Does the Planck length expand in a FRW universe?

[jcap]

By using particle physics natural units with $\hbar=c=1$ so that Planck's length $l_P=(8\pi G)^{1/2}$ we can express Einstein's field equations as
$$G_{\mu\nu}=l_P^2\ T_{\mu\nu},$$
where $G_{\mu\nu}$ has dimension $[\hbox{proper length}]^{-2}$, $l_P$ has dimension $[\hbox{proper length}]$, $T_{\mu\nu}$ has dimension $[\hbox{proper length}]^{-4}$.

In cosmology we assume the expanding FRW metric. If we assume flat space for simplicity and cartesian coordinates then we have the following line element
$$ds^2=-dt^2+a^2(t)\left(dx^2+dy^2+dz^2\right).$$
Therefore an interval of proper length in the x-direction for example is given by
$$ds=a(t)dx$$
If $l_P$ is a proper length then should it expand with the scale factor $a(t)$ or should it remain constant?

In order for it to remain constant then its corresponding comoving interval $dx \sim 1/a$ which seems unnatural to me.

Therefore I think that as $l_P$ is a proper length it should expand with the scale factor $a(t)$.

Does this make sense?

 

[PeterDonis]

If $l_P$ is a proper length

$l_P$ isn't a proper length. It's a physical constant that happens to have units of proper length.

The proper length $ds = a(t) dx$ is the proper distance between two events happening at the same time $t$ and separated by a spacelike coordinate interval $dx$. If we hold $dx$ constant (i.e., we have two comoving objects), this proper distance increases with $t$, since $a(t)$ does. So the ratio of $ds$ to $l_P$, i.e., the number of Planck lengths between the two comoving objects, will increase.