Time asymptotics in cosmological hydrodynamics

Upper bound on the radii of black-hole photonspheres

“…It is worth emphasizing that the two conditions (28) and (29) are indeed satisfied in all Einstein-matter models in which hairy black-hole configurations have been found (see [17] for details).…”

“…We consider static, spherically symmetric, asymptotically flat black-hole spacetimes. The line element may take the following form in Schwarzschild coordinates [17][18][19]…”

“…We shall now derive a generic upper bound on the radii of the innermost null circular geodesics. The conservation equation T µ ν;µ = 0 for the energy-momentum tensor has only one nontrivial component [17] T µ r;µ = 0 .…”

“…According to this theorem, the non-trivial (non-asymptotic) behavior of the hair must extend above the black-hole photonsphere. In this respect, the black-hole photonsphere provides a generic lower bound on the effective length of the black-hole hair [16,17]. In addition, it was recently proved [18] that the innermost null circular geodesic of a black-hole spacetime provides the fastest way (as measured by asymptotic observers) to circle the central black hole.…”

“…Furthermore, for hairy black-hole spacetimes a theorem was recently proved [16,17] that reveals the central role played by the black-hole photonsphere in determining the effective length of the hair outside the black-hole horizon. According to this theorem, the non-trivial (non-asymptotic) behavior of the hair must extend above the black-hole photonsphere.…”