Relativistic disks

Of course, the Shakura-Sunyaev disk models presented above are non-relativistic and, hence, can only be viewed as a crude approximation to real black hole accretion disks. The formal relativistic generalization of the steady-state Shakura-Sunyaev disk model to a relativistic accretion disk in a Kerr metric was made in 1974 by Kip Thorne, Don Page and Igor Novikov [114,115,116], whereas time-dependent relativistic disks were first treated by Douglas Eardley and Alan Lightman [117]. We shall refer to these relativistic, geometrically-thin and radiatively-efficient disks as the ``standard model'' for black hole accretion disks.

Several new features to the model arise once the fully relativistic situation is considered. Most importantly, there is a natural choice of inner boundary. In the standard model of a geometrically-thin relativistic accretion disk, the torque is assumed to vanish at the radius of marginal stability, $r=r_{\rm ms}$; we shall refer to this standard assumption as the zero-torque boundary condition (ZTBC). It is envisaged that, within $r=r_{\rm ms}$, the material plunges into the black hole conserving its energy and angular momentum. When the ZTBC is employed, the viscous dissipation per unit proper disk face area per unit proper time is [115]

$$D_{\rm PT}(r;\hbox{$a^*$})=\frac{\dot{M}}{4\pi r}{\cal F},$$ 15

where we have defined the function ${\cal F}$

with,

Here, we have defined the dimensionless spin parameter to be $\hbox{$a^*$}=a/M$. This is the relativistic generalization of eqn. 14. As we shall discuss in §3.2.4, there are likely situations in which the ZTBC does not apply and that the region in which angular momentum and energy transport occurs penetrates significantly within $r=r_{\rm ms}$. Nevertheless, the ``standard model'' is still a useful reference point against which more sophisticated disk models can be judged.

The properties of the inner edge of the disk are important in determining the efficiency of the accretion disk. We define the efficiency of the disk, $\eta$, by $L=\eta \dot{M}c^2$, where $\dot{M}$ is the mass accretion rate onto the black hole and $L$ is the total power output of the accretion disk (summing over all forms including radiation and the kinetic energy of any outflows). The assumptions of the standard model (and, in particular, the ZTBC), fixes the efficiency $\eta$ to be the change in specific binding energy as a fluid element moves from infinity down to $r=r_{\rm ms}$. For a non-rotating black hole, the radius of marginal stability is relatively distant ($r_{\rm ms}=6M$) and the radiative efficiency is only $\eta\approx 0.06$. On the other hand, a standard disk around a Kerr black hole with $\hbox{$a^*$}\equiv a/M=1$ has an efficiency of $\eta\approx 0.42$. For a realistic rapidly spinning black hole, with $\hbox{$a^*$}\approx 0.998$, the efficiency is $\eta\approx 0.30$.