Astronomy:Chandrasekhar–Page equations

Short description: A massive fermion wave equation in Kerr spacetime

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. 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 [math]\displaystyle{ e^{i(\omega t + m\phi)} }[/math] for the time and the azimuthal component of the spherical polar coordinates [math]\displaystyle{ (r,\theta,\phi) }[/math], 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 [math]\displaystyle{ R_{+\frac{1}{2}}(r) }[/math], [math]\displaystyle{ R_{-\frac{1}{2}}(r) }[/math] and [math]\displaystyle{ S_{+\frac{1}{2}}(\theta) }[/math], [math]\displaystyle{ S_{-\frac{1}{2}}(\theta) }[/math]. The energy as measured at infinity is [math]\displaystyle{ \omega }[/math] and the axial angular momentum is [math]\displaystyle{ m }[/math] which is a half-integer.

Chandrasekhar–Page angular equations

The angular functions satisfy the coupled eigenvalue equations,

[math]\displaystyle{ \begin{align} \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{align} }[/math]

where

[math]\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} }[/math]

and [math]\displaystyle{ Q= a\omega\sin\theta + m \csc\theta }[/math]. Here [math]\displaystyle{ a }[/math] is the angular momentum per unit mass of the black hole and [math]\displaystyle{ \mu }[/math] is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength). Eliminating [math]\displaystyle{ S_{+1/2}(\theta) }[/math] between the foregoing two equations, one obtains

[math]\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. }[/math]

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