The Kerr metric in (modified) Boyer-Lindquist coordinates (t ,r ,z=cos(Θ) , φ )
d
s
2
=
−
(
1
−
2
r
Σ
)
d
t
2
+
Σ
Δ
d
r
2
+
Σ
1
−
z
2
d
z
2
+
1
−
z
2
Σ
(
2
a
2
r
(
1
−
z
2
)
+
(
a
2
+
r
2
)
Σ
)
d
ϕ
2
−
4
a
r
(
1
−
z
2
)
Σ
d
t
d
ϕ
{\displaystyle ds^{2}=-(1-{\frac {2r}{\Sigma }})dt^{2}+{\frac {\Sigma }{\Delta }}dr^{2}+{\frac {\Sigma }{1-z^{2}}}dz^{2}+{\frac {1-z^{2}}{\Sigma }}(2a^{2}r(1-z^{2})+(a^{2}+r^{2})\Sigma )d\phi ^{2}-{\frac {4ar(1-z^{2})}{\Sigma }}dtd\phi }
with
Δ
=
r
(
r
−
2
)
+
a
2
{\displaystyle \Delta =r(r-2)+a^{2}}
Σ
=
r
2
+
a
2
z
2
{\displaystyle \Sigma =r^{2}+a^{2}z^{2}}
Teukolsky master equation:
{
(
(
r
2
+
a
2
)
2
Δ
−
a
2
(
1
−
z
2
)
)
∂
2
∂
t
2
+
4
a
r
Δ
∂
2
∂
t
∂
ϕ
+
(
a
2
Δ
−
1
1
−
z
2
)
∂
2
∂
ϕ
2
−
Δ
−
s
∂
∂
r
(
Δ
s
+
1
∂
∂
r
)
−
∂
∂
z
(
(
1
−
z
2
)
∂
∂
z
)
−
2
s
(
a
(
r
−
1
)
Δ
+
i
z
1
−
z
2
)
∂
∂
ϕ
−
2
s
(
r
2
−
a
2
Δ
−
r
−
i
a
z
)
∂
∂
t
+
s
(
s
+
1
)
z
2
−
1
1
−
z
2
}
Ψ
s
=
4
π
Σ
T
s
{\displaystyle {\begin{aligned}{\Bigg \{}&\left({\frac {(r^{2}+a^{2})^{2}}{\Delta }}-a^{2}(1-z^{2})\right){\frac {\partial ^{2}}{\partial t^{2}}}+{\frac {4ar}{\Delta }}{\frac {\partial ^{2}}{\partial t\partial \phi }}+\left({\frac {a^{2}}{\Delta }}-{\frac {1}{1-z^{2}}}\right){\frac {\partial ^{2}}{\partial \phi ^{2}}}-\Delta ^{-s}{\frac {\partial }{\partial r}}\left(\Delta ^{s+1}{\frac {\partial }{\partial r}}\right)-{\frac {\partial }{\partial z}}\left((1-z^{2}){\frac {\partial }{\partial z}}\right)\\&-2s\left({\frac {a(r-1)}{\Delta }}+{\frac {iz}{1-z^{2}}}\right){\frac {\partial }{\partial \phi }}-2s\left({\frac {r^{2}-a^{2}}{\Delta }}-r-iaz\right){\frac {\partial }{\partial t}}+s{\frac {(s+1)z^{2}-1}{1-z^{2}}}{\Bigg \}}\Psi _{s}=4\pi \Sigma T_{s}\end{aligned}}}
Radial equation
{
Δ
−
s
d
d
r
(
Δ
s
+
1
d
d
r
)
+
K
2
−
2
i
s
(
r
−
1
)
K
Δ
+
4
i
s
ω
r
−
λ
}
s
R
l
m
ω
(
r
)
=
s
T
l
m
ω
(
r
)
{\displaystyle \left\{\Delta ^{-s}{\frac {d}{dr}}\left(\Delta ^{s+1}{\frac {d}{dr}}\right)+{\frac {K^{2}-2is(r-1)K}{\Delta }}+4is\omega r-\lambda \right\}{_{s}R_{lm\omega }}(r)={_{s}T_{lm\omega }}(r)}
