Did Wald make a mistake here?

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  • Thread starter JonnyG
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  • #1
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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?
 

Answers and Replies

  • #2
haushofer
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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.
 
  • #3
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Thanks. I was being sloppy.
 
  • #4
haushofer
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When you can choose between "I am sloppy" or "Wald was sloppy", my experience is that the first explanation is more trustworthy.
 

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