Physical properties of some McVittie metrics

A class of McVittie's new nonquadratic solutions is investigated in some detail with regard to its physical properties. It is found that decreasing pressure and density are not compatible with center regularity for these perfect fluid spheres. It is further seen that for gaseous spheres (i.e., the density drops to zero at the outer boundary of the sphere together with the pressurep) oscillatory motions arenot possible. For these gaseous models the pressure and the density are both positive inside the outer surface, and their respective gradients are negative. For the outer gaseous shells models are constructed where for a certain time interval the pressure is increasing for contracting models. Without any restriction with respect to time, for these shell models it found that the density is increasing for contracting models, and the adiabatic speed of sound is less than the speed of light. It is also found that the trace of the energy-momentum tensor is positive, the total mass is negative, and for collapsing shells the rate of change of circumference as measured by an observer riding on the shell is an increasing function of time. However, all these models have the strange geometric feature that the physical radius is a decreasing function of comoving radial coordinate.