In linearised gravity, when dealing with gravitational waves it is common to consider just the spatial part of the trace-reversed metric perturbation $\bar{h}_{ij}$.

I know that in general the trace-reversed perturbation is defined,

$$ \bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} h$$

My question is, how explicitly do I convert e.g. $\bar{h}_{xx} \rightarrow h_{xx}$? I have a feeling that it is just a factor of $2$, e.g. Eq 6.26 of Carrol ?


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