  In this note several formulae that follow from W. Israel's thin shell
equation of motion are derived. We review the main results of Israel's seminal
papers. The Israel equation of motion is solved for the three regimes of total
energy E<0, E=0, and E>0. This family of solutions is analogous to the three
classes of solutions of the Lemaitre-Tolman-Bondi equations. We also derive the
constraint that the gravitational mass is always positive. The main result of
this note is the expression for the interior and exterior metrics in terms of
proper time on the shell. In these coordinates the time dilation formula is
readily derived. We also derive the time dilation formula by direct geometric
analysis. In particular we show that there is a differential time dilation of
clocks at rest inside the cavity and at rest just outside the cavity.
