Let us consider the FRW metric for flat space expressed in terms of conformal time ##\eta## and cartesian spatial co-ordinates ##x,y,z##:(adsbygoogle = window.adsbygoogle || []).push({});

$$ds^2=a^2(\eta)\{d\eta^2-dx^2-dy^2-dz^2\}.$$

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 constantproperdistance from the observer. But when one uses conformal time then one implicitly assumes that the mirror is at a constantco-movingdistance 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?

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

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