Formation of dynamically transversely trapping surfaces and the stretched hoop conjecture

Abstract

<jats:title>Abstract</jats:title> <jats:p>A dynamically transversely trapping surface (DTTS) is a new concept for 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.</jats:p>

Photon regions in stationary axisymmetric spacetimes and umbilic conditions

“…We tried to give a more clear and concise idea of the main geometric notions presented in [1], supplementing them with a number of new useful expressions and relations, which, in particular, turned out to be useful for analyzing their connection with Killing tensor fields [20]. We hope that this formalism will pave the way for obtaining new topological constraints, Penrose-type inequalities (and other estimates) [26][27][28], uniqueness theorems [23,[29][30][31][32][33], similar to ones known for photon spheres and transversally trapping surfaces [34,35].…”