Deriving Max Symmetric Space from Spherically Symmetric - Sean M. Carroll

[binbagsss]

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 .

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.

 

[Matterwave]

Maybe if you actually typed out the equations you are referring to, I could help. I don't have a copy of Carroll with me though.

 

[PeterDonis]

eq 7.13 has been derived from 8.4 by solving $R_{uv}=0$

Carroll is not talking in Chapter 8 about the process that leads from eq. 7.13 to eq. 7.20 (btw, it's eq. 7.20, not 7.13, that is derived by solving $R_{uv} = 0)$; eq. 7.13 is the form of the metric considered in Chapter 7 before that is done). The appearance of $\beta(r)$ in eq. 8.4 does not actually refer to eq. 7.20. Carroll leaves out some steps here; let's try to put them back in.

In eq. 8.1 Carroll gives the general form of the metric he will be considering, which is different from the general form he considers in Chapter 7. Eq. 8.1 has just plain $- dt^2$ in the line element, where in Chapter 7 there was a coefficient there, and eq. 8.1 splits out the time dependence of the spatial part of the metric into the function $a^2 (t)$, which multiplies the entire spatial metric, where in Chapter 7 the time dependence (before the vacuum assumption is made to remove it in deriving eq. 7.13) is only in the $dr^2$ part of the spatial metric (and is also in the $dt^2$ term).

So in eq. 8.4, the coefficient of the $dr^2$ term only depends on $r$ because the time dependence has already been factored out; note that eq. 8.4 is an expression for $\gamma_{ij} du^i du^j$, i.e., for the part that is multiplied by $a^2(t)$ in eq. 8.1. In other words, eq. 8.4 is the expression for the metric of a spatial slice of constant time, with the actual distance scale (the scale factor $a(t)$) removed. The form of eq. 8.4 is just the general metric of a spherically symmetric 3-space. The appearance of $\beta(r)$ in eq. 8.4 is just a similarity of notation with Chapter 7; it does not mean that eq. 8.4 was derived the way eq. 7.20 was derived.