Citaton wanted - curves of constant cosmological time can't be geodesics

From:spamme606 aka 'pervect' on physicsforums

I'm looking for a citation for the fact that a curve of constant
cosmological time, such as the curve of constant cosmological time
whose length defines the comoving distance between two points, is not a
geodesic.

Thus the comvoing distance is not measured along a geodesic curve.

This is reasonbly obvious if one writes the metric

dt^2 - a(t)^2 dx^2

and the corresponding geodesic equations

d^2 t / d tau^2 + a da/dt (dx/d tau)^2 = 0
d^2 x / d tau^2 + (2/a) ( da/dt ) (dx / d tau) (dt / dtau) = 0

A curve of constant time has d^2 t / dtau^2 = 0. Given a(t)>0 and an
expanding universe da/dt>0, the only solution is (dx / d tau) = 0,
which is a point, not a curve.

Since this will eventually wind up in a Wikipedia article, it won't
necessarily be obvious to the casual reader, and I think it would be
classifiable as "original research" :-(.

Therfore I would like, if possible, a citation to the literature which
points out this fact.

For my own info, I would also like to know if there is any standard
distance measure in cosmology which defines the "distance" between two
points along the geodesic curve connecting them, rather than a
coordinate-based defintion of a curve of constant cosmological time.
So far I haven't run accross any such defintion in my search of what
textbooks I could find.

.

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