# 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

## 1 Answer

See also https://en.wikipedia.org/wiki/Particle_horizon. The conformal time which you calculated is related to the particle horizon, today about 47 billion light years, and the conformal time is 47 billion years.

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.

The EFEs don't change.

This is related to a similar question you asked in 2013, question 67898, referenced in my second comment above, answered then.