Deriving the second Friedman equation from the first.

[thebiggerbang]

Hello!

My professors asked me why we don't usually consider the second Friedman equation. He told me that the solutions for the first always give a solution for the second. He told me that the first equation along with it's time derivative can imply the second equation. I have been trying the algebra for a long time now, but I am not being able to get ahead.

I wanted to know if the statement that the professor made is true in the first place.

 

[Orodruin]

You cannot just take the time derivative. You will also need to apply the continuity equation (i.e., the fact that the stress-energy tensor is divergence free).

 

[thebiggerbang]

Thank you, that helped!