Prologue

Black holes are the vacuum solutions of Einstein's field equations in general relativity. Classically, a black hole is conceived as a singularity in space time, censored from the rest of the Universe by a mathematically defined one way surface, the event horizon. Informations about the matter which formed the black hole or is falling into it, disappear behind the event horizon, are therefore permanently inaccessible to the external observer.

In astrophysics, black holes are the end point of gravitational collapse of massive celestial objects. Astrophysical black holes may be broadly classified into two categories, the stellar mass ($M_{BH}{\sim}$ a few $M_{\odot}$), and super massive ( $M_{BH}{\ge}{10^6}M_{\odot}$) black holes. Both kind of astrophysical black holes, the stellar mass and super massive, however, accrete matter from the surroundings. Depending on the intrinsic angular momentum content of accreting material, either spherically symmetric, or axisymmetric flow geometry is invoked to study an accreting black hole system.

For matter accreting with non zero intrinsic angular momentum, before the in-falling matter plunges through the event horizon, accreting fluid will be thrown into circular orbits around the hole, and such axisymmetric configuration of rotating matter (non axisymmetric disc accretion is also possible for certain misalignment is between the angular momentum of the accreting material and the black hole spin angular momentum) is known as accretion disc.

If the instantaneous dynamical velocity and local acoustic velocity of the accreting fluid, moving along a space curve parameterized by $r$, are $u(r)$ and $c_s(r)$, respectively, then the local Mach number $M(r)$ of the fluid can be defined as $M(r)={u(r)}/{c_s(r)}$. The flow will be locally subsonic or supersonic according to $M(r) < 1$ or $>1$, i.e., according to $u(r)<c_s(r)$ or $u(r)>c_s(r)$. The flow is transonic if at any moment it crosses $M=1$ hypersurface. This happens when a subsonic to supersonic or supersonic to subsonic transition takes place either continuously or discontinuously. The point(s) where such crossing takes place continuously is (are) called sonic point(s), and where such crossing takes place discontinuously are called shocks or discontinuities.

At a distance far away from the black hole, accreting material almost always remains subsonic (except for the supersonic stellar wind fed accretion) since it possesses negligible dynamical flow velocity. On the other hand, the flow velocity will approach the velocity of light ($c$) while crossing the event horizon, while the maximum possible value of sound speed (even for the steepest possible equation of state) would be $c/\sqrt{3}$, resulting $M>1$ close to the event horizon. In order to satisfy such inner boundary condition imposed by the event horizon, accretion onto black holes exhibit transonic properties in general.

For certain values of the intrinsic angular momentum density of accreting material, the number of sonic point, unlike spherical accretion, may exceed one, and accretion is called `multi-transonic'. Such situations may be observed for sub-Keplerian weakly rotating flows, which are commonly found in various astrophysical situations, such as detached binary systems fed by accretion from OB stellar winds, semi-detached low-mass non-magnetic binaries, and super-massive black holes fed by accretion from slowly rotating central stellar clusters. Even for a standard Keplerian accretion disc, turbulence may produce such low angular momentum flow.

In such supersonic astrophysical flows, shock waves may form resulting in a flow which becomes subsonic. This is because the repulsive centrifugal potential barrier experienced by such flows is sufficiently strong to brake the in-falling motion and a stationary solution could be introduced only through a shock. Rotating, transonic astrophysical fluid flows are thus believed to be `prone' to the shock formation phenomena. One also expects that a shock formation in black-hole accretion discs might be a general phenomenon because shock waves in rotating astrophysical flows potentially provide an important and efficient mechanism for conversion of a significant amount of the gravitational energy into radiation by randomizing the directed infall motion of the accreting fluid. Hence, the shocks play an important role in governing the overall dynamical and radiative processes taking place in astrophysical fluids and plasma accreting onto black holes. The study of steady, standing, stationary shock waves produced in black hole accretion has acquired an important status in recent days.

We address the issue of the formation of steady, standing shock waves in general relativistic black-hole accretion discs, and related phenomena.