1.1 Particle motion

Consider a body which has a size much smaller than the size of a black hole such that details connected with its internal structure are not important for the problem under consideration. Such an object is usually called a particle. Often we can consider as particles planet-like objects in their Keplerian motion near a black hole, small elements of accreting matter, stellar black holes and neutron stars falling into a massive black hole, and so on. We do not consider here the more complicated case when an object moving near a black hole has internal structure and its internal degrees of freedom can be excited by tidal forces.

Particle motion in the background black hole geometry is described by a solution of the geodesic equation

 (1.1)

Here u^μ = dx^μ/dτ is the four-velocity of a particle, τ is the proper time and Γ^μν_λ are the Christoffel symbols

 (1.2)

1.2 Schwarzschild metric

The geometry of a static non-rotating black hole is spherically symmetric and described by the Schwarzschild metric

 (1.3)

Here r_S = 2M is the Schwarzschild gravitational radius, M is the black hole mass, and dΩ² = dθ² + sin² θ dφ² is the line element on the unit sphere. The gravitational radius r_S is the only essential dimensional parameter. It determines all characteristic time and length scales. The metric can be rewritten in the following form:

 (1.4)

where x := r/r_S is a dimensionless radial coordinate and ˜t := t/r_S a dimensionless time coordinate. Since the geodesic equations are scale invariant, it is sufficient to solve them for only one value of black hole mass (say for r_S = 1). The solutions for other masses can be obtained simply by rescaling.

Particle trajectories near a non-rotating black hole can be found by solving the geodesic equation in the Schwarzschild geometry. A more effective way is to use the integrals of motion connected with the spacetime symmetries.