Formation of dynamically transversely trapping surfaces and the
  stretched hoop conjecture
  A dynamically transversely trapping surface (DTTS) is a new concept of an
extension of a photon sphere that appropriately represents a strong gravity
region and has close analogy with a trapped surface. We study formation of a
marginally DTTS in time-symmetric, conformally flat initial data with two black
holes, with a spindle-shaped source, and with a ring-shaped source, and clarify
that $\mathcal{C}\lesssim 6\pi GM$ describes the condition for the DTTS
formation well, where $\mathcal{C}$ is the circumference and $M$ is the mass of
the system. This indicates that an understanding analogous to the hoop
conjecture for the horizon formation is possible. Exploring the ring system
further, we find configurations where a marginally DTTS with the torus topology
forms inside a marginally DTTS with the spherical topology, without being
hidden by an apparent horizon. There also exist configurations where a
marginally trapped surface with the torus topology forms inside a marginally
trapped surface with the spherical topology, showing a further similarity
between DTTSs and trapped surfaces.
