}$
Proof. Construct the Riemann tensor of $\Sigma_t$ using $h$. One may view this as an endomorphism $L$ of the space of $2$-forms $W$. By the symmetry properties of the Riemann tensor, $L$ is symmetric, and by a theorem in linear algebra, $W$ has an orthonormal basis of eigenvectors of $L$. If the eigenvalues were distinct, one could pick out a preferred $2$-form on $\Sigma_t$. Using the Hodge star on $\Sigma_t$, one could then construct a preferred vector. Since this would violate isotropy, the eigenvalues must be equal. We call this value $K$: $$L=K\operatorname{id}_W$$ In other words, $${}^{(3)}R_{ab}{}^{cd}=K\delta^c{}_{[a}\delta^d{}_{b]}$$ where ${}^{(3)}R_{ab}{}^{cd}$ are the components of ${}^{(3)}\operatorname{Riem}$. Contracting everything appropriately gives $$R_\Sigma=3K$$ Homogeneity automatically fixes $K$ to be a constant. $\quad\Box$
This proof very closely follows the one given in Wald, R. M. 1984, General Relativity (Chicago University Press).