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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 $K_{\theta\theta}$.

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If I've understood what you did correctly, you took the expression for the 3-metric components from the beginning of our post and tried to use the evolution equation for the 3-metric ($\partial_t \gamma_{ij} = -2\alpha K_{ij}$) to compute the expressions that you gave for the initial data.

That is not the correct procedure. The initial data needs to be constructed such that it satisfies the Hamiltonian and momentum constraints. Expressions for those are easily found. They depend on only the 3-metric and the extrinsic curvatures (and their derivatives), but definitely not on the lapse or shift. Your description included some calculation that involved the lapse $\alpha$, so that's a hint that you used the wrong equations from the start.

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  • $\begingroup$ It makes sense that the constraints would give the correct initial data, but I guess I thought I could get away with just using one equation for each instead of trying to solve a system of two. I was a little surprised that doing it the "cheap" way didn't work. After all, the evolution equations are derived from the original Einstein equations so I assumed that they would always be valid, at least analytically. $\endgroup$ Sep 11 '21 at 19:43
  • $\begingroup$ Come to think of it, since the way the slices are connected is not unique, I understand now why this doesn't work. The lapse and the evolution equations just tell you how the hypersurfaces develop, and not how they must be shaped in order to be valid spacetimes, at least initially. $\endgroup$ Sep 11 '21 at 19:47
  • $\begingroup$ If $H$ is the Hamiltonian constraint and $P^i$ is the momentum constraint, then you don't quite have $\partial_t H=0$ and $\partial_t P^i=0$. What you have is that the time derivatives of the constraints are combinations of the constraints themselves. So if you start with something that satisfies the constraints, then you continue to satisfy the constraints and all of the Einstein equations are satisfied for all time. If you start with something that does not obey the constraints, then you started with a solution that didn't obey the Einstein equations and all bets are off for the evolution. $\endgroup$
    – Brick
    Sep 11 '21 at 20:23

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