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The FLRW metric is used to describe Universe expansion. Why for this purpose time component of the metric is chosen to be constant? Even other metrics which describe inhomogeneous Universe expansion still uses constant $g_{00}$. Could exists a solution to EFE a metric with $g_{00}$ dependent on time, similar to space components?

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2 Answers 2

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To add to G. Smith's answer, you can also use an arbitrary function $N(t)$, known as the Lapse function, in the $g_{00}$ component, which is still the same homogenous & isotropic FLRW metric. It's a choice people make to set $N=1$, as you're always free to do this, just as you're free to make coordinate transformations. In standard GR, the function acts as a Lagrange multiplier and is non-dynamical.

For a little more detail, in the ADM decomposition where you foliate your manifold into 3-dimensional spatial hypersurfaces of constant time $t$, also known as leaves, the Lapse roughly translates to the spacing between these leaves (see my answer here for more Hamiltonian Formulation of GR, 3+1 decomposition).

The takeaway is that having an arbitrary function of time in the $g_{00}$ component really doesn't make any physical difference as it's non-dynamical, but it can be helpful for some calculations.

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It was done this way so that $t$ is the time measured by a clock at fixed spatial coordinates. This is the time of a “comoving” observer. You can think of “comoving” as meaning “going with the flow” of the expansion of the universe. These observers have no “proper motion” of their own. They are fixed in space but space is growing between them.

You can use a different temporal coordinate if you want. For example, conformal time $\eta$ is another choice. It makes the metric look like this:

$$c^2 d\tau^2=a(\eta)^2(c^2 d\eta^2-d\Sigma^2).$$

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  • $\begingroup$ Thanks for the quick answer! It means that time from the FLRW metric would correspond actually to the time we measure, since we are a "comoving" observer. The distance however, is not "comoving", because we see the light sources from the past. Then we need the scale factor to represent expansion. Is that right? $\endgroup$
    – Andrei
    Jan 27, 2021 at 20:41
  • $\begingroup$ @Andrei Yes, that’s right. $\endgroup$
    – G. Smith
    Jan 28, 2021 at 1:18

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