# Schwarzschild metric

Light rays in the schwarzschild metric satisfy the differential equation

$$\frac {d^2u} {d\phi^2}+u=3Mu^2$$

u=1/r

I want to show that there is closed orbits with constant radius and also calculate the radius of the orbits as a function of the Schwarzchild radius.
Can anyone help me with this? Or give me some hits?
As simple as possible plz.

pervect
Staff Emeritus
JohanL said:
Light rays in the schwarzschild metric satisfy the differential equation

$$\frac {d^2u} {d\phi^2}+u=3Mu^2$$

u=1/r

I want to show that there is closed orbits with constant radius and also calculate the radius of the orbits as a function of the Schwarzchild radius.
Can anyone help me with this? Or give me some hits?
As simple as possible plz.

Well, if the orbit is one of constant radius, then du/dphi must be equal to zero, and so must du^2/dphi^2.

You are then left with the equation u = 3Mu^2. Solve for u.