I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, deriving the FRW metric, and I'm a little confused with the use of eq 8.4 .(adsbygoogle = window.adsbygoogle || []).push({});

I thought instead it should be using the general form of a spherically symmetric metric, not a vacuum spherical symmetric - eq 7.13 has been derived from 8.4 by solving ##R_{uv}=0## enabling us to loose the time dependence in the function ## \beta(t,r) ## as we don't want a vacuum solution here at all?

What has enabled us to conclude ## \beta(t,r) = \beta(r) ##

Thanks in advance.

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# Deriving maximally symmetric space from spherically sym st

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