 For a body orbiting in the θ = π/2 plane, we will have β = 1, and the body's world-line will maintain constant spatial coordinates for all time.Now, the metric is in the canonical form

From this canonical form, we can easily determine the rotational rate of a gyroscope in proper time

where the last equality is true only for free falling observers for which
there is no acceleration, and thus formula_9.This leads to

Solving this equation for ω yields

This is essentially Kepler's law of periods, which happens to be relativistically exact when expressed in terms of the time coordinate "t" of this particular rotating coordinate system.