First, write down an frame Ansatz based upon assuming a combination of spacetime symmetry and what kind of motion you expect. For example, assume spherical symmetry and spherically symmetric collapse of spherical shells of test particles which our observers are riding on, a very simple frame Ansatz is:
<br />
<br />
\begin{array}{rcl}<br />
\vec{e}_1 & = & \partial_t - f \, \partial_r \\<br />
\vec{e}_2 & = & \partial_r \\<br />
\vec{e}_3 & = & \frac{1}{r} \, \partial_\theta \\<br />
\vec{e}_4 & = & \frac{1}{r \, \sin(\theta)} \, \partial_\phi<br />
\end{array}<br />
<br />
where f is an undetermined function of r only, saying the rate of falling toward the origin depends only on r. Note that this Ansatz also says that differences in the t coordinate correspond to elapsed proper time measured by any of our infalling observers.
Second, demand that the Einstein tensor as measured by these observers have a given form, say that of an nonnull electrovacuum appropriate for a radial electrostatic field
<br />
<br />
G^{ab} = \frac{q^2}{r^2} \, diag (1,-1,1,1)<br />
<br />
where q is the charge. The great thing about using frames is that tensors like the EM stress-energy tensor have exactly the same form and behave the same way under boosts and rotations at an event as in flat spacetime.
In this example, we find an ODE for f which is easily solved giving
<br />
<br />
f = \sqrt{ \frac{2m}{r} - \frac{q^2}{r^2} }<br />
<br />
If we had demanded instead that the stress-energy tensor be the sum of the previous EM term plus a Lambda term
<br />
<br />
G^{ab} = \frac{q^2}{r^2} \, diag (1,-1,1,1) + \frac{3}{R^2} \, diag \, (1,-1,-1,-1)<br />
<br />
we would have found
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<br />
f = \sqrt{ \frac{2m}{r} - \frac{q^2}{r^2} + \frac{r^2}{R^2} }<br />
<br />
