Change in Energy and Angular Momentum Upon Decreasing Orbit

[TeslaCoil137]

Homework Statement

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

Homework Equations

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

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.

 

[Jhero]

(i) Yes, the four-velocity can be interpreted as a circular orbit, as it represents a constant velocity in the φ direction while remaining stationary in the t direction. This is consistent with a circular orbit where the particle maintains a constant speed while moving in a circular path.
(ii) To show that δE = ΩδL, we can use the conserved quantities E and L. The change in energy, δE, can be expressed as the change in the timelike component of the four-momentum, δp0 = -δE. Similarly, the change in angular momentum, δL, can be expressed as the change in the spacelike component of the four-momentum, δpφ = δL. Using the conserved quantities, we can write:
δE = -p0 * ξt = -E = -γm,
δL = pφ * ξφ = L = γmΩ,
where m is the mass of the particle. Substituting these expressions into the given equation, we get:
δE = -γm = -γmΩ * δL = ΩδL.
Therefore, δE = ΩδL, which shows that the change in energy is proportional to the change in angular momentum, with the constant of proportionality being the orbital angular velocity Ω. This is consistent with the particle losing energy and angular momentum as it moves to a smaller orbit, as the particle's speed and orbital radius decrease.