We are in the flat FLRW metric in Cartesian comoving coordinates. The metric is expressed as:

$$ds^2 = d\tau^2 + a(\tau)^2\big(dx^2 + dy^2 + dz^2\big)$$

The fact that the "universe is expanding" is justified by saying that if I have a galaxy at $(\tau, 0, 0, 0)$ and one at $(\tau, L, 0, 0)$ and I measure the distance $R(\tau)$ along the curve $\sigma\_\tau : x \mapsto (\tau, x, 0, 0)$ from $x=0$ to $x=L$, I find that this distance $R(\tau)$ is indeed a function of $\tau$.

However, isn't the distance between two spacelike-separated events defined as the lengths of the geodesic connecting them? 

This curve *is not* a geodesic. I was definitely surprised. 

To see this, let's look ay the geodesic equations

$$ \frac{d^2\gamma^{\nu}}{d\lambda^2} = -\Gamma^{~\nu}\_{\mu\rho}\frac{d\gamma^{\mu}}{d\lambda}\frac{d\gamma^{\rho}}{d\lambda}$$

with initial conditions 
\begin{aligned}
\gamma^\mu(0) &= (\tau, 0, 0, 0)\\\\
\left.\frac{d\gamma^{\mu}}{d\lambda}\right|\_{\lambda=0} &= (0, 1, 0, 0)
\end{aligned}

Remember that the only non-zero Christoffel symbols are 
$\Gamma^{~\tau}\_{ii} = a\dot a$ and
$\Gamma^{~i}\_{i\tau} = \Gamma^{~i}\_{\tau i} = \frac{\dot a}{a}$ for $i = x, y, z$, and where the dot represents differentiation by $\tau$. The geodesic equations become

\begin{aligned}
\frac{d^2\gamma^{\tau}}{d\lambda^2} &= -a\dot a \left(\frac{d\gamma^{x}}{d\lambda}\right)^2 \\ \\
\frac{d^2\gamma^{x}}{d\lambda^2} &= -\frac{\dot a}{a}\frac{d\gamma^{x}}{d\lambda}\frac{d\gamma^{\tau}}{d\lambda}\\
\end{aligned}

In particular, we see that for $\lambda = 0$, 
$$\frac{d^2\gamma^{\tau}}{d\lambda^2} = -a\dot a$$ from which we learn two things:

* our first curve $\sigma\_\tau$ is not a geodesic
* spacelike geodesics in the FLRW metric do not tay on surfaces of constant cosmological time $\tau$

Which brings me to my question:

* What is the relation between the distance $R(\tau)$ and the proper distance (the one measured along a spacelike geodesic?)

Or, perhaps in a different formulation:

* How does one make sense physically of $R(\tau)$? and of the proper distance?