An observer is really an army of observers stationed at all points in space. More formally, the worldlines of a family of observers foliate the spacetime. Each observer records what he or she sees, and the resulting logs are later compared. This process is often loosely described as “an observer seeing”; a better description would be “a family of observers recording”.

We briefly summarize the properties of the several families of observers we have so far discussed.

• Shell Observers: A shell observer stands still on a shell at $r=\hbox{constant}$, so that neither of $\theta$ or $\phi$ is changing. Shell observers therefore measure (infinitesimal) time using \begin{equation} \sigma^t = \sqrt{1-\frac{2m}{r}} \>dt \end{equation} Shell observers measure distance using the spatial part of the Schwarzschild line element, that is, they measure distance at constant time $t$. Shell observers therefore measure (infinitesimal) radial distance using \begin{equation} \sigma^r = \frac{dr}{\sqrt{1-\frac{2m}{r}}} \end{equation} and angular distance (in the equatorial plane, $\theta=\pi/2$) using \begin{equation} \sigma^\phi = r\,d\phi \end{equation}
• Freely Falling Observers: Freely-falling observers move along radial geodesics with $e=1$, that is, they start at rest at $r=\infty$. They therefore measure time and radial distance using “rain” coordinates, rather than Schwarzschild coordinates, that is, using $\{\sigma^T,\sigma^R\}$ (see §Rain Coordinates) rather than $\{\sigma^t,\sigma^r\}$. Since their motion is perpendicular to shells with $r=\hbox{constant}$, they measure distance on those shells the same way shell observers do, namely with $\sigma^\phi$.
• Far-Away “Observers”: Far-away “observers” are really bookkeepers, rather than observers. They notice that the Schwarzschild line element reduces to the Minkowski line element for $r\gg 2m$, so that $dt$ and $dr$ are good approximations to “shell time” and “shell (radial) distance”, respectively, so long as they are far away. But they then continue to use $dt$ and $dr$ to “measure” time and (radial) distance even when they are not in fact far away. One way to think of this is that far-away “observers” record the local values of the coordinates $t$ and $r$, then (incorrectly) use them to compute elapsed time and distance travelled for a passing object. Thus, rather than actually measure such times and distances, they are continually “phoning home” to ask a colleague far away what time it is… However, far-away observers have no difficulty with angular motion; spherical symmetry ensures that they, too, use $\sigma^\phi$ to measure arclength for circular motion.