  In recent years there has been growing interest in verifying the
horizon-scale homogeneity of the Universe that follows from applying the
Copernican Principle to the observed isotropy. This program has been stimulated
by the discovery that a very large void, centred near us, can explain supernova
luminosity distance measurements without dark energy. It is crucial to confront
such models with as wide a variety of data as possible. With this application
in mind, I develop the relativistic theory of linear scalar perturbations on
spherically symmetric dust (Lemaitre-Tolman-Bondi) spacetimes, using the
covariant 1 + 1 + 2 formalism. I show that the evolution of perturbations is
determined by a small set of new linear transfer functions. If decaying modes
are ignored (to be consistent with the standard inflationary paradigm), the
standard techniques of perturbation theory on homogeneous backgrounds, such as
harmonic expansion, can be applied, and results closely paralleling those of
familiar cosmological perturbation theory can be obtained.
