I Follow-up on the Expanding Universe Insight article

[cianfa72]

TL;DR Summary

About the form of RW comoving observer worldline in local Minkowski frame at event p.

Hi, reading this Insight raised a doubt regarding the section "Comoving observers in a local Minkowski frame".

Robertson-Walker (RW) comoving observers have constant $x$ in comoving coordinates (to take it simple assume a 1+1 RW spacetime). From the following coordinate transformation into local Minkowski coordinates at event $p$
$$\begin{align*}\tau &\simeq t + \frac{1}{2}H_0 a_0^2 x^2 = t + \frac{1}{2} a’^2_0 x^2, \\\xi &\simeq a_0 x (1 + H_0 t).\end{align*}$$
a comoving observer at proper distance $d_0$ from $\xi = 0$ at $\tau=0$ (i.e. on the spacelike hypersurface $\tau=0$) has $\xi = d_0$ coordinate, hence $x= d_0 / a_0$. Therefore such comoving observer's worldline in comoving coordinates is given by $x= d_0 / a_0$ constant and varying $t$.

Substituting it into the transformation above yields in $(\xi, \tau)$ local Minkowski coordinates
$$\xi \simeq d_0 (1 + H_0 t)$$
However in the Insight it is given by
$$\xi \simeq d_0 (1 + H_0 \tau)$$
From where the above come from ? Thanks.

 

[Orodruin]

They are the same to the ordered considered in the $\simeq$ relation.

 

[cianfa72]

They are the same to the ordered considered in the $\simeq$ relation.

Ah ok, basically for "small" $x$ the term involving $x^2$ in $$\tau \simeq t + \frac{1}{2}H_0 a_0^2 x^2 = t + \frac{1}{2} a’^2_0 x^2$$ can be neglected, hence $\tau \simeq t$.

It makes sense to pick "small" values for $x$ since the derivation of the transformation from RW coordinates to local Minkowski coordinates at point/event $p$ employs the assumption $x^{\alpha} = 0$ at $p$.