Gravity in general relativity, attractive and repulsive contributions

The gravitational equations of Einstein are solved for a sphere filled with a dust gas and floating in infinite empty space. It turns out that the radial acceleration of the gas has negative and positive contributions which may be interpreted as attractive and repulsive gravitational forces respectively. Two cases are considered: the collapse of a gas initially at rest and with uniform density, and the expansion of the gas with initial conditions appropriate to the observed Hubble diagram of Type Ia supernovae. The latter case may be seen as a proposal for a new type of cosmology.