An extreme critical spacetime: echoing and black-hole perturbations

A homothetic, static, spherically symmetric solution to the massless Einste in-Klein-Gordon equations is described. There is a curvature singularity wh ich is central, null, bifurcate, massless and marginally trapped. The space time is therefore extreme in the sense of lying at the threshold between bl ack holes and naked singularities, just avoiding both. A linear perturbatio n analysis reveals two types of dominant mode. One breaks the continuous se lf-similarity by periodic terms reminiscent of discrete self-similarity, wi th an echoing period within a few per cent of the value observed numericall y in near-critical gravitational collapse. The other dominant mode explicit ly produces a black hole, white hole, eternally naked singularity or regula r dispersal, the latter indicating that the background is critical. The bla ck hole is not static but has constant area, the corresponding mass being l inear in the perturbation amplitudes, explicitly determining a unit critica l exponent. It is argued that a central null singularity may be a feature o f critical gravitational collapse.