Null charts and naked singularities in spherically symmetric, homothetic spacetimes

Abstract

We give a unified derivation of a null chart for all spherically symmetric, homothetic spacetimes. These spacetimes contain an interesting class of naked singularities which we are also able to elucidate. Much use is made of graphical representation; in particular a chart of the spacetimes based on their homothetic group motions is introduced. Dust spacetimes, and two homogeneous examples with non-zero pressure (flat Robertson-Walker and a Kantowski-Sachs example) are studied in detail. We show the horizon structure in the null atlas, in comoving coordinates, in terms of the areal radius and comoving time, and in the homothetic diagrams. The critical delay between comoving observers for the onset of “nakedness” is interpreted in terms of a decreasing mass concentration in the spirit of Thorne's “hoop” conjecture. We also give a simple criterion for the existence of apparent horizons isolating the various singularities, and study in detail how this criterion is circumvented in the naked examples. We conclude that this type of naked singularity is a consequence of the imposed homothetic symmetry, by showing it to be generally present and timelike in the homothetic group chart even when it is not visible at comoving infinity (before the onset of criticality). It is the delayed final collapse of initially distant observers in inhomogeneous spacetimes that causes the initial singularity to become visible at comoving infinity. We conclude that these examples do not present an obstacle to the “Event Horizon Conjecture” as summarized by W. Israel (1984). That is, one can formulate criteria for the formation of apparent horizons that do not imply that all singularities are necessarily so enclosed. It is still possible that all singularities stronger than homothetic are isolated by an apparent horizon, in the spirit of Tipler's conjecture.