Closed Orbit Light Rays in Schwarzschild Metric - Help Needed

[JohanL]

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 Schwarzschild radius.
Can anyone help me with this? Or give me some hits?
As simple as possible please.

 

[pervect]

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 Schwarzschild radius.
Can anyone help me with this? Or give me some hits?
As simple as possible please.

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.

 

[R0man]

Sure, I can try to help with this problem. The first thing to note is that the equation you have provided is a second-order differential equation, which means it has two independent solutions. This is important because it allows us to have both outward and inward moving light rays in the Schwarzschild metric.

To find the closed orbits with constant radius, we need to look for solutions where the second derivative of u with respect to phi is equal to zero. This means that the first derivative of u with respect to phi is a constant, which we can call k. This leads to the following equation:

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

becomes

k^2+u=3Mu^2

We can now solve for u in terms of k:

u=\frac{k^2}{3M-k^2}

Now, since we know that u=1/r, we can rewrite this as:

r=\frac{3M-k^2}{k^2}

This is the equation for the closed orbits with constant radius in terms of the Schwarzschild radius (2GM/c^2). We can also use this equation to calculate the radius of the orbits for different values of k. For example, if k=0, we get:

r=\frac{3M}{0}=undefined

This means that there is no closed orbit at this value of k, which makes sense since this corresponds to a straight line trajectory. For any other values of k, we can calculate the radius of the orbit using the above equation.

I hope this helps and gives you some direction in solving this problem. Remember, the key is to look for solutions where the second derivative of u with respect to phi is equal to zero, since this will give us the closed orbits with constant radius. Good luck!