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I think Jerry's point is that if you have a metric that looks like:
$$ ds^2 = a(dx^0)^2 + f(dx^1, dx^2, dx^3) $$
where $a$ is a constant and $f$ is any function, then an observer moving in the $x^0$ direction follows a geodesic of the type you describe. An obvious example is the FLRW metric whe...