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When written in conventional coordinates, the temporal element of the FRW metric tensor is a constant over all space-time, and the Friedman equations that result from the tensor describe a universe that corresponds to the observed expansion of space. Assuming that Energy/Momentum can be viewed as a fluid whose pressure/density is conserved in this expanding space, I would expect this pressure/density to correspondingly decrease with time. I am under the impression that in a volume of space having higher Energy/Momentum, a clock will run slower than in a volume having lower density. If this is correct, the temporal element of the FRW metric tensor cannot be a constant over the age of the universe. Can someone explain where my reasoning is incorrect?

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You can have some $g_{00}(t)$, but you can always redefine your time coordinate so that you recover the (cosmic time) FLRW form of metric (where $g_{00}= $ constant).

Say you have $g_{00} = f(t)$ and you want to redefine coordinates into FLRW form. This just means solving

$$ \int \sqrt{f(t)} \ dt = \int dT $$

so that the metric expressed in the new time coordinate $T$ has the FLRW form.

Also, why would energy density ($\rho$) be conserved during the expansion of the universe? There is no timelike Killing vector; from Noether's theorem, the lack of time-translation symmetry implies that energy is not conserved during the universe's expansion. Perhaps what you're missing are the Christoffel symbols. These have non-trivial dependence on the scale factor $a(t)$, and you get the usual relation that $\rho \sim a^{-3}$ for matter and $\rho \sim a^{-4}$ for radiation.

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  • $\begingroup$ @safesphere It's just a coordinate transformation. These cosmology notes by Daniel Baumann are nice. See page 7, for instance - theory.uchicago.edu/~liantaow/my-teaching/dark-matter-472/… $\endgroup$ – Avantgarde Jul 1 '19 at 16:01
  • $\begingroup$ Your link explains how to move from physical to comoving coordinates, but this is not what the OP is asking. His point is that the formula 1.1.16 is not logically justified in your link (which is true). In fact, it is not justified for both temporal and spatial parts. Consider a hypothetical example of this uniform universe: physics.stackexchange.com/questions/424982/… - What is the scale factor there? $\endgroup$ – safesphere Jul 1 '19 at 16:27

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