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
(
σ
t
+
m
ϕ
)
{\displaystyle e^{i(\sigma t+m\phi )}}
(with
m
{\displaystyle m}
being a half integer and with the convention
R
e
{
σ
}
>
0
{\displaystyle \mathrm {Re} \{\sigma \}>0}
) for the time and the azimuthal component of the spherical polar coordinates
(
r
,
θ
,
ϕ
)
{\displaystyle (r,\theta ,\phi )}
, Chandrasekhar showed that the four bispinor components of the wave function,
[
F
1
(
r
,
θ
)
F
2
(
r
,
θ
)
G
1
(
r
,
θ
)
G
2
(
r
,
θ
)
]
e
i
(
σ
t
+
m
ϕ
)
{\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
f
1
=
(
r
−
i
a
cos
θ
)
F
1
{\displaystyle f_{1}=(r-ia\cos \theta )F_{1}}
,
f
2
=
(
r
−
i
a
cos
θ
)
F
2
{\displaystyle f_{2}=(r-ia\cos \theta )F_{2}}
,
g
1
=
(
r
+
i
a
cos
θ
)
G
1
{\displaystyle g_{1}=(r+ia\cos \theta )G_{1}}
and
g
2
=
(
r
+
i
a
cos
θ
)
G
2
{\displaystyle g_{2}=(r+ia\cos \theta )G_{2}}
(with
a
{\displaystyle a}
being the angular momentum per unit mass of the black hole) as in
f
1
(
r
,
θ
)
=
R
−
1
2
(
r
)
S
−
1
2
(
θ
)
,
f
2
(
r
,
θ
)
=
R
+
1
2
(
r
)
S
+
1
2
(
θ
)
,
{\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 ),}
g
1
(
r
,
θ
)
=
R
+
1
2
(
r
)
S
−
1
2
(
θ
)
,
g
2
(
r
,
θ
)
=
R
−
1
2
(
r
)
S
+
1
2
(
θ
)
.
{\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 ).}
Chandrasekhar–Page angular equations
edit
The angular functions satisfy the coupled eigenvalue equations,[ 3]
L
1
2
S
+
1
2
=
−
(
λ
−
a
μ
cos
θ
)
S
−
1
2
,
L
1
2
†
S
−
1
2
=
+
(
λ
+
a
μ
cos
θ
)
S
+
1
2
,
{\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
μ
{\displaystyle \mu }
is the particle's rest mass (measured in units so that it is the inverse of the Compton wavelength ),
L
n
=
d
d
θ
+
Q
+
n
cot
θ
,
L
n
†
=
d
d
θ
−
Q
+
n
cot
θ
{\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
Q
=
a
σ
sin
θ
+
m
csc
θ
{\displaystyle Q=a\sigma \sin \theta +m\csc \theta }
. Eliminating
S
+
1
/
2
(
θ
)
{\displaystyle S_{+1/2}(\theta )}
between the foregoing two equations, one obtains
(
L
1
2
L
1
2
†
+
a
μ
sin
θ
λ
+
a
μ
cos
θ
L
1
2
†
+
λ
2
−
a
2
μ
2
cos
2
θ
)
S
−
1
2
=
0.
{\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
S
+
1
2
{\displaystyle S_{+{\frac {1}{2}}}}
satisfies the adjoint equation, that can be obtained from the above equation by replacing
θ
{\displaystyle \theta }
with
π
−
θ
{\displaystyle \pi -\theta }
. The boundary conditions for these second-order differential equations are that
S
−
1
2
{\displaystyle S_{-{\frac {1}{2}}}}
(and
S
+
1
2
{\displaystyle S_{+{\frac {1}{2}}}}
) be regular at
θ
=
0
{\displaystyle \theta =0}
and
θ
=
π
{\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
σ
=
μ
{\displaystyle \sigma =\mu }
.[ 4]
Chandrasekhar–Page radial equations
edit
The corresponding radial equations are given by[ 3]
Δ
1
2
D
0
R
−
1
2
=
(
λ
+
i
μ
r
)
Δ
1
2
R
+
1
2
,
Δ
1
2
D
0
†
R
+
1
2
=
(
λ
−
i
μ
r
)
R
−
1
2
,
{\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
Δ
=
r
2
−
2
M
r
+
a
2
,
{\displaystyle \Delta =r^{2}-2Mr+a^{2},}
M
{\displaystyle M}
is the black hole mass,
D
n
=
d
d
r
+
i
K
Δ
+
2
n
r
−
M
Δ
,
D
n
†
=
d
d
r
−
i
K
Δ
+
2
n
r
−
M
Δ
,
{\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
K
=
(
r
2
+
a
2
)
σ
+
a
m
.
{\displaystyle K=(r^{2}+a^{2})\sigma +am.}
Eliminating
Δ
1
2
R
+
1
2
{\displaystyle \Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}}}
from the two equations, we obtain
(
Δ
D
1
2
†
D
0
−
i
μ
Δ
λ
+
i
μ
r
D
0
−
λ
2
−
μ
2
r
2
)
R
−
1
2
=
0.
{\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
Δ
1
2
R
+
1
2
{\displaystyle \Delta ^{\frac {1}{2}}R_{+{\frac {1}{2}}}}
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
λ
{\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
(
d
2
d
r
^
∗
2
+
σ
2
)
Z
±
=
V
±
Z
±
,
{\displaystyle \left({\frac {d^{2}}{d{\hat {r}}_{*}^{2}}}+\sigma ^{2}\right)Z^{\pm }=V^{\pm }Z^{\pm },}
where the Chandrasekhar–Page potentials
V
±
{\displaystyle V^{\pm }}
are defined by[ 3]
V
±
=
W
2
±
d
W
d
r
^
∗
,
W
=
Δ
1
2
(
λ
+
μ
2
r
2
)
3
/
2
ϖ
2
(
λ
2
+
μ
2
r
2
)
+
λ
μ
Δ
/
2
σ
,
{\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
r
^
∗
=
r
∗
+
tan
−
1
(
μ
r
/
λ
)
/
2
σ
{\displaystyle {\hat {r}}_{*}=r_{*}+\tan ^{-1}(\mu r/\lambda )/2\sigma }
,
r
∗
=
r
+
2
M
ln
(
r
/
2
M
−
1
)
{\displaystyle r_{*}=r+2M\ln(r/2M-1)}
is the tortoise coordinate and
ϖ
2
=
r
2
+
a
2
+
a
m
/
σ
{\displaystyle \varpi ^{2}=r^{2}+a^{2}+am/\sigma }
. The functions
Z
±
(
r
^
∗
)
{\displaystyle Z^{\pm }({\hat {r}}_{*})}
are defined by
Z
±
=
ψ
+
±
ψ
−
{\displaystyle Z^{\pm }=\psi ^{+}\pm \psi ^{-}}
, where
ψ
+
=
Δ
1
2
R
+
1
2
e
x
p
(
+
i
2
tan
−
1
μ
r
λ
)
,
ψ
−
=
R
−
1
2
e
x
p
(
−
i
2
tan
−
1
μ
r
λ
)
.
{\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
r
→
∞
{\displaystyle r\to \infty }
, but has the behaviour
V
±
=
μ
2
(
1
−
2
M
r
+
⋯
)
.
{\displaystyle V^{\pm }=\mu ^{2}\left(1-{\frac {2M}{r}}+\cdots \right).}
As a result, the corresponding asymptotic behaviours for
Z
±
{\displaystyle Z^{\pm }}
as
r
→
∞
{\displaystyle r\to \infty }
becomes
Z
±
=
e
x
p
{
±
i
[
(
σ
2
−
μ
2
)
1
/
2
r
+
M
μ
2
(
σ
2
−
μ
2
)
1
/
2
ln
r
2
M
]
}
.
{\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\}.}