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(adsbygoogle = window.adsbygoogle || []).push({});

$$G_{\mu\nu}=l_P^2\ T_{\mu\nu},$$

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?

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# I Does the Planck length expand in a FRW universe?

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