Initialize $\LaTeX$ output and load (frontend to) the differential forms package.
Enter a generic, spherically symmetric line element in orthogonal coordinates.
Compute the components of the Einstein tensor.
Simplify $G{}_{tr}$.
Check that the solution of $G_{tr}=0$ is $B(t,r)=b(r)$.
Simplify $G{}_{tt}+G{}_{rr}$, and insert above solution for $B$.
Check that the solution of $G{}_{tt}+G{}_{rr}=0$ is $A(t,r)=\alpha(t)/b(r)$.
Simplify $G{}_{tt}-G{}_{rr}$, and insert above solutions for $A$, $B$.
Check that the solution of $G{}_{tt}-G{}_{rr}=0$ is $b(r)=1/\sqrt{1-\kappa/r}$.
Substitute the above solutions for $A$, $B$, $b$ into the full Einstein tensor $G_{ij}$.
Enter your own code here for further computation if desired.