The spherically symmetric steady accretion of polytropic perfect fluids onto a black hole is the simplest flow model that can demonstrate the effects of backreaction. The analytic and numerical investigation reveals that backreaction keeps intact most of the characteristics of the sonic point. For any such system, with the only free parameter being the relative abundance of the fluid, the mass accretion rate achieves maximal value when the mass of the fluid is universally 1/3 of the total mass.The spherical steady accretion of gas onto a gravitational center has been investigated in the newtonian context by Bondi [1] and in the Schwarzschild space-time by Michel [2], Shapiro and Teukolsky [3], and others [4]. The generalrelativistic description including backreaction has been formulated in [5]. There are two reasons for inspecting steady flows. First, one can see the effects of backreaction in a simple but nontrivial accretion model. Second, gravitational collapse of the fluid which starts as a flow dominated by a steady accretion, can be better controlled, and that would allow one to have an insight into formation of gravitational singularities. This paper focuses on the issue of backreaction.We will consider a spherically symmetric compact ball of a fluid falling onto a non-rotating black hole. The black hole provides the simplest choice for the central object since one can neglect the occurence of shock waves. The fluid is regarded to be steady and by a black hole we understand the existence of an apparent horizon. For the detailed derivation of equations the reader should consult [5]. Here we give only a brief description, focusing attention on the physical assumptions. We will use comoving coordinateswhere the lapse N, α and the areal radius R are functions of a coordinate radius r and an asymptotic time variable t. The nonzero components of the extrinsic curvature K i j of the t = const slices readThe mean curvature of two-spheres of constant radius r, embedded in a Cauchy hypersurface is p =The energy-momentum tensor of the perfect fluid readswhere u µ denotes the four-velocity of the fluid,p is the pressure and ̺ the energy density in the comoving frame.The areal velocity U of a comoving particle designated by coordinates (r, t) is given by U(r, t) = and ∂ R (R 2 U) − R 2 trK = 0. We use here and in what follows the relation ∂ r = √ α(pR/2)∂ R in order to eliminate the differentiation with respect the comoving radius r. The quasilocal2 U(̺ +p). A more familiar form of that isṁ = −4πR 2 nU, where n is the baryonic density (see below). A standard condition for the steady collapsing fluid is that all its characteristics are constant at a fixed R (see [6]). In analytical terms By a black hole is meant a region within an apparent horizon to the future, i.e., a region enclosed by an outermost sphere S A on which the optical scalar θ + ≡
We integrate numerically axially symmetric stationary Einstein equations describing selfgravitating disks around spinless black holes. The numerical scheme is based on a method developed by Shibata, but contains important new ingredients. We derive a new general-relativistic Keplerian rotation law for self-gravitating disks around spinning black holes. Former results concerning rotation around spinless black holes emerge in the limit of a vanishing spin parameter. These rotation curves might be used for the description of rotating stars, after appropriate modification around the symmetry axis. They can be applied to the description of compact torus-black hole configurations, including active galactic nuclei or products of coalescences of two neutron stars.
We investigate spherical, isothermal and polytropic steady accretion models in the presence of the cosmological constant. Exact solutions are found for three classes of isothermal fluids, assuming the test gas approximation. The cosmological constant damps the mass accretion rate andabove certain limit -completely stops the steady accretion onto black holes. A "homoclinic-type" accretion flow of polytropic gas has been discovered in AdS spacetimes in the test-gas limit. These results can have cosmological connotation, through the Einstein-Straus vacuole model of embedding local structures into Friedman-Lemaitre-Robertson-Walker spacetimes. In particular one infers that steady accretion would not exist in the late phases of the Penrose's scenario of the evolution of the Universe, known as the Weyl curvature hypothesis.
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