An extreme critical space-time: echoing and black-hole perturbations

A homothetic, static, spherically symmetric solution to the massless Einstein- Klein-Gordon equations is described. There is a curvature singularity which is central, null, bifurcate and marginally trapped. The space-time is therefore extreme in the sense of lying at the threshold between black holes and naked singularities, just avoiding both. A linear perturbation analysis reveals two types of dominant mode. One breaks the continuous self-similarity by periodic terms reminiscent of discrete self-similarity, with echoing period within a few percent of the value observed numerically in near-critical gravitational collapse. The other dominant mode explicitly produces a black hole, white hole, eternally naked singularity or regular dispersal, the latter indicating that the background is critical. The black hole is not static but has constant area, the corresponding mass being linear in the perturbation amplitudes, explicitly determining a unit critical exponent. It is argued that a central null singularity may be a feature of critical gravitational collapse.