Astronomy:Chandrasekhar–Page equations

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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]\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,[3]

[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 rest mass of the particle. 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].[4]

References

  1. Chandrasekhar, S. (1976-06-29). "The solution of Dirac's equation in Kerr geometry". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (The Royal Society) 349 (1659): 571–575. doi:10.1098/rspa.1976.0090. ISSN 2053-9169. Bibcode1976RSPSA.349..571C. 
  2. Page, Don N. (1976-09-15). "Dirac equation around a charged, rotating black hole". Physical Review D (American Physical Society (APS)) 14 (6): 1509–1510. doi:10.1103/physrevd.14.1509. ISSN 0556-2821. Bibcode1976PhRvD..14.1509P. 
  3. Chandrasekhar, S.,(1983). The mathematical theory of black holes. Clarenden Press, Section 104
  4. Chakrabarti, S. K. (1984-01-09). "On mass-dependent spheroidal harmonics of spin one-half". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (The Royal Society) 391 (1800): 27–38. doi:10.1098/rspa.1984.0002. ISSN 2053-9169. Bibcode1984RSPSA.391...27C.