Abstract:
Paczynski realized that a properly chosen gravitational potential may accurately model (in a "pseudo Newtonian" theory) general relativistic effects that determine motion of matter near a non-rotating black hole. Paczynski's choice, known today as the "Paczynski-Wiita potential", proved to be very practical. It was used by numerous researchers in the black hole accretion theory, and became a standard tool in relativistic astrophysics. The model is an example of Paczynski's admired ability to invent "out of nowhere" simple ideas that were brilliant, deep and useful. Paczynski has guessed intuitively the form of the potential. However, it could be also derived by a a step-by-step formal procedure. I show the derivation here. My derivation is based on a standard definition of the relativistic "effective potential" in the Schwarzschild spacetime. The relativistic effective potential may be uniquely divided into its "gravitational" and "centrifugal" part. The gravitational part differs from the Paczynski-Wiita potential only by a constant.

Abstract:
Three radii are associated with a circle: the "geodesic radius" R_1 which is the distance from circle's center to its perimeter, the "circumferential radius" R_2 which is the length of the perimeter divided by 2 pi and the "curvature radius" R_3 which is circle's curvature radius in the Frenet sense. In the flat Euclidean geometry it is R_1 = R_2 = R_3, but in a curved space these three radii are different. I show that although Newton's dynamics uses Euclidean geometry, its equations that describe circular motion in spherical gravity always unambiguously refer to one particular radius of the three --- geodesic, circumferential, or curvature. For example, the gravitational force is given by F = -GMm/(R_2)^2, and the centrifugal force by mv^2/R_3. Building on this, I derive a Newtonian formula for the perihelion of Mercury advance.

Abstract:
We apply Feynman's principle, ``The same equations have the same solutions'', to Kepler's problem and show that Newton's dynamics in a properly curved 3-D space is identical with that described by Einstein's theory in the 3-D optical geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly, Newton's formulae for Kepler's problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at the marginally stable orbit.

Abstract:
We present an analysis of the behaviour of the electromagnetic self-force for charged particles in a conformally static spacetime, interpreting the results with the help of optical geometry. Some conditions for the vanishing of the local terms in the self-force are derived and discussed.

Abstract:
This review covers the main aspects of black hole accretion disk theory. We begin with the view that one of the main goals of the theory is to better understand the nature of black holes themselves. In this light we discuss how accretion disks might reveal some of the unique signatures of strong gravity: the event horizon, the innermost stable circular orbit, and the ergosphere. We then review, from a first-principles perspective, the physical processes at play in accretion disks. This leads us to the four primary accretion disk models that we review: Polish doughnuts (thick disks), Shakura-Sunyaev (thin) disks, slim disks, and advection-dominated accretion flows (ADAFs). After presenting the models we discuss issues of stability, oscillations, and jets. Following our review of the analytic work, we take a parallel approach in reviewing numerical studies of black hole accretion disks. We finish with a few select applications that highlight particular astrophysical applications: measurements of black hole mass and spin, black hole vs. neutron star accretion disks, black hole accretion disk spectral states, and quasi-periodic oscillations (QPOs).

Abstract:
We consider a new version of the twin paradox. The twins move along the same circular free photon path around the Schwarzschild center. In this case, despite their different velocities, all twins have the same non-zero acceleration. On the circular photon path, the symmetry between the twins situations is broken not by acceleration (as it is in the case of the classic twin paradox), but by the existence of an absolute standard of rest (timelike Killing vector). The twin with the higher velocity with respect to the standard of rest is younger on reunion. This closely resembles the case of periodic motions in compact (non-trivial topology) 3-D space recently considered in the context of the twin paradox by Barrow and Levin, except that there accelerations of all twins were equal to zero, and that in the case considered here, the 3-D space has trivial topology.

Abstract:
Thermal instability is examined for advection-dominated one-temperature accretion disks. We consider axisymmetric perturbations with short wavelength in the radial direction. The viscosity is assumed to be sufficiently small for the vertical hydrostatic balance to hold in perturbed states. The type of viscosity is given either by the $\alpha$-viscosity or by a diffusion-type stress tensor. Optically thick disks are found to be in general more unstable than optically thin ones. When the thermal diffusion is present, the optically thin disks become stable, but the optically thick disks are still unstable. The instability of the advection-dominated disks is different from that of the geometrically thin disks without advection. In the case of no advection, the thermal mode behaves under no appreciable surface density change. In the case of advection-dominated disks, however, the thermal mode occurs with no appreciable pressure change (compared with the density change), when local perturbations are considered. The variations of angular momentum and of surface density associated with the perturbations lead to a thermal instability. The astrophysical implications of this instability are briefly discussed.

Abstract:
We present a systematic numerical study of two-dimensional axisymmetric accretion flows around black holes. The flows have no radiative cooling and are treated in the framework of the hydrodynamical approximation. The models calculated in this study cover the large range of the relevant parameter space. There are four types of flows, determined by the values of the viscosity parameter $\alpha$ and the adiabatic index $\gamma$: convective flows, large-scale circulations, pure inflows and bipolar outflows. Thermal conduction introduces significant changes to the solutions, but does not create a new flow type. Convective accretion flows and flows with large-scale circulations have significant outward-directed energy fluxes, which have important implications for the spectra and luminosities of accreting black holes.

Abstract:
Recent hydrodynamical simulations of radiatively inefficient black hole accretion flows with low viscosity have demonstrated that these flows differ significantly from those described by an advection-dominated model. The black hole flows are advection-dominated only in their inner parts, but convectively dominated at radii R>100R_g. In such flows, the radiative output comes mostly from the convection part, and the radiative efficiency is independent of accretion rate and equals ~0.001. This value gives a limit for how dim an accreting black hole could be. It agrees with recent Chandra observations which indicate that accreting black holes in low-mass X-ray binaries are by factor about 100 dimmer that neutron stars accreting with the same accretion rates.

Abstract:
Recent numerical simulations of radiatively inefficient accretion flows onto compact objects have shown that convection is a general feature in such flows. Dissipation of rotational and gravitational energies in the accretion flows results in inward increase of entropy and development of efficient convective motions. Convection-dominated accretion flows (CDAFs) have a structure that is modified significantly in comparison with the canonical advection-dominated and Bondi-like accretion flows. The flows are characterized by the flattened radial density profiles, ~R^{-1/2}, and have reduced mass accretion rates. Convection transports outward a significant amount of the released binding energy of the accretion flow. We discuss basic dynamical and observational properties of ADAFs using numerical models and self-similar analytical solutions.