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 [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] ????????
