Chandrasekhar–Page equations

Chandrasekhar–Page equations describe the wave function of the spin-1/2 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. Later, Don Page extended this work to Kerr–Newman metric, that is applicable to charged black holes. 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 ${\displaystyle e^{i(\sigma t+m\phi )}}$ (with ${\displaystyle m}$ being a half integer and with the convention ${\displaystyle \mathrm {Re} \{\sigma \}>0}$) for the time and the azimuthal component of the spherical polar coordinates ${\displaystyle (r,\theta ,\phi )}$, Chandrasekhar showed that the four bispinor components of the wave function,

${\displaystyle {\begin{bmatrix}F_{1}(r,\theta )\\F_{2}(r,\theta )\\G_{1}(r,\theta )\\G_{2}(r,\theta )\end{bmatrix}}e^{i(\sigma t+m\phi )}}$

can be expressed as product of radial and angular functions. The separation of variables is effected for the functions ${\displaystyle f_{1}=(r-ia\cos \theta )F_{1}}$, ${\displaystyle f_{2}=(r-ia\cos \theta )F_{2}}$, ${\displaystyle g_{1}=(r+ia\cos \theta )G_{1}}$ and ${\displaystyle g_{2}=(r+ia\cos \theta )G_{2}}$ (with ${\displaystyle a}$ being the angular momentum per unit mass of the black hole) as in

${\displaystyle f_{1}(r,\theta )=R_{-{\frac {1}{2}}}(r)S_{-{\frac {1}{2}}}(\theta ),\quad f_{2}(r,\theta )=R_{+{\frac {1}{2}}}(r)S_{+{\frac {1}{2}}}(\theta ),}$

${\displaystyle g_{1}(r,\theta )=R_{+{\frac {1}{2}}}(r)S_{-{\frac {1}{2}}}(\theta ),\quad g_{2}(r,\theta )=R_{-{\frac {1}{2}}}(r)S_{+{\frac {1}{2}}}(\theta ).}$