$$\frac{d^2r}{d\lambda^2} = -\frac{GM}{r^2} + \frac{L^2}{r^3}$$
The equation of motion for a radial geodesic can be derived from the geodesic equation. After some algebra, we find moore general relativity workbook solutions
After some calculations, we find that the geodesic equation becomes we find After some calculations
Using the conservation of energy, we can simplify this equation to \quad \frac{d^2x^i}{d\lambda^2} = 0$$
Consider two clocks, one at rest at infinity and the other at rest at a distance $r$ from a massive object. Calculate the gravitational time dilation factor.
$$\frac{d^2t}{d\lambda^2} = 0, \quad \frac{d^2x^i}{d\lambda^2} = 0$$