Chandrasekhar–Page equations

The Chandrasekhar–Page equations govern the wave function for massive spin-1/2 particles. They arise from finding a separable solution to the Dirac equation within either the Kerr metric or the Kerr–Newman metric. Subrahmanyan Chandrasekhar first demonstrated in 1976 that the Dirac equation in the Kerr metric admits such a separable solution. Don Page then expanded this result to the Kerr–Newman metric, which describes black holes carrying an electric charge. In his paper, Page noted that N. Toop had independently arrived at the same results, a fact communicated to him by Chandrasekhar. While working on this problem, Chandrasekhar also found, for the first time, a separable solution to the Dirac equation in flat spacetime using oblate spheroidal coordinates.

By assuming a normal mode decomposition with a factor \( e^{i(m\phi - \omega t)} \)—where \( m \) is the azimuthal angular momentum component (taking half-integer values) and \( \omega \) is the frequency—for the time and azimuthal angle in spherical polar coordinates \((r, \theta, \phi)\), Chandrasekhar showed that the four components of the Dirac spinor, written as \([F_1(r,\theta), F_2(r,\theta), G_1(r,\theta), G_2(r,\theta)] e^{i(m\phi - \omega t)}\), can be separated into products of radial and angular functions. This separation is achieved by defining the functions \( f_1 = (r - i a \cos\theta) F_1 \), \( f_2 = (r - i a \cos\theta) F_2 \), \( g_1 = (r + i a \cos\theta) G_1 \), and \( g_2 = (r + i a \cos\theta) G_2 \), where \( a \) is the black hole's angular momentum per unit mass. These are then expressed as \( f_1(r,\theta) = R_{-}(r) S_{-}(\theta) \), \( f_2(r,\theta) = R_{+}(r) S_{+}(\theta) \), \( g_1(r,\theta) = R_{+}(r) S_{-}(\theta) \), and \( g_2(r,\theta) = R_{-}(r) S_{+}(\theta) \).