General Relativity - Schwarzschild Metric

[bon]

Homework Statement

A spaceship is moving without power in a circular orbit about an object with mass M. The radius of the orbit is R = 7GM/c^2

(1) Find the relation between the rate of change of angular position of the spaceship and the proper time and radius of the orbit.

Homework Equations

The Attempt at a Solution

So I wrote down the Lagranian from the Schw. metric and finding the EOM for phi i get

d/dtau (2r^2 sin^theta dphi/dtau) = 0

so i can conclude r^2 dphi/dtau = constant

so r^2 dphi/dt times dt/dtau = constant

is this right? and is this what they want? I'm confused!

Thanks

 

[mark726]

for your question! It seems like you are on the right track with your approach. However, I would suggest writing out the full Lagrangian and EOM for this problem, as it will help clarify the relationship between the rate of change of angular position and the proper time and radius of the orbit.

The Lagrangian for this system would be:

L = 1/2 m (dx/dt)^2 + 1/2 m (dy/dt)^2 + 1/2 m (dz/dt)^2 - GMm/r

where m is the mass of the spaceship, dx/dt and dy/dt are the velocities in the x and y directions, and r is the distance between the spaceship and the object with mass M.

To find the EOM for phi, we can use the Euler-Lagrange equation:

d/dt (dL/d(phi dot)) - dL/d(phi) = 0

Plugging in the Lagrangian and simplifying, we get:

d/dt (m r^2 dphi/dt) = 0

This means that the quantity m r^2 dphi/dt is conserved, which is what you found in your attempt. However, since we are dealing with the proper time (tau) rather than coordinate time (t), we need to use the chain rule to relate dphi/dt to dphi/dtau:

dphi/dtau = dphi/dt * dt/dtau

Using the relation between dt/dtau and the radius of the orbit (R = 7GM/c^2), we can rewrite the conserved quantity as:

m r^2 dphi/dtau = constant * (c^2/7GM)

So, to answer the original question, the rate of change of angular position (dphi/dtau) is inversely proportional to the radius of the orbit (r) and is directly proportional to the square root of the mass of the spaceship (m) and the square root of the mass of the object it is orbiting (M).

I hope this helps clarify the relationship between the rate of change of angular position, proper time, and radius of the orbit! Let me know if you have any further questions.