I have a fairly simple question about perturbations of Schwarzchild spacetime in general relativity but cannot seem to find the answer anywhere.
Start with the standard Schwarzchild metric
$g^{SCH}= -\phi^2 dt^2 + \Bigg(1 + \frac{m_0}{2 r} \Bigg)^4 \delta$
where $\delta$ is the Euclidean metric and $\phi$ is defined to be the warping factor
$\phi = \frac{1 - \frac{m_0}{2r} } {1 + \frac{m_0}{2r}}$
and then add a perturbation so that you have a perturbed metric $g_{\alpha \beta} = g^{SCH}_{\alpha \beta} + h_{\alpha \beta}$, where the size of the components of $h_{\alpha \beta}$ is small (in some suitable sense or norm). Now if one takes the $t=0$ hypersurface in this perturbed spacetime, what would the induce metric $g_{ij}$ be and what would the extrinsic curvature $k_{ij}$ be? In the case of regular Schwarzchild, the extrinsic curvature of the $t=0$ hypersurface would vanish due to the spacetime being static, but now it has to take some general form.
Throughout, Roman indices run over spacial coordinates while Greek indices run over all coordinates.
Since you're always working in an adapted coordinate system with $\partial_0$ orthogonal to the hypersurface, the induced metric is quite simple: just the restriction of the ambient metric to the spatial coordinates (which is why the notation $g_{ij}$ is justified).
The second fundamental form simplifies similarly in adapted coordinates. In codimension one, we can write the scalar second fundamental form with respect to the future unit normal vector $n$, and expand in terms of Christoffel symbols. $$ k_{ij}=\left\langle n,\nabla_i\partial_j\right\rangle=\left\langle n,\Gamma^\alpha{}_{ij}\partial_\alpha\right\rangle=\Gamma^\alpha{}_{ij}\langle n,\partial_\alpha\rangle=n^\beta\Gamma_{\beta ij} $$ For actually computing the normal, one can use the fact that $(dx^0)^\sharp$ is orthogonal to the hypersurface, and only needs to be normalized. $$ n=\frac{1}{\sqrt{-g^{00}}}g^{0\alpha}\partial_\alpha $$ And thus $$ k_{ij}=\frac{\Gamma^{0}{}_{ij}}{\sqrt{-g^{00}}} $$ Does it make sense this computation would proceed for your metric?