1.1 Orbits are planar

Consider a particle moving in the Schwarzschild metric. Using spherical symmetry, one can always choose coordinates so that at the initial moment τ₀ one has θ₀ = π/2 and (dθ/dτ)|₀ = 0. Since the only non-vanishing components of Γ^θ_μν are

 (1.1)

the θ-component of the geodesic equation of motion takes the form

 (1.2)

A solution of this equation for the given initial data is θ = π/2. Thus, the trajectory of a particle is planar and we can assume it to lie in the equatorial plane θ = π/2.

1.2 Effective potential

The Schwarzschild metric is invariant under time t and angular coordinate φ translations. The corresponding conserved quantities are (θ = π/2):

 (1.3)

 (1.4)

Ẽ = E/m is the specific energy of a particle (E being the energy, and m being the mass of the particle). The quantity l = L/m is the specific angular momentum of a particle (L being its angular momentum). For the motion in the equatorial plane, the total angular momentum coincides with the azimuthal angular momentum. For a motion of the particle in the black hole exterior, both t and φ are monotonic functions of τ .

Using these relations and the normalization condition for the four-velocity u^μ

 (1.5)

one gets :

 (1.6)

where we introduced the effective potential

 (1.7)

Figure 1.1. Effective potential U(x) as a function of x = r/r_S and for a specific angular momentum ˜L of a particle.

1.3 Properties of the effective potential

Different types of particle trajectory can be classified by studying the turning points of its radial motion where

V (r ) = Ẽ².    (1.8)

The only scale parameter in the problem is the gravitational radius of the black hole. Using again the dimensionless coordinate x and ˜L = l/r_S we can rewrite V (x) according to

 (1.9)

The effective potential U(x) is shown in figure 1.1. For fixed |˜L | ≥ √3, U as a function of x has a maximum at x₊ and a minumum at x₋, where

 (1.10)

For ˜L = √3, x₋ = x₊ = 3. A heavy full line in figure 5.1 shows the position of the extrema.