Exact solutions with w modes

An explicit necessary condition for the internal trapping of null geodesics, along with the occurrence of resonance scattering of axial gravitational waves, is proposed for static spherically symmetric perfect fluid solutions of Einstein's equations. Some exact inhomogeneous solutions which exhibit this trapping are given with special attention to boundary conditions and the physical acceptability of the space-times. In terms of the tenuity (α=R/M at the boundary) all the examples given here lie in the narrow range 2.8<α<2.9. The tenuity can be raised to more interesting values by the addition of an envelope without altering the trapping.