Asymptotic behavior of null geodesics near future null infinity. III. Photons towards inward directions

A new sufficient condition for photons emitted near future null infinity to reach is derived by studying geodesics in the Bondi coordinates asymptotically flat spacetimes. In our previous works [M. Amo et al., Phys. Rev. D 104, 064025 (2021).; M. 105, 064074 (2022).], such a was established outward or tangential directions constant radial surfaces. This paper improves result including inward directions. four dimensions, imposing same assumptions on metric functions as previously, we prove that if their initial values of $|dr/du|$ are smaller than certain quantity, where $r$ and $u$ retarded time coordinates, respectively. quantity determined asymptotic properties connected conjectured maximal luminosity. higher with $dr/du>\ensuremath{-}(1\ensuremath{-}1/\sqrt{3})\ensuremath{\approx}\ensuremath{-}0.423$ shown without functions.