Asymptotic behavior of null geodesics near future null infinity II:
  curvatures, photon surface and dynamically transversely trapping surface
  Bearing in mind our previous study on asymptotic behavior of null geodesics
near future null infinity, we analyze the behavior of geometrical quantities
such as a certain extrinsic curvature and Riemann tensor in the Bondi
coordinates. In the sense of asymptotics, the condition for an $r$-constant
hypersurface to be a photon surface is shown to be controlled by a key quantity
that determines the fate of photons initially emitted in angular directions. As
a consequence, in four dimensions, such a non-expanding photon surface can be
realized even near future null infinity in the presence of enormous energy flux
for a short period of time. By contrast, in higher-dimensional cases, no such a
photon surface can exist. This result also implies that the dynamically
transversely trapping surface, which is proposed as an extension of a photon
surface, can have an arbitrarily large radius in four dimensions.
