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Chandrasekhar–Page equations

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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.[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 (with being a half integer and with the convention ) for the time and the azimuthal component of the spherical polar coordinates , Chandrasekhar showed that the four bispinor components of the wave function,

can be expressed as product of radial and angular functions. The separation of variables is effected for the functions , , and (with being the angular momentum per unit mass of the black hole) as in

Chandrasekhar–Page angular equations

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The angular functions satisfy the coupled eigenvalue equations,[3]

where is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength),

and . Eliminating between the foregoing two equations, one obtains

The function satisfies the adjoint equation, that can be obtained from the above equation by replacing with . The boundary conditions for these second-order differential equations are that (and ) be regular at and . The eigenvalue problem presented here in general requires numerical integrations for it to be solved. Explicit solutions are available for the case where .[4]

Chandrasekhar–Page radial equations

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The corresponding radial equations are given by[3]

where is the black hole mass,

and Eliminating from the two equations, we obtain

The function satisfies the corresponding complex-conjugate equation.

Reduction to one-dimensional scattering problem

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The problem of solving the radial functions for a particular eigenvalue of of the angular functions can be reduced to a problem of reflection and transmission as in one-dimensional Schrödinger equation; see also Regge–Wheeler–Zerilli equations. Particularly, we end up with the equations

where the Chandrasekhar–Page potentials are defined by[3]

and , is the tortoise coordinate and . The functions are defined by , where

Unlike the Regge–Wheeler–Zerilli potentials, the Chandrasekhar–Page potentials do not vanish for , but has the behaviour

As a result, the corresponding asymptotic behaviours for as becomes

References

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  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. 349 (1659). The Royal Society: 571–575. Bibcode:1976RSPSA.349..571C. doi:10.1098/rspa.1976.0090. ISSN 2053-9169. S2CID 122791570.
  2. ^ Page, Don N. (1976-09-15). "Dirac equation around a charged, rotating black hole". Physical Review D. 14 (6). American Physical Society (APS): 1509–1510. Bibcode:1976PhRvD..14.1509P. doi:10.1103/physrevd.14.1509. ISSN 0556-2821.
  3. ^ a b c 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. 391 (1800). The Royal Society: 27–38. Bibcode:1984RSPSA.391...27C. doi:10.1098/rspa.1984.0002. ISSN 2053-9169. JSTOR 2397528. S2CID 120673756.