# Induced metric on the brane

by atrahasis
Tags: brane, induced, metric
 P: 11 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 $h_{00}=-1\neq -\Bigl(1-\dot a^2\Bigr)$ ????????
 P: 11 Ok I have half of the answer, the normal vector is wrong, because $r=a(t)$, we have $dr-\dot a dt=0$, which gives for the normal vector $n^\mu=n(-\dot a,1,0,0)$ with $n$ a normalization factor in the goal to have $g_{\mu\nu}n^\mu n^\nu=+1$. But I still don't have the right induced metric
 P: 11 Induced metric on the brane Thanks for the reply, 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