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?