To simplify the equations of motion of a test particle we used a special choice of coordinates, namely we oriented the z-axis (that is the direction θ = 0, π) to be orthogonal to the plane of the orbit. Let us now check how the equations of motion are modified if the z-axis is tilted and not orthogonal to the orbit plane. This exercise is instructive for the discussion of particle motion in the Kerr geometry where there exists a preferred direction of the z-axis determined by the direction of the angular momentum of the rotating black hole.

The expression for Ẽ remains the same, while the specific azimuthal angular momentum, which we denote now l_z, is

 (1.1)

One also needs the expression for the conserved total angular momentum, l,

 (1.2)

Using these relations and the normalization condition u_μu^μ = −1 one can obtain the following set of equations:

(1.3)

 (1.4)

 (1.5)

 (1.6)

The equation for θ(τ) shows that the angle θ changes between θ₀ and π −θ₀, where sin θ₀ = l_z/l. This means that the angle between the normal to the trajectory plane and z-axis is π/2 − θ₀.