Self-gravitating razor-thin discs around black holes via multi-hole seeds

Abstract

We construct self-gravitating razor-thin discs of counterrotating matter around Schwarzschild black holes (BHs) by applying the ‘displace, cut, and reflect’ method to known seed solutions representing multi-holes. All but one of the sources of the seed solution generate the surrounding annular disc, whereas the remaining BH is mapped onto a Schwarzschild BH which lies at the disc centre after the transformation. The discs are infinite in extent, have annular character, and are linearly stable up to the innermost stable circular orbit (ISCO) of the system. The spacetime is asymptotically flat, having finite Arnowitt–Deser–Misner mass. Moreover, all energy conditions for the disc are satisfied for radii larger than the ISCO radius; the method, however, produces counterrotating streams with superluminal velocities in the vicinity of the central BH. We also comment on charged discs around extremal Reissner–Nordström BHs constructed from a Majumdar–Papapetrou N-BH seed solution. These simple examples can be extended to more general ‘BH + disc’ solutions, obtained by the same method from seeds of the type ‘BH + arbitrary axisymmetric source’. A natural follow-up of this work would be to construct discs around Reissner–Nordström BHs with arbitrary charge-to-mass ratio and around Kerr BHs.

Black hole encircled by a thin disk: fully relativistic solution

“…(5) Such a potential can be constructed by considering some mass distribution along the negative half of the axis z < 0, then cutting the solution along the equatorial plane z = 0 and reflecting its upper part z > 0 to the negative part z < 0 of the axis -the so-called 'displace, cut and reflect' method (Kuzmin 1956;Vieira 2020). The resulting field is then symmetric with respect to the equatorial plane z = 0 and given by…”

Black Hole Encircled by a Thin Disk: Fully Relativistic Solution*

“…Such a potential can be constructed by considering some mass distribution along the negative half of the axis z < 0, then cutting the solution along the equatorial plane z = 0, and reflecting its upper part z > 0 to the negative part z < 0 of the axis-the so-called "displace, cut, and reflect" method (Kuzmin 1956;Vieira 2020). The resulting field is then symmetric with respect to the equatorial plane z = 0 and given by…”

Post-Newtonian Hamiltonian dynamics: applications to stationary spacetimes and statistical mechanics

“…where the expressions for the momenta (19) were used in the last equation. We can now solve the Hamiltonian constraint π µ π µ = −c 2 and write π 0 in terms of the spatial momenta, and essentially we reproduce the isoenergetic reduction construction and get (13). The freedom in choosing different evolution parameters will be important in order to properly compare different Hamiltonian and Lagrangian formulations of the CBE in the following sections.…”

“…The energy-momentum tensor for a razor-thin disk has the form T µν ∝ δ(z), see for instance [11,12,13] and references therein. We can write it as [2,3] n…”