Does an observer on an expanding three-sphere naturally have a hyperbolic sense of time? One can consider the 3-sphere of radius a
  as being embedded in a four-dimensional Euclidean space. One has in this view the condition for any coordinate system with origin at the center of the 3-sphere:
$$a^{2}=g_{\mu\nu}x^{\mu}x^{\nu}$$
Note that the metric here is Riemannian (not pseudoriemannian). In standard Cartesian coordinates for example this reads:
$$a^{2}=x^{2}+y^{2}+z^{2}+w^{2}$$
This is simply the condition that the coordinates lie somewhere on the 3-sphere. For a 3-sphere of changing radius a
 , one can write this infinitesimally as:
$$da^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$$
Where it is clear that a
  will be a function of some external parameter (time or conformal time). Let us now allow a body to move around on the three-sphere while it is expanding. For a coordinate choice with three variables (\chi
 ) on the three sphere, one has an infinitesimal displacement given by:
$$dl^{2}=g_{\mu\nu}\frac{d\chi^{\mu}}{dt}dt\frac{d\chi^{\nu}}{dt}dt$$
Our total displacement condition then becomes:
$$dL(t,\dot{\chi})^{2}=da(t)^{2}+g_{\mu\nu}d\chi(t)^{\mu}d\chi(t)^{\nu}$$
But $L$
  is not invariant; different observers would argue over the value of $L$
 . $a$
   however is invariant, being agreed upon regardless of position on the three sphere. In this manner it is somewhat more natural to rewrite:
$$-da(t)^{2}=-dL^{2}+g_{\mu\nu}d\chi(t)^{\mu}d\chi(t)^{\nu}$$
It makes sense here to designate $L=\chi^{0}$
  and a pseudoriemannian metric such that:
$$-da(t)^{2}=g_{\mu\nu}d\chi^{\mu}d\chi^{\nu}$$
Note this argument also applies for two arbitrary points on $S^{3}$
 at two differing values of $a$
 . Given the different geometric roles of a
  and L
  (as compared to dimensions on $S^{3}$
 ), one might choose different units such that $a=c\tau$
  and $\chi^{0}=ct$
  where c
  is a constant. Note one then has the natural condition:
$$-c^{2}=g_{\mu\nu}\frac{d\chi^{\mu}}{d\tau}\frac{d\chi^{\nu}}{d\tau}$$
Locally, for large a, this looks a lot like the conditions of Lorentz invariance as in Special relativity. 
 anyway,  I thought it was interesting.
 A: Conformal transformation of a metric $g_{\mu\nu}~\rightarrow~\Omega^2g_{\mu\nu}$ which for the line element 
$$
ds^2~=~g_{\mu\nu}dx^\mu dx^\nu~\rightarrow~\Omega^2g_{\mu\nu}dx^\mu dx^\nu
$$
which for a diagonal metric is
$$
ds'^2~=~\Omega^2(d\chi^2~-~d\Sigma^{(3)}),
$$
where $\chi$ conformal time and $\Sigma^{(3)}$ the metric of the spatial surface. For a flat surface $d\Sigma^{(3)}~=~dx^2~+~dy^2~+~dz^2$. In general 
$$
d\Sigma^{(3)}~=~\left(\frac{dr^2}{1 - kr^2} + r^2d\theta^2 + r^2\sin^2\theta d\phi^2\right).
$$
that is mapped by the conformal transformation $d\Sigma^{(3)}~\rightarrow~\Omega^2d\Sigma^{(3)}$.
We now let 
$$
d\chi~=~\frac{d\chi}{dt}dt~=~\Omega^{-1}dt,
$$
which gives the spacetime metric
$$
ds^2~=~dt^2~-~\Omega^2\left(\frac{dr^2}{1 - kr^2} + r^2d\theta^2 + r^2\sin^2\theta d\phi^2\right).
$$
The conformal transformation is a time dependent function which we write as $\Omega~=~a(t)$ so this gives the FLRW metric
$$
ds^2~=~dt^{2}~-~a(t)^{2}\left(\frac{dr^2}{1 - kr^2} + r^2d\theta^2 + r^2\sin^2\theta d\phi^2\right).
$$
