# Why is the temporal element of the FRW metric tensor constant?

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?

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.