= Solution
A <fundamental observer> follows the coarse-grained cosmological matter, with fixed <comoving coordinates>. The <cosmological principle> asserts <spatial homogeneity> and <cosmological isotropy> on large scales for this family of observers. A moving observer generally sees a preferred direction, for example a Doppler dipole in the <cosmic microwave background>, even when the underlying cosmology is isotropic.
In an <FLRW metric>, these worldlines are orthogonal to homogeneous spatial slices. Synchronize their clocks to define <cosmic time> $t$, which is their <proper time>. Along a fixed spatial label, $ds^2=c^2dt^2$, giving $g_{tt}=c^2$. Orthogonality of the spatial coordinate directions to the observers' <four-velocity> gives $g_{ti}=0$. Thus, in the $+---$ <metric signature>,
$$
\boxed{ds^2=c^2dt^2+g_{ij}(t,\mathbf x)\,dx^idx^j,}
$$
where the spatial matrix $g_{ij}$ is negative definite. <Spatial homogeneity> and <cosmological isotropy> further restrict it to $-a(t)^2\gamma_{ij}(\mathbf x)$, with $\gamma$ a constant-curvature spatial <metric>.
Back to article page
