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Angular velocity calculation from Schwarzschild metric

  1. Mar 3, 2015 #1

    I need to find the angular velocity using Schwarzschild metric.
    At first I wrote the metric in general form and omitted the co-latitude:

    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. jcsd
  3. Mar 6, 2015 #2


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    2017 Award

    Staff: Mentor

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