# Change in Energy and Angular Momentum Upon Decreasing Orbit

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1. Dec 29, 2016

### TeslaCoil137

1. The problem statement, all variables and given/known data
A stationary, axisymmetric, spacetime has two Killing vector fields [ξt, ξφ] corresponding to translation along t or φ directions. A particle of unit mass moving in this spacetime has a four-velocity u = γ[ξt + Ωξφ].
(i) Explain why we can interpret this as a particle moving in a circular orbit.
(ii) Suppose the particle is acted upon by some dissipative forces making it lose energy by the amount δE and orbital angular momentum by the amount δL and move to another circular orbit of smaller radius. Show that δE = ΩδL

2. Relevant equations
Conserved quantities: E = -p * ξt, L = p*ξφ, the energy and angular momentum of the orbit.

3. The attempt at a solution
(i) The particle can be interpreted as moving in a circular orbit since the four velocity, u, can be rewritten as u= γ(1,0,0,Ω); clearly a circular orbit about the axis of symmetry.
(ii) I was thinking maybe an argument similar to the Penrose process but I cannot find an expression for the change in four velocity during this orbital transfer.