# Should Einstein's Field Equations be modified when using conformal time?

Let us consider the FRW metric for flat space expressed in terms of conformal time $\eta$ and cartesian spatial co-ordinates $x,y,z$: $$ds^2=a^2(\eta)\{d\eta^2-dx^2-dy^2-dz^2\}.$$

As in the standard FRW co-ordinate system one can see that if two observers are separated by a constant co-moving interval $dx$ then the interval of proper distance between them, $ds$, is given by: $$ds=a(\eta)\ dx.$$ Thus we have an expanding universe as expected.

But, contrary to the standard FRW co-ordinates, an interval of proper time $d\tau$ measured by a co-moving observer using conformal time $\eta$ is given by: $$d\tau=a(\eta)\ d\eta.$$ Thus the co-moving observer's clock is going slower as the universe expands. This can be understood if one imagines that the co-moving observer uses a lightclock that measures a unit of time by bouncing a pulse of light off a mirror placed some distance away. When one uses the standard time co-ordinate one assumes that such a mirror is at a constant proper distance from the observer. But when one uses conformal time then one implicitly assumes that the mirror is at a constant co-moving distance from the observer. Thus he is using a clock whose unit of time is getting longer as the Universe expands.

Now this may sound odd but I think this should be a perfectly consistent view. One can certainly express a metric using any arbitrary co-ordinate system.

But my question is this: should the EFE be modified if one is using the FRW metric with conformal time $\eta$?

Einstein's Field equations (EFE) are given in SI units by: $$G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}.$$

Let us define a characteristic time called the reduced Planck time, $t_{pl}$, given by: $$t_{pl}=\sqrt{\frac{8\pi G \hbar}{c^5}}.$$

We can express the EFE is Natural units by setting $\hbar=c=1$ giving: $$G_{\mu\nu}=t_{pl}^2\ T_{\mu\nu}.$$

Now the Planck time $t_{pl}$ is a constant interval of proper time. As described above, in order to measure conformal time a co-moving observer's clock ticks slower and slower as the Universe expands. Thus a constant interval of proper time, like the Planck time $t_{pl}$, will be represented by fewer ticks of that clock as the Universe expands.

Thus in conformal co-ordinates the EFE should be written as: $$G_{\mu\nu}=\Big(\frac{t_{pl}}{a(\eta)}\Big)^2\ T_{\mu\nu}.$$

Is this correct?

• The metric used is already included in the equation - the Einstein Metric is defined as $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$. Commented Sep 5, 2016 at 21:33
• The EFEs don't change. $t_{pl}$ is simply a constant. You just changed coordinate systems and the EFEs are still satisfied. You are then interpreting wrong, eta is not comoving time. It's conformal time. Commented Sep 6, 2016 at 3:23
• And you asked a similar question in 2013. physics.stackexchange.com/q/67898 Commented Sep 6, 2016 at 3:30
• What is the point of using Planck units? This is all classical.
– user4552
Commented Feb 24, 2019 at 3:55

As in my comment above, $t_{pl}$ is simply a constant. You just changed coordinate systems with a different time coordinate, it is not a comoving time, that would be t which you scaled using a, a variable. No longer the same.