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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 [itex]g_{\mu\nu}[/itex] hence the one in the brane is [itex]h_{\mu\nu}=g_{\mu\nu}-n_\mu n_{\nu}[/itex] where [itex]n_{\mu}[/itex] is the normalized spacelike normal vector to the brane. I agree that it defines a projection tensor since [itex]h_{\mu\nu}n^{\mu}=0[/itex] 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:

[itex]ds^2=-dt^2+dr^2+r^2\Bigl(d\theta^2+sin^2\theta d\phi^2\Bigr)[/itex]

and we consider the surface defined by the equation [itex]r=a(t)[/itex], hence we have

[itex]ds^2=-\Bigl(1-\dot a^2\Bigr)dt^2+a^2\Bigl(d\theta^2+sin^2\theta d\phi^2\Bigr)[/itex]

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

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

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# Induced metric on the brane

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