Review of some basic theoretical issues

Throughout this review, we shall assume that standard General Relativity is valid for all regions of interest. Under this conservative assumption, it is a remarkable fact that the relatively simple Kerr metric [26] precisely describes the spacetime outside of almost any astrophysical black hole⁵. The Kerr metric is described by only two parameters which can be chosen to be the mass $M$ and angular momentum $J$ of the black hole. The no-hair theorem of General Relativity tells us that black holes can also possess electrical charge. However, in any astrophysical setting, a black hole with appreciable charge will rapidly discharge via vacuum polarization. In Boyer-Lindquist coordinates (the natural generalization of spherical polar coordinates), the line element of the Kerr metric is given as

where $a=J/M$, $\Delta=r^2-2Mr+a^2$, $\Sigma = r^2+a^2\cos^2\theta$ and we have chosen units such as to set $G=c=1$. The event horizon of the black hole is given by the outer root of $\Delta=0$, i.e. $r_+=(M+\sqrt{M^2-a^2})$. The special case of $a=0$ corresponds to non-rotating black holes, reducing the metric to the Schwarzschild (1916) metric [27]

$$ds^2=-\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2\,d\theta^2+ r^2\sin^2\theta\,d\phi^2,$$ 1

with the event horizon at $r=2M$.

A feature of the black hole spacetime that is crucially important for accretion disk models is the existence of a radius of marginal stability, $r_{\rm ms}$. This is the radius within which circular test particle orbits are no longer stable. We shall refer to the region $r<r_{\rm ms}$ as the plunging region since, in most astrophysical settings, it will be occupied by material that is in the process of plunging into the black hole. For a given black hole mass, this radius is a function of the black hole's angular momentum and, for orbits in the equatorial ($\theta=\pi/2$) plane, is given by

where the $\mp$ sign is for test particles in prograde and retrograde orbits, respectively, relative to the spin axis of the black hole and we have defined,

Effects of the black hole spin that arise as a result of the $dt\,d\phi$ term in the line element correspond to the dragging of inertial frames. An extreme manifestation of frame-dragging is the existence of a region (the ergosphere; given by $r<M+\sqrt{M^2-a^2\cos^2\theta}$) in which all time-like particles must rotate in the same sense as the black hole as seen by an observer at infinity. The frame-dragging becomes more extreme as one approaches the event horizon until, at the horizon, all time-like particles orbit with an angular velocity of $\Omega_{\rm H}=a/2Mr_+$. This is referred to as the angular velocity of the event horizon. Another peculiar feature of the Kerr metric is the existence of negative energy orbits within the ergosphere. If a particle is placed on such an orbit via some interaction within the ergosphere, it will fall into the black hole and diminish the mass and angular momentum of the black hole. This gives rise to the Penrose-process for extracting the rotational energy of a spinning black hole [28]. A consideration of black hole thermodynamics readily shows that, in principle, an energy of up to 29% of the mass energy of the hole, i.e. 0.29M, can be extracted from a maximally-rotating black hole, thereby reducing its angular momentum to zero (e.g., see the treatment in [29]).

In principle, a Kerr black hole can possess a spin parameter arbitrarily close to $a/M=1$ without violating the cosmic censorship hypothesis. However, as argued by Kip Thorne [30], a rapidly spinning black hole that is actively accreting matter will preferentially capture photons with negative angular momentum, thereby limiting the spin parameter to $a/M\approx 0.998$ (depending somewhat on the angular distribution of the photons emitted by the accreting matter). For this reason, most of the astrophysical literature uses the term ``extremal'' or ``near-extremal'' Kerr black hole to refer to a black hole with $a/M=0.998$. We shall follow this convention.