fix more typos; realize that the spherical accretion initial data likely does...

fix more typos; realize that the spherical accretion initial data likely does not satisfy the geometric constraints

tex/grhd.tex

@@ -1639,8 +1639,7 @@ The 3+1 decomposition of the metric yields
 \end{align}
 in particular, the shift vector $\beta^\mu$ does not vanish in this case.
 
-For the evaluation of the source terms in the GRHD equations we need the following
-quantities
+For the evaluation of the source terms in the GRHD equations we need the following quantities
 \begin{align}
   T^{tt} &= h \rho (u^t)^2 - (\frac{2M}{r} + 1) p \, ,
   &
@@ -1651,9 +1650,9 @@ quantities
   T^{\theta\theta} &= \sin(\theta)^2 T^{\phi\phi} = \frac{p}{r^2} \, .
 \end{align}
 \begin{align}
-  K_{rr} &= \frac{2M(M+r)}{\sqrt{r^5( 2M+r )}} \, ,
+  K_{rr} &= -\frac{2M(M+r)}{\sqrt{r^5( 2M+r )}} \, ,
   &
-  K_{\theta\theta} &= \frac{K_{\varphi\varphi}}{\sin(\theta)^2} = - 2M \sqrt{\frac{r}{2M+r}} \, ,
+  K_{\theta\theta} &= \frac{K_{\varphi\varphi}}{\sin(\theta)^2} = 2M \sqrt{\frac{r}{2M+r}} \, ,
 \end{align}
 \begin{align}
   \tensor{\hGa}{^r_r_r} &= - \frac{M}{r(2M+r)} \, ,
@@ -1669,6 +1668,7 @@ quantities
   \tensor{\hGa}{^\phi_\theta_\phi} &= \frac{\cos(\theta)}{\sin(\theta)} \, ,
 \end{align}
 where $\tensor{\hGa}{^i_j_k}$ are the Christoffels of the induced metric $\tensor{\hat{\gamma}}{_i_j}$.
+The expression for the extrinsic curvature is also given in \cite[Table 2.1]{baumgarte2010}.
 
 For the choice of reference metric and conformal metric we use
 $\tensor{\gamma}{_i_j} = \tensor{\bar{\gamma}}{_i_j} = \tensor{\hat{\gamma}}{_i_j}$,
@@ -1681,13 +1681,21 @@ with $\vec{g}^{i}, \vec{r}$ being the SRHD values.
 
 
 
+\detail{
+Here follow unnecessary details which I wrote down before I realized the following:
+The initial data construction for this test, outlined in \autoref{ssec:spherical-accretion},
+assumes the Schwarzschild solution as a fixed background spacetime.
+Consequently, the geometric constraint equations need not be satisfied, because the
+Schwarzschild solution is a solution to the vacuum EFEs.
+Nevertheless, it is possible to have a stationary or static state solution in the hydro
+sector only.
+
 To evaluate the geometric constraints we need to compute the following values too,
 \begin{align}
-  \partial_r K_{rr} &= \frac{2M}{\sqrt{r^5(2M+r)}} - \frac{5M(M+r)}{\sqrt{r^7(2M+r)}} - \frac{M(M+r)}{\sqrt{r^5(2M+r)^3}} \, ,
-  \\
-  \partial_r K_{\theta\theta} &= \partial_r \frac{K_{\phi\phi}}{\sin(\theta)^2} =
-  - \frac{M}{\sqrt{r(2M+r)}}  + M \sqrt{\frac{r}{(2M+r)^3}} \, ,
-  \\
+  \partial_r K_{rr} &= \frac{2M(5M^2+2r^2+6Mr)}{\sqrt{r^7(2M+r)^3}} \, ,
+  &
+  \partial_r K_{\theta\theta} &= \partial_r \frac{K_{\phi\phi}}{\sin(\theta)^2} = \frac{2M^2}{\sqrt{r(2M+r)^3}} \, ,
+  &
   \partial_\theta \tensor{K}{_\phi_\phi} \sim \cos(\theta) &\xrightarrow{\theta = \pi/2} 0 \, ,
 \end{align}
 
@@ -1756,7 +1764,7 @@ With this we can evaluate the $r$ component of the momentum constraint
 \begin{align}
   D_j (K^{rj} - \gamma^{rj} K) = 8\pi S^r \, .
 \end{align}
-
+}
 
 
 
@@ -1991,6 +1999,7 @@ computed in the following references
 
 
 \subsubsection{Spherical accretion}
+\label{ssec:spherical-accretion}
 
 
 The construction of the initial data is discussed in \cite[section 4.2]{bugner2015} and