# What happens if we let time expand in the FLRW metric?

If we multiplied the time differential (dt) by a scale factor that depends on time in the FLRW metric, what would this imply on cosmology? In particular, what are its implications on the cosmological principle?

• Assuming a flat space, the FLRW metric is $ds^2=dt^2-a^2(t)(dr^2+r^2 d\Omega^2)$. Is your question "what happens when we have $a(t) \leadsto a(f(t))$ for some function $f$?" If so, rename $a \circ f$ into just $a'$. Maybe you have a particular example of the FRLW metric in mind... Commented Dec 16, 2023 at 9:19
• No, I mean what happens if the dt was multiplied by a scale factor that either expands or contracts with time. Commented Dec 16, 2023 at 9:21
• I edited the question to clarify Commented Dec 16, 2023 at 9:26

If you have the metric, $$$$ds^2=b^2(t) dt^2-a^2(t) d\vec{x}^2$$$$ You can make a coordinate transformation, $$$$\tau=\int b(t) dt$$$$
Then you can rewrite the metric in the usual FLRW form $$$$ds^2=d\tau^2-a^2(t(\tau)) d\vec{x}^2$$$$ I.e. you have not transitioned any new physics, just a coordinate transformation of the known spacetime.
In fact it is quite useful to consider different times. E.g. you may consider the so-called conformal time for which, $$$$ds^2=a^2(\eta) (d\eta^2-d\vec{x}^2)$$$$ You may notice that it is proportional to the Minkowski metric. So the lightcones $$ds^2=0$$ are the same in this coordinates as in the Minkowski spacetime. This make it very easy to study the causal structure of the spacetime in such coordinates.