# I Should Einstein's Field Equations be modified for cosmology?

1. Sep 6, 2016

### jcap

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?

2. Sep 6, 2016

### vanhees71

How do come to that conclusion? From your derivar tion you see that you just write your coupling constant in terms of the Planck time. What has this to do with any coordinate time?

3. Sep 6, 2016

### jcap

The Planck time $t_{pl}$ is normally expressed in units of proper (or equivalently cosmological) time $t$ i.e. we use a lightclock of constant proper length. But if we are using conformal time $\eta$ then we are using units of time measured by an expanding lightclock of constant co-moving length. Therefore in order to express the Planck time in units of conformal time we need to compensate for the expanding lightclock by dividing the proper time value $t_{pl}$ by $a(\eta)$ to give the value of the Planck time as $t_{pl}/a(\eta)$ in conformal time units.

4. Sep 6, 2016

### vanhees71

No, Planck time is just a combination of fundamental constants of the dimension of time. It is a conversion factor like $c$ (or even worse $\epsilon_0$ and $\mu_0$ in the SI), $\hbar$, between men-made artificial and natural units.

5. Sep 6, 2016

### jcap

The reduced Planck time $t_{pl}$ in $G_{\mu\nu}=t_{pl}^2\ T_{\mu\nu}$ has the value of $2.7\times 10^{-43}$ seconds of cosmological (or proper) time $t$. But if we choose to use conformal time $\eta$, defined by $dt=a(\eta)d\eta$, then the proper size of each $\eta$-second expands with the scale factor $a(\eta)$. Therefore we should replace the value $2.7\times 10^{-43}$ with $2.7\times 10^{-43}/a$ to arrive at the correct amount of proper time when using $\eta$-seconds.

Last edited: Sep 6, 2016
6. Sep 6, 2016

### vanhees71

Again, your ideas are wrong. Planck time is not a coordinate time but a fundamental constant of nature built with Newton's gravitational constant (the universal gauge coupling of the gravitational field to the energy-momentum tensor of matter + radiation), Planck's constant $\hbar$ and the speed of light $c$. See

https://en.wikipedia.org/wiki/Planck_units

7. Sep 6, 2016

### jcap

I accept that the Planck time is a fundamental constant but its actual value is always expressed in terms of some arbitrary time-unit. I'm asking what happens if we take a time-unit that increases with the Universal scale factor $a$. Do we have to alter the EFEs accordingly?

8. Sep 6, 2016

### pervect

Staff Emeritus
Using units in which planck's constant changes with time is every bit as physically significant as using units in which the number of centimeters per inch varies with time.

That said, a correct application of tensor methods is capable of handling a system where one has time-varying centimeters, inches, planck's constant, or whatever. There would be no need to modify a tensor equation to change units, even screwball ones.

If there is a physical argument to be made here, changing the units just obscures and obfuscates the issue, just as it would doing physics with a time-variable number of centimeters per inch.

9. Sep 6, 2016

### Staff: Mentor

Yes. This is correct.

No, the EFE doesn't change regardless of your coordinates. That is the whole point of using tensors.

You can use whatever units you like. Just plug in the corresponding values for G and c, no changes needed.

10. Sep 6, 2016

### Staff: Mentor

No, tensors take care of that for you. The best (in my opinion) way to do this is to consider the coordinates to be unitless, I.e. just numbers that label events. Then you can keep track of the units at the tensor level instead of component by component.

Other people prefer to assign units to components, which can be done also but requires more care.