Chandrasekhar–Page equations

Chandrasekhar–Page equations describe the wave function of the spin-½ massive particles, that resulted by seeking a separable solution to the Dirac equation in Kerr metric or Kerr–Newman metric. In 1976, Subrahmanyan Chandrasekhar showed that a separable solution can be obtained from the Dirac equation in Kerr metric.^[1] Later, Don Page extended this work to Kerr-Newman metric, that is applicable to charged black holes.^[2] In his paper, Page notices that N. Toop also derived his results independently, as informed to him by Chandrasekhar.

By assuming a normal mode decomposition of the form ${\displaystyle e^{i(\omega t+m\phi )}}$ for the time and the azimuthal component of the spherical polar coordinates ${\displaystyle (r,\theta ,\phi )}$, Chandrasekhar showed that the four bispinor components can be expressed as product of radial and angular functions. The two radial and angular functions, respectively, are denoted by ${\displaystyle R_{+{\frac {1}{2}}}(r)}$, ${\displaystyle R_{-{\frac {1}{2}}}(r)}$ and ${\displaystyle S_{+{\frac {1}{2}}}(\theta )}$, ${\displaystyle S_{-{\frac {1}{2}}}(\theta )}$. The energy as measured at infinity is ${\displaystyle \omega }$ and the axial angular momentum is ${\displaystyle m}$ which is a half-integer.

Chandrasekhar–Page angular 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]

References[]