"Foundations of Black Hole Accretion Disk Theory"
Marek A. Abramowicz and P. Chris Fragile 
1 Introduction
2 Three Destinations in Kerr’s Strong Gravity
2.1 The event horizon
2.2 The ergosphere
2.3 ISCO: the orbit of marginal stability
2.4 The Paczyński–Wiita potential
2.5 Summary: characteristic radii and frequencies
3 Matter Description: General Principles
3.1 The fluid part
3.2 The stress part
3.3 The Maxwell part
3.4 The radiation part
4 Thick Disks, Polish Doughnuts, & Magnetized Tori
4.1 Polish doughnuts
4.2 Magnetized Tori
5 Thin Disks
5.1 Equations in the Kerr geometry
5.2 The eigenvalue problem
5.3 Solutions: Shakura–Sunyaev & Novikov–Thorne
6 Slim Disks
7 Advection-Dominated Accretion Flows (ADAFs)
8 Stability
8.1 Hydrodynamic stability
8.2 Magneto-rotational instability (MRI)
8.3 Thermal and viscous instability
9 Oscillations
9.1 Dynamical oscillations of thick disks
9.2 Diskoseismology: oscillations of thin disks
10 Relativistic Jets
11 Numerical Simulations
11.1 Numerical techniques
11.2 Matter description in simulations
11.3 Polish doughnuts (thick) disks in simulations
11.4 Novikov–Thorne (thin) disks in simulations
11.5 ADAFs in simulations
11.6 Oscillations in simulations
11.7 Jets in simulations
11.8 Highly magnetized accretion in simulations
12 Selected Astrophysical Applications
12.1 Measurements of black-hole mass and spin
12.2 Black hole vs. neutron star accretion disks
12.3 Black-hole accretion disk spectral states
12.4 Quasi-Periodic Oscillations (QPOs)
12.5 The case of Sgr A*
13 Concluding Remarks

List of Footnotes

1 An interesting, unanswered question is if there are any missing populations of black holes, e.g., “intermediate” mass black holes in the range 6 100M ⊙ ≲ M ≲ 10 M ⊙ [180, 203, 90Jump To The Next Citation Point, 67Jump To The Next Citation Point] or “primordial” black holes [56].
2 M ⊙ = 1.99 × 1033 [g] denotes the mass of the Sun, used in astrophysics as a mass unit.
3 Radiant power in astrophysics is traditionally called “luminosity.” At the Eddington luminosity, LEdd ≡ 1.2× 1038M ∕M⊙ [erg∕s], radiation force balances the gravity of the central object (with mass M). In the case of stars, “super-Eddington” luminosities, L > LEdd, are not possible, as this would mean radiation pressure would blow the star apart.
4 Observations of black holes may eventually cast light on the quantum gravity structure of the physical vacuum [3Jump To The Next Citation Point], possibly constraining string theory [64] and the hypothesis of extra dimensions [21, 251, 140].
5 Strictly speaking, this statement is only true for nearly spherical gravity sources. Higher order (octopole) moments allow for the formation of an ISCO, even in Newtonian theory [15].
6 Astrophysical black holes do not themselves radiate. The temperature associated with Hawking radiation is TH = (ℏc3)∕(8πGM kB). For a stellar-mass black hole TH ∼ 10−8[∘ K ]. Thus, Hawking radiation is completely suppressed by the thermal bath of the 3[∘ K] cosmic background radiation. For supermassive black holes, the Hawking temperature is at least five orders of magnitude smaller still.
7 The rate M˙∗ at which matter accretes is therefore regulated by internal torques and radiative processes. Only if M˙∗ = M˙0 = const everywhere, with M˙0 being the outside mass supply, can the accretion process be stationary. Since the internally determined M˙∗ may change due to instabilities, limit cycles, etc., an occurrence of a really long-term steady accretion flow should be considered a fine-tuned eigenstate. Note, too, that in many astrophysical situations M˙0 is also genuinely variable.
8 This makes black hole accretion the most efficient energy generation process in the universe, short of matter-antimatter annihilation.
9 Note that there is an analogy with stars, where there are three timescales – dynamical, thermal, and nuclear – which for most of the life of a star obey tdyn ≪ tth ≪ tnuc.
10 The slow rotation cases are sometimes referred to as “Bondi flows” in honor of Bondi’s pioneering works on spherically-symmetric (non-rotating) accretion [50].
11 “Well, in our country,“ said Alice, still panting a little, “you’d generally get to somewhere else if you run very fast for a long time, as we’ve been doing.” “A slow sort of country!” said the Queen. “Now, here, you see, it takes all the running you can do, to keep in the same place.”
12 For a congruence of observers (or particles or photons) with four velocity Uμ, the kinematic invariants fully describe their relative motion. Consider those that, in a particular moment s0, occupy the surface of an infinitesimally small sphere. Now, consider the deformation of that surface at a later moment s0 + ds. The volume change dV∕ds = Θ is called expansion. The shear tensor σμν measures the ellipsoidal distortion of this sphere, and the vorticity tensor ωμν describes its rotation (i.e., three independent rotations around three perpendicular axes). Expansion, shear, and vorticity are determined by the tensor X μν = ∇ μUν in the following way: Θ = (1∕3)Xμμ, σ μν = hαμhβν(1∕2)(X αβ +X βα)− Θh μν, and ωμν = hαμhβν((1∕2)(X αβ − X βα). Here hαμ = δαμ + U αUμ is the projection tensor. The acceleration aμ is also considered a kinematic invariant.
13 For the benefit of readers who are unfamiliar with the phenomenology of relativistic jets we mention that many jets demonstrate incredibly consistent collimation, even along hundreds or thousands of kiloparsecs (e.g. [114]). From this we can infer that such jets have maintained their orientation and outflow for millions of years.
14 Radio loud AGN are ∼ 103– 104 times brighter in radio than radio quiet AGN of comparable optical luminosities [289].
15 It is quite remarkable that the mathematical theory describing resonances makes some very general predictions about the behavior of frequencies and amplitudes, even if the specific physical properties of the oscillating objects are not known. To some extent it is possible to accurately describe how things oscillate without even knowing what is oscillating.
16 The mathematical foundations of the resonance model are described and discussed in a special issue of the Astronomische Nachrichten [2].