Chandrasekhar–Page equations

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. Incidentally, while solving this problem, Chandrasekhar discovered a separable solution to the Dirac equation in flat space-time in oblate spheroidal coordinates for the first time.

By assuming a normal mode decomposition of the form ${\displaystyle e^{i(\sigma t+m\phi )}}$ (with ${\displaystyle m}$ being a half integer and with the convention ${\displaystyle \mathrm {Re} \{\sigma \}>0}$) for the time and the azimuthal component of the spherical polar coordinates ${\displaystyle (r,\theta ,\phi )}$, Chandrasekhar showed that the four bispinor components of the wave function,

${\displaystyle {\begin{bmatrix}F_{1}(r,\theta )\\F_{2}(r,\theta )\\G_{1}(r,\theta )\\G_{2}(r,\theta )\end{bmatrix}}e^{i(\sigma t+m\phi )}}$

can be expressed as product of radial and angular functions. The separation of variables is effected for the functions ${\displaystyle f_{1}=(r-ia\cos \theta )F_{1}}$, ${\displaystyle f_{2}=(r-ia\cos \theta )F_{2}}$, ${\displaystyle g_{1}=(r+ia\cos \theta )G_{1}}$ and ${\displaystyle g_{2}=(r+ia\cos \theta )G_{2}}$ (with ${\displaystyle a}$ being the angular momentum per unit mass of the black hole) as in

${\displaystyle f_{1}(r,\theta )=R_{-{\frac {1}{2}}}(r)S_{-{\frac {1}{2}}}(\theta ),\quad f_{2}(r,\theta )=R_{+{\frac {1}{2}}}(r)S_{+{\frac {1}{2}}}(\theta ),}$

${\displaystyle g_{1}(r,\theta )=R_{+{\frac {1}{2}}}(r)S_{-{\frac {1}{2}}}(\theta ),\quad g_{2}(r,\theta )=R_{-{\frac {1}{2}}}(r)S_{+{\frac {1}{2}}}(\theta ).}$

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

${\displaystyle {\begin{aligned}{\mathcal {L}}_{\frac {1}{2}}S_{+{\frac {1}{2}}}&=-(\lambda -a\mu \cos \theta )S_{-{\frac {1}{2}}},\\{\mathcal {L}}_{\frac {1}{2}}^{\dagger }S_{-{\frac {1}{2}}}&=+(\lambda +a\mu \cos \theta )S_{+{\frac {1}{2}}},\end{aligned}}}$

where ${\displaystyle \mu }$ is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength),

${\displaystyle {\mathcal {L}}_{n}={\frac {d}{{d}\theta }}+Q+n\cot \theta ,\quad {\mathcal {L}}_{n}^{\dagger }={\frac {d}{{d}\theta }}-Q+n\cot \theta }$

and ${\displaystyle Q=a\sigma \sin \theta +m\csc \theta }$. Eliminating ${\displaystyle S_{+1/2}(\theta )}$ between the foregoing two equations, one obtains

${\displaystyle \left({\mathcal {L}}_{\frac {1}{2}}{\mathcal {L}}_{\frac {1}{2}}^{\dagger }+{\frac {a\mu \sin \theta }{\lambda +a\mu \cos \theta }}{\mathcal {L}}_{\frac {1}{2}}^{\dagger }+\lambda ^{2}-a^{2}\mu ^{2}\cos ^{2}\theta \right)S_{-{\frac {1}{2}}}=0.}$

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

The corresponding radial equations are given by^[3]

${\displaystyle {\begin{aligned}\Delta ^{\frac {1}{2}}{\mathcal {D}}_{0}R_{-{\frac {1}{2}}}&=(\lambda +i\mu r)\Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}},\\\Delta ^{\frac {1}{2}}{\mathcal {D}}_{0}^{\dagger }R_{+{\frac {1}{2}}}&=(\lambda -i\mu r)R_{-{\frac {1}{2}}},\end{aligned}}}$

where ${\displaystyle \Delta =r^{2}-2Mr+a^{2},}$ ${\displaystyle M}$ is the black hole mass,

${\displaystyle {\mathcal {D}}_{n}={\frac {d}{{d}r}}+{\frac {iK}{\Delta }}+2n{\frac {r-M}{\Delta }},\quad {\mathcal {D}}_{n}^{\dagger }={\frac {d}{{d}r}}-{\frac {iK}{\Delta }}+2n{\frac {r-M}{\Delta }},}$

and ${\displaystyle K=(r^{2}+a^{2})\sigma +am.}$ Eliminating ${\displaystyle \Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}}}$ from the two equations, we obtain

${\displaystyle \left(\Delta {\mathcal {D}}_{\frac {1}{2}}^{\dagger }{\mathcal {D}}_{0}-{\frac {i\mu \Delta }{\lambda +i\mu r}}{\mathcal {D}}_{0}-\lambda ^{2}-\mu ^{2}r^{2}\right)R_{-{\frac {1}{2}}}=0.}$

The function ${\displaystyle \Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}}}$ satisfies the corresponding complex-conjugate equation.

The problem of solving the radial functions for a particular eigenvalue of ${\displaystyle \lambda }$ 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

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

where the Chandrasekhar–Page potentials ${\displaystyle V^{\pm }}$ are defined by^[3]

${\displaystyle V^{\pm }=W^{2}\pm {\frac {dW}{d{\hat {r}}_{*}}},\quad W={\frac {\Delta ^{\frac {1}{2}}(\lambda +\mu ^{2}r^{2})^{3/2}}{\varpi ^{2}(\lambda ^{2}+\mu ^{2}r^{2})+\lambda \mu \Delta /2\sigma }},}$

and ${\displaystyle {\hat {r}}_{*}=r_{*}+\tan ^{-1}(\mu r/\lambda )/2\sigma }$, ${\displaystyle r_{*}=r+2M\ln(r/2M-1)}$ is the tortoise coordinate and ${\displaystyle \varpi ^{2}=r^{2}+a^{2}+am/\sigma }$. The functions ${\displaystyle Z^{\pm }({\hat {r}}_{*})}$ are defined by ${\displaystyle Z^{\pm }=\psi ^{+}\pm \psi ^{-}}$, where

${\displaystyle \psi ^{+}=\Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}}\mathrm {exp} \left(+{\frac {i}{2}}\tan ^{-1}{\frac {\mu r}{\lambda }}\right),\quad \psi ^{-}=R_{-{\frac {1}{2}}}\mathrm {exp} \left(-{\frac {i}{2}}\tan ^{-1}{\frac {\mu r}{\lambda }}\right).}$

Unlike the Regge–Wheeler–Zerilli potentials, the Chandrasekhar–Page potentials do not vanish for ${\displaystyle r\to \infty }$, but has the behaviour

${\displaystyle V^{\pm }=\mu ^{2}\left(1-{\frac {2M}{r}}+\cdots \right).}$

As a result, the corresponding asymptotic behaviours for ${\displaystyle Z^{\pm }}$ as ${\displaystyle r\to \infty }$ becomes

${\displaystyle Z^{\pm }=\mathrm {exp} \left\{\pm i\left[(\sigma ^{2}-\mu ^{2})^{1/2}r+{\frac {M\mu ^{2}}{(\sigma ^{2}-\mu ^{2})^{1/2}}}\ln {\frac {r}{2M}}\right]\right\}.}$