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A dynamically transversely trapping surface (DTTS) is a new concept of anextension of a photon sphere that appropriately represents a strong gravityregion and has close analogy with a trapped surface. We study formation of amarginally DTTS in time-symmetric, conformally flat initial data with two blackholes, with a spindle-shaped source, and with a ring-shaped source, and clarifythat $\mathcal{C}\lesssim 6\pi GM$ describes the condition for the DTTSformation well, where $\mathcal{C}$ is the circumference and $M$ is the mass ofthe system. This indicates that an understanding analogous to the hoopconjecture for the horizon formation is possible. Exploring the ring systemfurther, we find configurations where a marginally DTTS with the torus topologyforms inside a marginally DTTS with the spherical topology, without beinghidden by an apparent horizon. There also exist configurations where amarginally trapped surface with the torus topology forms inside a marginallytrapped surface with the spherical topology, showing a further similaritybetween DTTSs and trapped surfaces.

Formation of dynamically transversely trapping surfaces and the stretched hoop conjecture