Chandrasekhar–Page equations explained

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

e ^{i(\sigma} (with m being a half integer and with the convention Re \{\sigma\}> 0 ) for the time and the azimuthal component of the spherical polar coordinates (r,\theta,\phi) , Chandrasekhar showed that the four bispinor components of the wave function,\begin{bmatrix} F _{1(r,\theta)} \ F _{2(r,\theta)} \ G _{1(r,\theta)} \ G ^{i(\sigma t + m\phi)} _{2 (r,\theta)\end{bmatrix}e}

can be expressed as product of radial and angular functions. The separation of variables is effected for the functions

f _{1=(r-ia\cos\theta)F} _{1} , f _{2=(r-ia\cos\theta)F} _{2} , g _{1=(r+ia\cos\theta)G} _{1} and g _{2=(r+ia\cos\theta)G} _{2} (with a being the angular momentum per unit mass of the black hole) as in f _{1(r,\theta)} = R _{- (1)/(2)} (r)S _{- (1)/(2)} (\theta), f _{2(r,\theta)} = R _{+ (1)/(2)} (r)S _{+ (1)/(2)} (\theta), g _{1(r,\theta)} = R _{+ (1)/(2)} (r)S _{- (1)/(2)} (\theta), g _{2(r,\theta)} = R _{- (1)/(2)} (r)S _{+ (1)/(2)} (\theta).

Chandrasekhar–Page angular equations

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

\begin{align} l{L} _{(1)/(2)} S _{+ (1)/(2)} & = - ( λ - a\mu \cos\theta )S _{- (1)/(2)} , \\ l{L} ^{\dagger} _{(1)/(2)} S _{- (1)/(2)} & = + ( λ + a\mu \cos\theta )S _{+ (1)/(2)} , \end{align} where \mu is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength), l{L} _{n} = ({d)/() } + Q + n\cot \theta, \quad \mathcal_n^ = \frac - Q + n\cot\theta

and

Q = a\sigma\sin\theta + m \csc\theta . Eliminating S _{+1/2} (\theta) between the foregoing two equations, one obtains\left( l{L} _{(1)/(2)} l{L} ^{\dagger} _{(1)/(2)} + (a\mu\sin\theta)/(λ+ a\mu\cos\theta) l{L} ^{\dagger} _{(1)/(2)} + λ ^{2} - a ^{2\mu} ^{2\cos} ^{2\theta\right)} S _{- (1)/(2)} = 0 .

The function

S _{+ (1)/(2)} satisfies the adjoint equation, that can be obtained from the above equation by replacing \theta with \pi - \theta . The boundary conditions for these second-order differential equations are that S _{- (1)/(2)} (and S _{+ (1)/(2)} ) be regular at \theta =0 and \theta = \pi . The eigenvalue problem presented here in general requires numerical integrations for it to be solved. Explicit solutions are available for the case where \sigma = \mu .^[4]

Chandrasekhar–Page radial equations

The corresponding radial equations are given by

\begin{align} \Delta ^{(1)/(2)} l{D} _{0} R _{- (1)/(2)} & = ( λ + i\mu r)\Delta ^{(1)/(2)} R _{+ (1)/(2)} , \\ \Delta ^{(1)/(2)} l{D} ^{\dagger} _{0} R _{+ (1)/(2)} & = ( λ - i\mu r)R _{- (1)/(2)} , \end{align} where \Delta = r ^{2} - 2 Mr + a ^{2,} M is the black hole mass,l{D} _{n} = ({d)/() } + \frac + 2n \frac, \quad \mathcal_n^\dagger = \frac - \frac + 2n \frac,

and

K = (r ^{2+a} ^{2)\sigma} + am. Eliminating \Delta ^{(1)/(2)} R _{+ (1)/(2)} from the two equations, we obtain\left(\Delta l{D} ^{\dagger l{D}} _{0} - (i\mu \Delta)/(λ+ i\mu r) l{D} _{0} -λ ^{2} - \mu ^{2r} ^{2\right)} R _{- (1)/(2)} = 0 .

The function

\Delta ^{(1)/(2)} R _{+ (1)/(2)} satisfies the corresponding complex-conjugate equation.

Reduction to one-dimensional scattering problem

The problem of solving the radial functions for a particular eigenvalue of

λ 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\left( (d ^{2})/(d\hat r ^{2} _{*}) + \sigma ^{2\right)} Z ^{\pm} = V ^{\pm} Z ^{\pm} ,

where the Chandrasekhar–Page potentials

V ^{\pm} are defined byV ^{\pm} = W ^{2} \pm (dW)/(d\hat r _{*}) , W = (\Delta ^{(1)/(2)} ( λ + \mu ^{2r} ^{2)} ^{3/2})/(\varpi ^{2(λ} ^{2+\mu} ^{2r} ^{2)} +λ \mu\Delta/ 2 \sigma) ,

and

\hat r _{*=r} ^{-1} ^{* + \tan} (\mu r/ λ )/ 2 \sigma , r _{*=r+2Mln(r/2M-1)} is the tortoise coordinate and \varpi ^{2} = r ^{2+a} ^{2} + am/\sigma . The functions Z ^{\pm} (\hat r _{*)} are defined by Z ^{\pm} = \psi ^{+} \pm \psi ^{-} , where\psi ^{+} = \Delta ^{(1)/(2)} R _{+ (1)/(2)} exp \left( + (i)/(2) \tan ^{-1} (\mu r)/(λ) \right), \psi ^{-} = R _{- (1)/(2)} exp \left( - (i)/(2) \tan ^{-1} (\mu r)/(λ) \right).

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

r\to inf ty , but has the behaviourV ^{\pm} = \mu ^{2\left(1} - (2 M)/(r) + … \right).

As a result, the corresponding asymptotic behaviours for

Z ^{\pm} as r\to inf ty becomesZ ^{\pm} = exp \left\{\pm i \left[(\sigma ^{2-\mu} ^{2)} ^{1/2} r + (M\mu ^{2})/((\sigma ^{2-\mu} ^{2)} ^{1/2}) ln (r)/(2 M) \right]\right\}.

Notes and References

1. 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 . 1976-06-29 . 2053-9169 . 10.1098/rspa.1976.0090 . 571–575. 1976RSPSA.349..571C . Chandrasekhar . S. . 122791570 .
2. Page . Don N. . Dirac equation around a charged, rotating black hole . Physical Review D . American Physical Society (APS) . 14 . 6 . 1976-09-15 . 0556-2821 . 10.1103/physrevd.14.1509 . 1509–1510. 1976PhRvD..14.1509P .
3. Chandrasekhar, S.,(1983). The mathematical theory of black holes. Clarenden Press, Section 104
4. Chakrabarti. S. K.. 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 . 1984-01-09 . 2053-9169 . 10.1098/rspa.1984.0002 . 27–38. 2397528. 1984RSPSA.391...27C. 120673756.