The FRW metric can be written in conformal co-ordinates to give: $$ds^2=a^2(\eta)(-d\eta^2+d\mathbf{\Sigma^2}),$$ where $\eta$ is the conformal time and $\mathbf{\Sigma}$ ranges over 3-dimensional space of uniform curvature.

Using the above metric one can state that an interval of proper time $d\tau$ is given by: $$d\tau = a(\eta)\ d\eta.$$ Additionally an interval of proper distance $ds$ is given by: $$ds=a(\eta)\ d\mathbf{\Sigma}.$$

Now one normally interprets the FRW metric in standard co-ordinates $(t,\mathbf{\Sigma})$ as describing a space that expands with respect to cosmological time $t$.

But the choice of co-ordinates is arbitrary.

By using conformal co-ordinates $(\eta,\mathbf{\Sigma})$ is one equally justified in asserting that the FRW metric describes a spacetime whose scale $a(\eta)$ changes with respect to conformal time $\eta$? In other words the units of length and time change with respect to conformal time $\eta$.

Now the Einstein Field Equations can be written in natural units ($\hbar=c=1$): $$G_{\mu\nu}=l_P^2\ T_{\mu\nu},$$ where $l_P$ is the reduced Planck length and the cosmological constant is assumed to be the vacuum energy component of $T_{\mu\nu}$.

If I decide to use the FRW metric expressed in conformal spacetime co-ordinates then I think this implies that the units of length and time scale with the universal scale factor $a(\eta)$.

As the proper reduced Planck length $l_P$ must always be constant as my units of length expand then perhaps Einstein's Field Equations should be modified to give:



This is right with respect to the FLRW metric being a conformal coodinate system. The metric $$ ds^2 = \Omega^2(u)(du^2 - d\Sigma^{(3)}) $$ is with the change of variables $du^2 = \Omega(u)^{-2}dt^2$

lets us identify the above metric as the conformal time dependent metric $$ ds^2 = dt^2 - \Omega^2(u)d\Sigma^{(3)} $$ with the expansion factor $a(t) = \Omega^2(u)$.

I am not sure about $G^{\mu\nu} = \ell_p^2T^{\mu\nu}$. How did $\hbar$ creep into Einstein's field equations that are purely classical? I suppose one could write the Einstein field equations $$ G^{\mu\nu} = 8\pi\frac{\ell_p^2}{\hbar c}T^{\mu\nu} $$ The inclusion of $a(\eta)$ in the Einstein field equation is complicated, for it depends upon how $T^{\mu\nu}$ behaves.


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