  We use a full general relativistic framework to study the self-similar
expansion of bubbles of the stable phase into a flat
Friedmann-Lema\^itre-Robertson-Walker Universe in a first order phase
transition in the early Universe. With a simple linear barotropic equation of
state in both phases, and in the limit of a phase boundary of negligible width,
we find that self-similar solutions exist, which are qualitatively similar to
the analogous solutions in Minkowski space, but with distinguishing features.
Rarefaction waves extend to the centre of the bubble, while spatial sections
near the centre of the bubble have negative curvature. Gravitational effects
redistribute the kinetic energy of the fluid around the bubble, and can change
the kinetic energy fraction significantly. The kinetic energy fraction of the
gravitating solution can be enhanced over the analogous Minkowski solution by
as much as $\mathcal{O}(1)$, and suppressed by a factor as larger as
$\mathcal{O}(10)$ in case of fast detonations. The amount of negative spatial
curvature at the centre of the bubble is of the same order of magnitude of the
naive expectation based on considerations of the energy density perturbation in
Minkowski solutions, with gravitating deflagrations less negatively curved, and
detonations more. We infer that general relativistic effects might have a
significant impact on accurate calculations of the gravitational wave power
spectrum when the bubble size becomes comparable to the cosmological Hubble
radius, affecting the primary generation from the fluid shear stress, and
inducing secondary generation by scalar perturbations.
