Thread: Induced metric on the brane View Single Post

## Induced metric on the brane

Hello,

I have a problem to understand what people say by "induced metric". In many papers, it is written that for brane models, if we consider the metric on the bulk as $g_{\mu\nu}$ hence the one in the brane is $h_{\mu\nu}=g_{\mu\nu}-n_\mu n_{\nu}$ where $n_{\mu}$ is the normalized spacelike normal vector to the brane. I agree that it defines a projection tensor since $h_{\mu\nu}n^{\mu}=0$ but I don't understand how this can be the induced metric on the brane.

For example, if we consider a flat spacetime in spherical coordinates:

$ds^2=-dt^2+dr^2+r^2\Bigl(d\theta^2+sin^2\theta d\phi^2\Bigr)$

and we consider the surface defined by the equation $r=a(t)$, hence we have

$ds^2=-\Bigl(1-\dot a^2\Bigr)dt^2+a^2\Bigl(d\theta^2+sin^2\theta d\phi^2\Bigr)$

which is for me the induced metric on the surface. But it doesn't match with the metric $h_{\mu\nu}$ where $n_\mu=(0,1,0,0)$

which would give $h_{00}=-1\neq -\Bigl(1-\dot a^2\Bigr)$ ????????
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