PhDConformal methods have proven to be very useful in the analysis global properties
and stability of vacuum spacetimes in general relativity. These methods transform
the physical spacetime into a different Lorentzian manifold known as the unphysical
spacetime where the ideal points at infinity are located at a finite position. This thesis
makes use of conformal methods and applies them to various problems involving
trace-free matter models. In particular, it makes progress towards the understanding
of the evolution of unphysical spacetimes perturbed by trace-free matter as well as
the behaviour of the the matter itself. To this end, evolution equations (wave equations)
are derived and analyzed for both the unphysical spacetime and the matter.
To investigate the relation between solutions of these wave equations to the Einstein
field equations, a suitable system of subsidiary evolution equations is also derived.
Furthermore, this thesis looks in detail at the behaviour of an unphysical spacetime
coupled to the simplest matter trace free model: the confomally invariant scalar
field. Finally, the system of conformal wave equations is used to show that the
deSitter spacetime is non-linearly stable under perturbations by trace-free matter
