# Angular velocity calculation from Schwarzschild metric

1. Mar 3, 2015

### fourvector

Hello,

I need to find the angular velocity using Schwarzschild metric.
At first I wrote the metric in general form and omitted the co-latitude:
ds2=T*dt2+R*dr2+Φ*dφ2

and wrote a Lagrangian over t variable:
L = √(T+R*(dr/dt)2+Φ*(dφ/dt)2)

now I can use the Euler–Lagrange equations for φ variable and note that L does not depend on φ.
dL/d(dφ/dt) = const => Φ*(dφ/dt) / L = const => dφ/dt = const * L / Φ

The result is that dφ/dt depends on dr/dt because L contains dr/dt term.

But there is one more way that I can calculate the dφ/dt.
I can write the Lagrangian over new τ variable:
L = √(T*(dt/dτ)2+R*(dr/dτ)2+Φ*(dφ/dτ)2)

I can do the same calculations for dφ/dτ and dt/dτ variables:

dL/d(dφ/dτ) = const => Φ*(dφ/dτ) / L = const => dφ/dτ = const * L / Φ
dL/d(dt/dτ) = const => T*(dt/dτ) / L = const => dt/dτ = const * L / T

Now I can divide one over another and get
dφ/dt = const * T / Φ

The result is that the angular velocity does not depend on dr/dt.

Could someone help me what is wrong with one of my calculation?

2. Mar 6, 2015

### Staff: Mentor

The angular velocity of what?
Please give the full and exact problem statement.