Consider a 3+1 spherically symmetric vacuum spacetime with an arbitrary initial slice $h(r)$. The slice can be thought of as the curve of Minkowski coordinates: $$t_{M} = h(r_{M})$$ The evolution of the system will be over the universal time $t$, of which the hypersurfaces are the level sets of the foliation. The initial data $(t=0)$ for the spatial metric components can be found using the above to be $$\gamma_{rr} = 1 - h'^{2} \\ \gamma_{\theta\theta} = r^{2} $$ since $r$ coincides with $r_{M}$ initially and the only difference is due to the radial slice function so the angular components are the same as with the Minkowski metric.
From the evolution equations, $$ K_{ij} = -\frac{1}{2\alpha}\partial_{t}\gamma_{ij} $$ where $\alpha$ is the lapse and the shift is zero.
Now, this is probably a dumb question but for some reason I just can't figure out how this leads to the following initial data from the source I am reading: $$ K_{rr} = -\frac{h''}{\sqrt{\gamma_{rr}}} \\ K_{\theta\theta} = -\frac{rh'}{\sqrt{\gamma_{rr}}} $$
My attempts give me $$ K_{rr} = \frac{h'h''}{\alpha}\frac{\mathrm{d}r}{\mathrm{d}t} = \frac{h'h''}{\sqrt{\gamma_{rr}}}\\ K_{\theta\theta} = -\frac{r}{\alpha}\frac{\mathrm{d}r}{\mathrm{d}t} = -\frac{r}{\sqrt{\gamma_{rr}}} $$ where I have made use of the fact that $\alpha\gamma_{rr}^{-\frac{1}{2}}$ is like the speed of light in the 3+1 metric $g_{\mu\nu}$. I don't understand what happened to the $h'$ in $K_{rr}$ and how it ended up in $K_{\theta\theta}$.
