Английская Википедия:Chandrasekhar–Page equations
Шаблон:Short description 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 <math>e^{i(\omega t + m\phi)}</math> for the time and the azimuthal component of the spherical polar coordinates <math>(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>R_{+\frac{1}{2}}(r)</math>, <math>R_{-\frac{1}{2}}(r)</math> and <math>S_{+\frac{1}{2}}(\theta)</math>, <math>S_{-\frac{1}{2}}(\theta)</math>. The energy as measured at infinity is <math>\omega</math> and the axial angular momentum is <math>m</math> which is a half-integer.
Chandrasekhar–Page angular equations
The angular functions satisfy the coupled eigenvalue equations,[3]
- <math>
\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>\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>Q= a\omega\sin\theta + m \csc\theta</math>. Here <math>a</math> is the angular momentum per unit mass of the black hole and <math>\mu</math> is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength). Eliminating <math>S_{+1/2}(\theta)</math> between the foregoing two equations, one obtains
- <math>\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>S_{+\frac{1}{2}}</math> satisfies the adjoint equation, that can be obtained from the above equation by replacing <math>\theta</math> with <math>\pi-\theta</math>. The boundary conditions for these second-order differential equations are that <math>S_{-\frac{1}{2}}</math>(and <math>S_{+\frac{1}{2}}</math>) be regular at <math>\theta=0</math> and <math>\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>\omega=\mu</math>.[4]
References
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Chandrasekhar, S.,(1983). The mathematical theory of black holes. Clarenden Press, Section 104
- ↑ Шаблон:Cite journal
