Read the full article for free* in Classical and Quantum Gravity:
Bondi-type accretion in the Reissner-Nordström-(anti-)de Sitter spacetime
Filip Ficek 2015 Class. Quantum Grav. 32 235008
arXiv: 1509.07005
*until 30/12/15

Filip Ficek is a graduate student in Theoretical Physics at Jagiellonian University.

In spite of numerous investigations, accretion flows onto the Kerr black hole are still not fully understood, especially for radially dominated flows, where aside from a very specific case of an ultra-hard fluid, general solutions are not known. Some insight may be provided by considering a simpler problem instead, namely spherically symmetric, steady accretion in Reissner-Nordström spacetimes. It is well known that rotating Kerr black holes and charged Reissner-Nordström black holes feature similar horizon and causal structures. In fact, it is common to treat a Reissner-Nordström black hole as a toy model of an astrophysical black hole. If we also take into account the cosmological constant, we may suppose, that accretion solutions in Reissner-Nordström-(anti-)de Sitter spacetime will share qualitative similarities with characteristic for Kerr-(anti-)de Sitter spacetime.

Considering an accretion of a perfect fluid one obtains two conservation laws, that need to be supplemented by the equation of state and boundary conditions. Assuming that a fluid is barotropic, one can reduce the problem to investigation of an autonomous, hamiltonian, two-dimensional dynamical system. In this approach sonic points can be understood as critical points of this system, although the details depend on the chosen parameterization. For better readability of the results I work in (u^r/u_t)² vs. r variables, while keeping in mind that this choice generates additional critical points at horizons.

In the figure, I present exemplary solutions for polytropic test fluid with Γ=4/3, in a presence of negative cosmological constant. Vertical, dashed lines represent the Cauchy horizon (the inner one) and the event horizon (the outer one). Different values of B represent solutions with different entropy. The solution with B=B₀ passes through a saddle point, in which fluid velocity is equal to the local speed of sound. The other types of critical points that can be seen in the picture are centre-type points. Their existence is connected with the existence of homoclinic orbits. These orbits are closed under Cauchy horizon (similar effect may be observed in pure the Reissner-Nordström spacetime) and far away from the black hole (an effect caused by a non-zero cosmological constant).

A physical consequence of the existence of homoclinic solutions is the fact that they are not global. The problem of existence of non-global solutions has been investigated recently. It occurs, that it strongly depends on both the spacetime and the equation of state of an accreting fluid. For example, in the Reissner-Nordström-anti-de Sitter spacetime certain isothermal test fluids admit global solutions. On the other hand it is possible to get homoclinic solutions even in pure Schwarzschild spacetime. The specific relation between spacetime, fluid and type of solution is not yet known, but it seems to be a promising research topic.

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