In general relativity one has the Schwarzchild metric for a non-rotating black hole
$g_{SC} = -\phi^2 \: dt^2 + \Bigg(1 + \frac{m_0}{2r} \Bigg)^4 \delta $
and from this one has the spacelike Schwarzchild metric
$g = \Bigg(1 + \frac{m_0}{2r} \Bigg)^4 \delta $
which corresponds to a spacelike slice of the 4D manifold with vanishing extrinsic curvature where $t$ has been set to $0$. If one takes instead a perturbation of the 4D manifold by some metric $h_{\alpha \beta}$ with small components, is the perturbation of the spacelike Schwarzchild metric by the spatial components $h_{i j}$ of the above perturbation a spacelike slice of the perturbed 4D Schwarzchild spacetime and if so, what is the extrinsic curvature of the hypersurface?
In the perturbed case the extrinsic curvature is zero, but in this case it must take some general form. For example, the 3D Riemann tensor on a hypersurface $\Sigma$ can be expressed in terms of the 4D tensor evaluated on $\Sigma$, so I'm wondering if something similar can be stated here.
