  Shape dynamics is a classical theory of gravity which agrees with general
relativity in many important cases, but possesses different gauge symmetries
and constraints. Rather than spacetime diffeomorphism invariance, shape
dynamics takes spatial diffeomorphism invariance and spatial Weyl invariance as
the fundamental gauge symmetries associated with the gravitational field.
Despite these differences, shape dynamics and general relativity generically
predict the same dynamics---there exist gauge-fixings of each theory that
ensure agreement with the other. However, these gauge-fixing conditions are not
necessarily globally well-defined and it is therefore possible to find
solutions of the shape dynamics equations of motion that agree with general
relativity on some open neighborhoods, but which have different global
structures. In particular, the black hole solutions of the two theories
disagree globally. Understanding these novel "shape dynamic black holes" is the
primary goal of this thesis.
