Did Wald make a mistake here?

[JonnyG]

Main Question or Discussion Point

On page 97, Wald is computing the component $G_{ii}$ where $1 \leq i \leq 3$ of the Einstein tensor, assuming that the metric is given by $g = -d\tau^2 + a^2(\tau)\big(dx^2 + dy^2 + dz^2)$ where $a$ is the time evolution function. He writes:

$G_{ii} = R_{ii} - \frac{1}{2}R$. But if $G_{ii} = R_{ii} - \frac{1}{2}R$ then that must mean that $g_{ii} = 1$, which isn't necessarily true, as $g_{ii} = a^2(\tau)$. Did he make a mistake?

 

[haushofer]

In my copy he is calculating the projection of the Einstein tensor on a homogeneous hypersurface, using a unit (!) vector tangent $s$ to this hypersurface. That is something else than just calculating the spatial components! So instead of $g_{ii}=1$, he obtains (in his notation) $g_{**}=1$, which is the condition of unity $g_{**} \equiv g_{ab}s^a s^b=1$ defining this projection.

 

[JonnyG]

Thanks. I was being sloppy.

 

[haushofer]

When you can choose between "I am sloppy" or "Wald was sloppy", my experience is that the first explanation is more trustworthy.