Transversely trapping surfaces: Dynamical version
  We propose new concepts, a dynamically transversely trapping surface (DTTS)
and a marginally DTTS, as indicators for a strong gravity region. A DTTS is
defined as a two-dimensional closed surface on a spacelike hypersurface such
that photons emitted from arbitrary points on it in transverse directions are
acceleratedly contracted in time, and a marginally DTTS is reduced to the
photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close
analogy with (marginally) trapped surfaces in many aspects. After preparing the
method of solving for a marginally DTTS in the time-symmetric initial data and
the momentarily stationary axisymmetric initial data, some examples of
marginally DTTSs are numerically constructed for systems of two black holes in
the Brill-Lindquist initial data and in the Majumdar-Papapetrou spacetimes.
Furthermore, the area of a DTTS is proved to satisfy the Penrose-like
inequality, $A_0\le 4\pi (3GM)^2$, under some assumptions. Differences and
connections between a DTTS and the other two concepts proposed by us
previously, a loosely trapped surface [arXiv:1701.00564] and a
static/stationary transversely trapping surface [arXiv:1704.04637], are also
discussed. A (marginally) DTTS provides us with a theoretical tool to
significantly advance our understanding of strong gravity fields. Also, since
DTTSs are located outside the event horizon, they could possibly be related
with future observations of strong gravity regions in dynamical evolutions.
