  Birkhoff's theorem is a classic result that characterizes locally spherically
symmetric solutions of the Einstein equations. In this paper, we illustrate the
consequences of its local nature for the cases of vacuum and positive
cosmological constant. We construct several examples of initial data for
spherically symmetric spacetimes on Cauchy surfaces of different topology than
that of the maximal analytic extension of Schwarzschild and Schwarzschild-de
Sitter spacetimes. The spacetimes formed from the evolution of these initial
data sets also have very different physical properties; in particular they need
not contain a static region or be asymptotically flat or asymptotically de
Sitter. We also present locally spherically symmetric initial data sets for de
Sitter spacetimes that are not covered by the maximal analytic extension of de
Sitter spacetime itself. Finally we illustrate the utility of Birkhoff's
theorem in identifying the spacetimes associated with two spherically symmetric
initial data sets; one proven to exist but not explicitly exhibited and one
which has negative ADM mass.
