I Deriving the second Friedman equation from the first.

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The discussion centers on the relationship between the first and second Friedman equations in cosmology. It highlights a professor's assertion that solutions to the first equation inherently provide solutions for the second, suggesting that taking the time derivative of the first equation can imply the second. However, it is noted that simply taking the time derivative is insufficient; the continuity equation, which ensures the stress-energy tensor is divergence-free, must also be applied. This indicates a more complex interplay between the equations than initially suggested. The conversation emphasizes the importance of understanding the underlying principles in deriving cosmological equations.
thebiggerbang
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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.
 
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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).
 
Thank you, that helped!
 

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