A massive fermion wave equation in Kerr spacetime
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,
![{\displaystyle {\begin{bmatrix}F_{1}(r,\theta )\\F_{2}(r,\theta )\\G_{1}(r,\theta )\\G_{2}(r,\theta )\end{bmatrix}}e^{i(\sigma t+m\phi )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f65f8c7e71f09880af3e5e503f47fb984652a90)
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
![{\displaystyle f_{1}(r,\theta )=R_{-{\frac {1}{2}}}(r)S_{-{\frac {1}{2}}}(\theta ),\quad f_{2}(r,\theta )=R_{+{\frac {1}{2}}}(r)S_{+{\frac {1}{2}}}(\theta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21047affac874db923b1a1203d729266f5430806)
![{\displaystyle g_{1}(r,\theta )=R_{+{\frac {1}{2}}}(r)S_{-{\frac {1}{2}}}(\theta ),\quad g_{2}(r,\theta )=R_{-{\frac {1}{2}}}(r)S_{+{\frac {1}{2}}}(\theta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4d0556397bdbccb733dba1289ade4df915e787)
Chandrasekhar–Page angular equations[edit]
The angular functions satisfy the coupled eigenvalue equations,[3]
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6773d618b3ac5c0d381ac7926b1a67a1cecd3d9)
where
is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength),
![{\displaystyle {\mathcal {L}}_{n}={\frac {d}{{d}\theta }}+Q+n\cot \theta ,\quad {\mathcal {L}}_{n}^{\dagger }={\frac {d}{{d}\theta }}-Q+n\cot \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d3043f45ac3d6e00136b4e7cbe57bf3e9d17ed)
and
. Eliminating
between the foregoing two equations, one obtains
![{\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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33a196826058fbd265f6fcc705dd6ec2cf0b8b83)
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[edit]
The corresponding radial equations are given by[3]
![{\displaystyle {\begin{aligned}\Delta ^{\frac {1}{2}}{\mathcal {D}}_{0}R_{-{\frac {1}{2}}}&=(\lambda +i\mu r)\Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}},\\\Delta ^{\frac {1}{2}}{\mathcal {D}}_{0}^{\dagger }R_{+{\frac {1}{2}}}&=(\lambda -i\mu r)R_{-{\frac {1}{2}}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6daae95240f88359270dc5e8f4d564737d82093)
where
is the black hole mass,
![{\displaystyle {\mathcal {D}}_{n}={\frac {d}{{d}r}}+{\frac {iK}{\Delta }}+2n{\frac {r-M}{\Delta }},\quad {\mathcal {D}}_{n}^{\dagger }={\frac {d}{{d}r}}-{\frac {iK}{\Delta }}+2n{\frac {r-M}{\Delta }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3004d7937ba20281ee7f61a6e370d52967aa67e9)
and
Eliminating
from the two equations, we obtain
![{\displaystyle \left(\Delta {\mathcal {D}}_{\frac {1}{2}}^{\dagger }{\mathcal {D}}_{0}-{\frac {i\mu \Delta }{\lambda +i\mu r}}{\mathcal {D}}_{0}-\lambda ^{2}-\mu ^{2}r^{2}\right)R_{-{\frac {1}{2}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4520e4f716c60ddf99e57c3c64cec57299a6fb6)
The function
satisfies the corresponding complex-conjugate equation.
Reduction to one-dimensional scattering problem[edit]
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
![{\displaystyle \left({\frac {d^{2}}{d{\hat {r}}_{*}^{2}}}+\sigma ^{2}\right)Z^{\pm }=V^{\pm }Z^{\pm },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1180bd34f978fe0b2be07359ef0def20adfe5029)
where the Chandrasekhar–Page potentials
are defined by[3]
![{\displaystyle V^{\pm }=W^{2}\pm {\frac {dW}{d{\hat {r}}_{*}}},\quad W={\frac {\Delta ^{\frac {1}{2}}(\lambda +\mu ^{2}r^{2})^{3/2}}{\varpi ^{2}(\lambda ^{2}+\mu ^{2}r^{2})+\lambda \mu \Delta /2\sigma }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ff229e530eacac9dfbcfba6c7f64b4e6e4e6af)
and
,
is the tortoise coordinate and
. The functions
are defined by
, where
![{\displaystyle \psi ^{+}=\Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}}\mathrm {exp} \left(+{\frac {i}{2}}\tan ^{-1}{\frac {\mu r}{\lambda }}\right),\quad \psi ^{-}=R_{-{\frac {1}{2}}}\mathrm {exp} \left(-{\frac {i}{2}}\tan ^{-1}{\frac {\mu r}{\lambda }}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1b73a06fab2aec16c730d1022d7f81352773a3)
Unlike the Regge–Wheeler–Zerilli potentials, the Chandrasekhar–Page potentials do not vanish for
, but has the behaviour
![{\displaystyle V^{\pm }=\mu ^{2}\left(1-{\frac {2M}{r}}+\cdots \right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06a7b5a5ecae37130a5722aaed7a15e8881e6323)
As a result, the corresponding asymptotic behaviours for
as
becomes
![{\displaystyle Z^{\pm }=\mathrm {exp} \left\{\pm i\left[(\sigma ^{2}-\mu ^{2})^{1/2}r+{\frac {M\mu ^{2}}{(\sigma ^{2}-\mu ^{2})^{1/2}}}\ln {\frac {r}{2M}}\right]\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb00acb5b14bbad412577c4a97cccfd7e11ff2bf)
References[edit]