On spherically symmetric shear-free perfect fluid configurations (neutral and charged). II. Equation of state and singularities

Abstract

Geometrical and physical properties of the solutions derived and classified in Part I [J. Math. Phys. 28, 1118 (1987)] are examined in detail. It is shown how the imposition of zero shear restricts the possible choices of equations of state. Two types of singular boundaries arising in these solutions are examined by verifying the local behavior of causal curves approaching these boundaries. For this purpose, a criterion due to C. J. S. Clarke (private communication) is given, allowing one to test the completeness of arbitrary accelerated timelike curves in terms of their acceleration and proper time. One of these boundaries is a spacelike singularity at which causal curves terminate as pressure diverges but matter-energy and charge densities remain finite. At the other boundary, which is timelike if the expansion Θ is finite, proper volume of local fluid elements vanishes as all state variables diverge but causal curves are complete. If Θ diverges at this boundary, a null singularity arises as the end product of the collapse of a two-sphere generated by a given class of timelike curves. The gravitational collapse of bounded spheres matched to a Schwarzschild or Reissner–Nordstro/m exterior is also examined in detail. It is shown that the spacelike singularity mentioned above could be naked under certain parameter choices. Solutions presenting the other boundary produce very peculiar black holes in which the ‘‘surface’’ of the sphere collapses into the above mentioned null singularity, while the ‘‘interior’’ fluid layers avoid this singularity and evolve towards their infinite future.