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 ${\displaystyle e^{i(m\phi -\omega t)}}$ (with ${\displaystyle m}$ being the azimuthal component of the particle angular momentum and takes half integer values and with ${\displaystyle \omega }$ being the frequency) 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,

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

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

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

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

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

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

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

${\displaystyle {\mathcal {L}}_{n}={\frac {d}{{d}\theta }}+(m\csc \theta -a\omega \sin \theta )+n\cot \theta ,}$

${\displaystyle {\mathcal {L}}_{n}^{\dagger }={\frac {d}{{d}\theta }}-(m\csc \theta -a\omega \sin \theta )+n\cot \theta .}$

Eliminating ${\displaystyle S_{+}(\theta )}$ between the two equations, one obtains

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

The function ${\displaystyle S_{+}}$ 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 ${\displaystyle S_{-}}$(and ${\displaystyle S_{+}}$) 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.

The eigenvalue problem depends on two continuous parameters, namely ${\displaystyle a\omega }$ and ${\displaystyle a\mu }$. For a given ${\displaystyle a\omega }$ and ${\displaystyle a\mu }$, the eigenstates are characterised by three discrete numbers: the particle angular momentum, ${\displaystyle j=1/2,3/2,\dots }$, its azimuthal component ${\displaystyle m=-j,-j+1,\dots ,j-1,j}$ and the parity ${\displaystyle {\mathcal {P}}=\pm 1.}$ The spectrum elements may then be explicitly labelled as^[4]

${\displaystyle S_{\pm }={}_{s=\pm {\frac {1}{2}}}S_{j,m,{\mathcal {P}}}^{(a\omega ,a\mu )},\lambda =\lambda _{j,m,{\mathcal {P}}}^{(a\omega ,a\mu )}.}$

The eigenvalue ${\displaystyle \lambda }$ has the physical interpretation of being the square root of the generalised total anagular momentum squared.^[5] The knowledge of the spectrum in the positive quadrant ${\displaystyle a\omega >0,a\mu >0}$ is sufficient to determine the full spectrum, as implied by the symmetry:

${\displaystyle \lambda _{j,m,{\mathcal {P}}}^{(a\omega ,a\mu )}=-\lambda _{j,-m,-{\mathcal {P}}}^{(-a\omega ,a\mu )}=-\lambda _{j,m,-{\mathcal {P}}}^{(a\omega ,-a\mu )}=\lambda _{j,-m,{\mathcal {P}}}^{(-a\omega ,-a\mu )}.}$

Furthermore

${\displaystyle {}_{s}S_{j,m,{\mathcal {P}}}^{(a\omega ,a\mu )}(\theta )=(-1)^{s-1/2}{}_{s}S_{j,m,-{\mathcal {P}}}^{(a\omega ,-a\mu )}(\Theta )={\mathcal {P}}(-1)^{m-1/2}{}_{-s}S_{j,-m,-{\mathcal {P}}}^{(-a\omega ,a\mu )}(\theta )={\mathcal {P}}(-1)^{j+m}{}_{-s}S_{j,m,{\mathcal {P}}}^{(a\omega ,a\mu )}(\pi -\theta )}$

and the combinations thereof.

Non-rotating black hole (Schwarzschild black hole) ${\displaystyle a=0}$: The problem can be solved explicitly. The eigenvalues and eigenfunctions are given by

${\displaystyle \lambda _{j,m,{\mathcal {P}}}^{(0,0)}={\mathcal {P}}(j+1/2),}$

${\displaystyle {\begin{bmatrix}{}_{+1/2}S_{j,m,{\mathcal {P}}}^{(0,0)}(\theta )\\{}_{-1/2}S_{j,m,{\mathcal {P}}}^{(0,0)}(\theta )\end{bmatrix}}=A{\begin{bmatrix}\cos {\frac {\theta }{2}}&\sin {\frac {\theta }{2}}\\-\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{bmatrix}}{\begin{bmatrix}P_{j+{\mathcal {P}}/2}^{m+1/2}(\cos \theta )\\c_{j,m,{\mathcal {P}}}^{(0,0)}P_{j+{\mathcal {P}}/2}^{m-1/2}(\cos \theta )\end{bmatrix}}}$

where ${\displaystyle P}$ is the associated Legendre polynomials and

${\displaystyle c_{j,m,{\mathcal {P}}}^{(0,0)}={\mathcal {P}}(j+{\mathcal {P}}/2+1/2)-m,\quad A={\sqrt {\frac {(j-m)!}{2\pi (j+m)!}}}.}$

Special case ${\displaystyle \omega =\pm \mu }$: For the special case where ${\displaystyle \omega =+\mu }$, the solutions are given by^[6]

${\displaystyle \lambda _{j,m,{\mathcal {P}}}^{(a\omega ,a\omega )}=-{\frac {1}{2}}+{\mathcal {P}}{\sqrt {(j+{\mathcal {P}}/2+1/2)^{2}-2ma\omega +a^{2}\omega ^{2}}},}$

${\displaystyle {\begin{bmatrix}{}_{+1/2}S_{j,m,{\mathcal {P}}}^{(0,0)}(\theta )\\{}_{-1/2}S_{j,m,{\mathcal {P}}}^{(0,0)}(\theta )\end{bmatrix}}=A{\begin{bmatrix}\cos {\frac {\theta }{2}}&\sin {\frac {\theta }{2}}\\-\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{bmatrix}}{\begin{bmatrix}P_{j+{\mathcal {P}}/2}^{m+1/2}(\cos \theta )\\c_{j,m,{\mathcal {P}}}^{(a\omega ,a\omega )}P_{j+{\mathcal {P}}/2}^{m-1/2}(\cos \theta )\end{bmatrix}}}$

where

${\displaystyle c_{j,m,{\mathcal {P}}}^{(a\omega ,a\omega )}=[(j+{\mathcal {P}}/2+1/2)^{2}-m^{2}]/(\lambda _{j,m,{\mathcal {P}}}^{(a\omega ,a\omega )}+m+1/2-a\omega ),\quad A={\sqrt {{\frac {(j+{\mathcal {P}}/2+1/2)}{2\pi }}{\frac {(j+{\mathcal {P}}/2-m-1/2)!}{(j+{\mathcal {P}}/2+m+1/2)!}}\left(1+{\frac {{\mathcal {P}}(m-a\omega )}{\sqrt {(j+{\mathcal {P}}/2+1/2)^{2}-2ma\omega +a^{2}\omega ^{2}}}}\right)}}.}$

When ${\displaystyle \omega =-\mu }$, we have

${\displaystyle \lambda _{j,m,{\mathcal {P}}}^{(a\omega ,-a\omega )}={\frac {1}{2}}+{\mathcal {P}}{\sqrt {(j+{\mathcal {P}}/2+1/2)^{2}-2ma\omega +a^{2}\omega ^{2}}},}$

${\displaystyle {\begin{bmatrix}{}_{+1/2}S_{j,m,{\mathcal {P}}}^{(a\omega ,-a\omega )}(\theta )\\{}_{-1/2}S_{j,m,{\mathcal {P}}}^{(0,0)}(\theta )\end{bmatrix}}=A{\begin{bmatrix}\cos {\frac {\theta }{2}}&-\sin {\frac {\theta }{2}}\\\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{bmatrix}}{\begin{bmatrix}P_{j+{\mathcal {P}}/2}^{m+1/2}(\cos \theta )\\c_{j,m,{\mathcal {P}}}^{(a\omega ,-a\omega )}P_{j+{\mathcal {P}}/2}^{m-1/2}(\cos \theta )\end{bmatrix}}}$

where

${\displaystyle c_{j,m,{\mathcal {P}}}^{(a\omega ,-a\omega )}=-[(j-{\mathcal {P}}/2+1/2)^{2}-m^{2}]/(-\lambda _{j,m,{\mathcal {P}}}^{(a\omega ,-a\omega )}+m+1/2-a\omega ),\quad A={\sqrt {{\frac {(j-{\mathcal {P}}/2+1/2)}{2\pi }}{\frac {(j-{\mathcal {P}}/2-m-1/2)!}{(j-{\mathcal {P}}/2+m+1/2)!}}\left(1-{\frac {{\mathcal {P}}(m-a\omega )}{\sqrt {(j-{\mathcal {P}}/2+1/2)^{2}-2ma\omega +a^{2}\omega ^{2}}}}\right)}}.}$

For convenience, let us write ${\displaystyle \sigma =-\omega .}$ The 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

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