Chandrasekhar–Page equations

The angular functions satisfy the coupled eigenvalue equations,[3]

${\displaystyle {\begin{aligned}{\mathcal {L}}_{\frac {1}{2}}S_{+{\frac {1}{2}}}&=-(\lambda -a\mu \cos \theta )S_{-{\frac {1}{2}}},\\{\mathcal {L}}_{\frac {1}{2}}^{\dagger }S_{-{\frac {1}{2}}}&=+(\lambda +a\mu \cos \theta )S_{+{\frac {1}{2}}},\end{aligned}}}$

where

${\displaystyle {\mathcal {L}}_{\frac {1}{2}}={\frac {\mathrm {d} }{\mathrm {d} \theta }}+Q+{\frac {\cot \theta }{2}},\quad {\mathcal {L}}_{\frac {1}{2}}^{\dagger }={\frac {\mathrm {d} }{\mathrm {d} \theta }}-Q+{\frac {\cot \theta }{2}}}$

and ${\displaystyle Q=a\omega \sin \theta +m\csc \theta }$. Here ${\displaystyle a}$ is the angular momentum per unit mass of the black hole and ${\displaystyle \mu }$ is the rest mass of the particle. Eliminating ${\displaystyle S_{+1/2}(\theta )}$ between the foregoing two equations, one obtains

${\displaystyle \left({\mathcal {L}}_{\frac {1}{2}}{\mathcal {L}}_{\frac {1}{2}}^{\dagger }+{\frac {a\mu \sin \theta }{\lambda +a\mu \cos \theta }}{\mathcal {L}}_{\frac {1}{2}}^{\dagger }+\lambda ^{2}-a^{2}\mu ^{2}\cos ^{2}\theta \right)S_{-{\frac {1}{2}}}=0.}$

The function ${\displaystyle S_{+{\frac {1}{2}}}}$ satisfies the adjoint equation, that can be obtained from the above equation by replacing ${\displaystyle \theta }$ with ${\displaystyle \pi -\theta }$. The boundary conditions for these second-order differential equations are that ${\displaystyle S_{-{\frac {1}{2}}}}$(and ${\displaystyle S_{+{\frac {1}{2}}}}$) be regular at ${\displaystyle \theta =0}$ and ${\displaystyle \theta =\pi }$. The eigenvalue problem presented here in general requires numerical integrations for it to be solved. Explicit solutions are available for the case where ${\displaystyle \omega =\mu }$.[4]