Thread: Induced metric on the brane View Single Post
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## Induced metric on the brane

I checked on Poisson's book and also Gourgoulhon's review but I couldn't found the reason.

I finally understood my mistake, $h_{\mu\nu}$ is not the induced metric but only the projection tensor. For to have the induced metric we have to look to the tangential components of the tensor and not to $h_{00}$.
In fact the 3 vectors orthogonal to the normal vector and which define a basis on the hypersurface are
$V1^\mu=(1,\dot a,0,0)$
$V2^\mu=(0,0,1,0)$
$V3^\mu=(0,0,0,1)$

so it is perfectly fine to look for $h_{22}$ and $h_{33}$. But the last component is not $h_{00}=h_{tt}$ but $h_{V1 V1}$

So now we have $\partial_{V1}=\partial_t+\dot a \partial_\rho$ which implies that

$h_{V1V1}=h_{00}+2\dot a h_{01}+\dot a^2 h_{11}$ which gives the correct result $h_{V1V1}=-1+\dot a^2$.

So it is a modification of the coordinates ...

Thanks