# Closed Orbit Light Rays in Schwarzschild Metric - Help Needed

• JohanL
In summary, the conversation discusses the differential equation and conditions for closed orbits with constant radius in the Schwarzschild metric. The equation u=3Mu^2 is mentioned as a way to find solutions for u, and help or hints are requested in solving the equation.
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?

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 Schwarzschild radius.
Can anyone help me with this? Or give me some hits?

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.

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!

## 1. What is the Schwarzschild Metric?

The Schwarzschild Metric is a mathematical description of the curved spacetime around a non-rotating, spherically symmetric mass, such as a black hole or a massive star.

## 2. What are closed orbit light rays?

Closed orbit light rays are paths that light can take in the curved spacetime described by the Schwarzschild Metric, where the light returns to its starting point after completing a closed loop.

## 3. Why are closed orbit light rays important?

Closed orbit light rays are important because they provide information about the curvature of spacetime and the effects of gravity near massive objects.

## 4. How are closed orbit light rays calculated?

Closed orbit light rays are calculated using the equations of motion for light in the Schwarzschild Metric, which take into account the mass and radius of the object creating the curvature.

## 5. What can we learn from studying closed orbit light rays in the Schwarzschild Metric?

Studying closed orbit light rays can help us understand the properties of black holes and other massive objects, as well as test the predictions of Einstein's theory of general relativity.

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