Miêu tả
Equations
Line-element in Boyer-Lindquist-coordinates:
d
τ
2
=
(
1
−
2
r
−
℧
2
Σ
)
d
t
2
−
Σ
Δ
d
r
2
−
Σ
d
θ
2
−
χ
Σ
sin
2
θ
d
ϕ
2
+
2
Λ
Σ
d
t
d
ϕ
{\displaystyle {\rm {d\tau ^{2}\ =\ \left(1-{\frac {2r-\mho ^{2}}{\Sigma }}\right)\mathrm {d} t^{2}\ -\ {\frac {\Sigma }{\Delta }}\ \mathrm {d} r^{2}\ -\ \Sigma \ d\theta ^{2}\ -\ {\frac {\chi }{\Sigma }}\ \sin ^{2}\theta \ d\phi ^{2}\ +\ 2\ {\frac {\Lambda }{\Sigma }}\ dt\ d\phi }}}
Shorthand terms:
Δ
=
r
2
−
2
r
+
a
2
+
℧
2
,
Σ
=
r
2
+
a
2
cos
2
θ
,
χ
=
(
a
2
+
r
2
)
2
−
a
2
sin
2
θ
Δ
,
Λ
=
a
(
2
r
−
℧
2
)
sin
2
θ
{\displaystyle {\rm {\Delta =r^{2}-2r+a^{2}+\mho ^{2}\ ,\ \Sigma =r^{2}+a^{2}\ \cos ^{2}\theta \ ,\ \chi =(a^{2}+r^{2})^{2}-a^{2}\ \sin ^{2}\theta \ \Delta \ ,\ \ \Lambda =a\ (2r-\mho ^{2})\ \sin ^{2}\theta }}}
with the dimensionless spin parameter a=Jc/G/M² and the dimensionless electric charge parameter ℧=Q ₑ/M·√(K/G). Here G=M=c=K=1 so that a=J und ℧=Q ₑ, with lengths in GM/c² and times in GM/c³.
Co- and contravariant metric:
g
μ
ν
=
(
1
−
2
r
−
℧
2
Σ
0
0
Λ
Σ
0
−
Σ
Δ
0
0
0
0
−
Σ
0
Λ
Σ
0
0
−
χ
sin
2
θ
Σ
)
→
g
μ
ν
=
(
χ
Δ
Σ
0
0
−
a
(
℧
2
−
2
r
)
Σ
(
℧
2
−
2
r
+
Σ
)
χ
−
a
Λ
0
−
Δ
Σ
0
0
0
0
−
1
Σ
0
−
a
(
℧
2
−
2
r
)
Σ
(
℧
2
−
2
r
+
Σ
)
χ
−
a
Λ
0
0
−
Δ
−
a
2
sin
2
θ
Δ
Σ
sin
2
θ
)
{\displaystyle {g_{\mu \nu }={\rm {\left({\begin{array}{cccc}{\rm {1-{\frac {2r-\mho ^{2}}{\Sigma }}}}&0&0&{\frac {\Lambda }{\Sigma }}\\0&{\rm {-{\frac {\Sigma }{\Delta }}}}&0&0\\0&0&{\rm {-\Sigma }}&0\\{\frac {\Lambda }{\Sigma }}&0&0&-{\frac {\chi \sin ^{2}\theta }{\Sigma }}\ \end{array}}\right)}}\ \to \ g^{\mu \nu }={\rm {\left({\begin{array}{cccc}{\rm {\frac {\chi }{\Delta \Sigma }}}&0&0&{\rm {-{\frac {a\left({\rm {\mho ^{2}-2r}}\right)\Sigma }{\rm {\left({\rm {\mho ^{2}-2r+\Sigma }}\right)\chi -a\Lambda }}}}}\\0&{\rm {-{\frac {\Delta }{\Sigma }}}}&0&0\\0&0&{\rm {-{\frac {1}{\Sigma }}}}&0\\{\rm {-{\frac {a\left({\rm {\mho ^{2}-2r}}\right)\Sigma }{\rm {\left({\rm {\mho ^{2}-2r+\Sigma }}\right)\chi -a\Lambda }}}}}&0&0&{\rm {-{\frac {\Delta -a^{2}\sin ^{2}\theta }{\Delta \Sigma \sin ^{2}\theta }}}}\\\end{array}}\right)}}}}
Contravariant Maxwell tensor:
F
μ
ν
=
(
0
−
4
(
a
2
+
r
2
)
℧
(
cos
(
2
θ
)
a
2
+
a
2
−
2
r
2
)
(
cos
(
2
θ
)
a
2
+
a
2
+
2
r
2
)
3
−
8
a
2
r
℧
sin
(
2
θ
)
(
cos
(
2
θ
)
a
2
+
a
2
+
2
r
2
)
3
0
4
(
a
2
+
r
2
)
℧
(
cos
(
2
θ
)
a
2
+
a
2
−
2
r
2
)
(
cos
(
2
θ
)
a
2
+
a
2
+
2
r
2
)
3
0
0
a
℧
(
a
2
cos
2
θ
−
r
2
)
(
r
2
+
a
2
cos
2
θ
)
3
8
a
2
r
℧
sin
(
2
θ
)
(
cos
(
2
θ
)
a
2
+
a
2
+
2
r
2
)
3
0
0
16
a
r
℧
cot
θ
(
cos
(
2
θ
)
a
2
+
a
2
+
2
r
2
)
3
0
a
℧
(
r
2
−
a
2
cos
2
θ
)
(
r
2
+
a
2
cos
2
θ
)
3
−
16
a
r
℧
cot
θ
(
cos
(
2
θ
)
a
2
+
a
2
+
2
r
2
)
3
0
)
{\displaystyle {\rm {F}}^{\mu \nu }=\left({\begin{array}{cccc}0&-{\frac {\rm {4(a^{2}+r^{2})\ \mho \ (\cos(2\theta )\ a^{2}+a^{2}-2r^{2})}}{\rm {(\cos(2\theta )\ a^{2}+a^{2}+2r^{2})^{3}}}}&-{\frac {\rm {8a^{2}r\ \mho \sin(2\theta )}}{\rm {(\cos(2\theta )\ a^{2}+a^{2}+2r^{2})^{3}}}}&0\\{\frac {\rm {4(a^{2}+r^{2})\ \mho \ (\cos(2\theta )\ a^{2}+a^{2}-2r^{2})}}{\rm {(\cos(2\theta )\ a^{2}+a^{2}+2r^{2})^{3}}}}&0&0&{\frac {a\ \mho \ (a^{2}\cos ^{2}\theta -r^{2})}{(r^{2}+a^{2}\cos ^{2}\theta )^{3}}}\\{\frac {\rm {8a^{2}r\ \mho \sin(2\theta )}}{\rm {(\cos(2\theta )\ a^{2}+a^{2}+2r^{2})^{3}}}}&0&0&{\frac {\rm {16a\ r\ \mho \cot \theta }}{\rm {(\cos(2\theta )\ a^{2}+a^{2}+2r^{2})^{3}}}}\\0&{\frac {\rm {a\ \mho \ (r^{2}-a^{2}\cos ^{2}\theta )}}{\rm {(r^{2}+a^{2}\cos ^{2}\theta )^{3}}}}&-{\frac {\rm {16a\ r\ \mho \cot \theta }}{\rm {(\cos(2\theta )\ a^{2}+a^{2}+2r^{2})^{3}}}}&0\\\end{array}}\right)}
The coordinate acceleration of a test-particle with the specific charge q is given by
x
¨
i
=
−
∑
j
=
1
4
∑
k
=
1
4
x
˙
j
x
˙
k
Γ
j
k
i
+
q
F
i
k
x
˙
j
g
j
k
{\displaystyle {\rm {{\ddot {x}}^{i}=-\sum _{j=1}^{4}\sum _{k=1}^{4}{\dot {x}}^{j}\ {\dot {x}}^{k}\ \Gamma _{jk}^{i}+q\ {F^{ik}}\ {{\dot {x}}^{j}}}}\ {g_{\rm {jk}}}}
with the Christoffel-symbols
Γ
j
k
i
=
∑
s
=
1
4
g
i
s
2
(
∂
g
s
j
∂
x
k
+
∂
g
s
k
∂
x
j
−
∂
g
j
k
∂
x
s
)
{\displaystyle \Gamma _{\rm {jk}}^{\rm {i}}=\sum _{\rm {s=1}}^{4}{\frac {g^{\rm {is}}}{2}}\left({\frac {\partial {g}_{\rm {sj}}}{\partial {\rm {x^{k}}}}}+{\frac {\partial {g}_{\rm {sk}}}{\partial {\rm {x^{j}}}}}-{\frac {\partial {g}_{\rm {jk}}}{\partial {\rm {x^{s}}}}}\right)}
So the second proper time derivatives are
t
¨
=
−
(
a
2
θ
˙
(
sin
(
2
θ
)
(
q
℧
r
+
(
℧
2
−
2
r
)
t
˙
)
−
2
a
sin
3
θ
cos
θ
(
℧
2
−
2
r
)
ϕ
˙
)
+
{\displaystyle {\rm {{\ddot {t}}=-(a^{2}\ {\dot {\theta }}\ (\sin(2\theta )(q\ \mho \ r+(\mho ^{2}-2r)\ {\dot {t}})-2a\sin ^{3}\theta \cos \theta \ (\mho ^{2}-2r)\ {\dot {\phi }})+}}}
(
r
˙
(
(
a
2
+
r
2
)
(
a
2
cos
2
θ
(
q
℧
−
2
t
˙
)
+
r
(
2
(
r
−
℧
2
)
t
˙
−
q
℧
r
)
)
+
a
sin
2
θ
ϕ
˙
(
2
a
4
cos
2
θ
+
{\displaystyle {\rm {({\dot {r}}\ ((a^{2}+r^{2})(a^{2}\cos ^{2}\theta \ (q\mho -2{\dot {t}})+r(2\ (r-\mho ^{2}){\dot {t}}-q\ \mho \ r))+a\sin ^{2}\theta \ {\dot {\phi }}\ (2a^{4}\cos ^{2}\theta +}}}
a
2
℧
2
r
(
cos
(
2
θ
)
+
3
)
−
a
2
r
2
(
cos
(
2
θ
)
+
3
)
+
4
℧
2
r
3
−
6
r
4
)
)
)
/
(
a
2
+
(
r
−
2
)
r
+
℧
2
)
)
/
(
(
a
2
cos
2
θ
+
r
2
)
2
)
{\displaystyle {\rm {a^{2}\mho ^{2}r\ (\cos(2\theta )+3)-a^{2}r^{2}(\cos(2\theta )+3)+4\mho ^{2}r^{3}-6r^{4})))/(a^{2}+(r-2)r+\mho ^{2}))/((a^{2}\cos ^{2}\theta +r^{2})^{2})}}}
for the time component,
r
¨
=
(
a
2
θ
˙
sin
(
2
θ
)
r
˙
)
/
(
a
2
cos
2
θ
+
r
2
)
+
r
˙
2
(
(
r
−
1
)
/
(
a
2
+
(
r
−
2
)
r
+
℧
2
)
−
r
/
(
a
2
cos
2
θ
+
r
2
)
)
+
{\displaystyle {\rm {{\ddot {r}}=(a^{2}{\dot {\theta }}\sin(2\theta )\ {\dot {r}})/(a^{2}\cos ^{2}\theta +r^{2})+{\dot {r}}^{2}((r-1)/(a^{2}+(r-2)\ r+\mho ^{2})-r/(a^{2}\cos ^{2}\theta +r^{2}))+}}}
(
(
a
2
+
(
r
−
2
)
r
+
℧
2
)
(
8
a
sin
2
θ
ϕ
˙
(
a
2
cos
2
θ
(
q
℧
−
2
t
˙
)
+
r
(
2
(
r
−
℧
2
)
t
˙
−
q
℧
r
)
)
+
{\displaystyle {\rm {((a^{2}+(r-2)\ r+\mho ^{2})(8a\sin ^{2}\theta \ {\dot {\phi }}\ (a^{2}\cos ^{2}\theta \ (q\ \mho -2{\dot {t}})+r(2(r-\mho ^{2}){\dot {t}}-q\ \mho \ r))+}}}
8
t
˙
(
a
2
cos
2
θ
(
t
˙
−
q
℧
)
+
r
(
q
℧
r
+
(
℧
2
−
r
)
t
˙
)
)
+
8
r
θ
˙
2
(
a
2
cos
2
θ
+
r
2
)
2
+
{\displaystyle {\rm {8{\dot {t}}\ (a^{2}\cos ^{2}\theta \ ({\dot {t}}-q\ \mho )+r\ (q\ \mho \ r+(\mho ^{2}-r)\ {\dot {t}}))+8r\ {\dot {\theta }}^{2}\ (a^{2}\cos ^{2}\theta +r^{2})^{2}+}}}
sin
2
θ
ϕ
˙
2
(
2
a
4
sin
2
(
2
θ
)
+
r
(
a
2
(
a
2
cos
(
4
θ
)
+
3
a
2
+
4
(
a
−
℧
)
(
a
+
℧
)
cos
(
2
θ
)
+
4
℧
2
)
+
{\displaystyle {\rm {\sin ^{2}\theta \ {\dot {\phi }}^{2}\ (2a^{4}\sin ^{2}(2\theta )+r\ (a^{2}(a^{2}\cos(4\theta )+3a^{2}+4\ (a-\mho )(a+\mho )\cos(2\theta )+4\mho ^{2})+}}}
8
r
(
−
a
2
sin
2
θ
+
2
a
2
r
cos
2
θ
+
r
3
)
)
)
)
)
/
(
8
(
a
2
cos
2
θ
+
r
2
)
3
)
{\displaystyle {\rm {8r\ (-a^{2}\sin ^{2}\theta +2a^{2}r\cos ^{2}\theta +r^{3})))))/(8\ (a^{2}\cos ^{2}\theta +r^{2})^{3})}}}
for the radial component,
θ
¨
=
−
(
2
r
θ
˙
r
˙
)
/
(
a
2
cos
2
θ
+
r
2
)
−
(
a
2
sin
θ
cos
θ
r
˙
2
)
/
(
(
a
2
+
(
r
−
2
)
r
+
{\displaystyle {\rm {{\ddot {\theta }}=-(2r\ {\dot {\theta }}\ {\dot {r}})/(a^{2}\cos ^{2}\theta +r^{2})-(a^{2}\sin \theta \cos \theta \ {\dot {r}}^{2})/((a^{2}+(r-2)\ r+}}}
℧
2
)
(
a
2
cos
2
θ
+
r
2
)
)
+
(
sin
(
2
θ
)
(
a
2
(
8
θ
˙
2
(
a
2
cos
2
θ
+
r
2
)
2
−
8
t
˙
(
2
q
℧
r
+
{\displaystyle {\rm {\mho ^{2})(a^{2}\cos ^{2}\theta +r^{2}))+(\sin(2\theta )(a^{2}(8{\dot {\theta }}^{2}(a^{2}\cos ^{2}\theta +r^{2})^{2}-8{\dot {t}}(2q\ \mho \ r+}}}
(
℧
2
−
2
r
)
t
˙
)
)
+
16
a
(
a
2
+
r
2
)
ϕ
˙
(
q
℧
r
+
(
℧
2
−
2
r
)
t
˙
)
+
ϕ
˙
2
(
3
a
6
+
11
a
4
r
2
+
10
a
4
r
−
{\displaystyle {\rm {(\mho ^{2}-2r)\ {\dot {t}}))+16a\ (a^{2}+r^{2})\ {\dot {\phi }}(q\ \mho \ r+(\mho ^{2}-2r)\ {\dot {t}})+{\dot {\phi }}^{2}(3a^{6}+11a^{4}r^{2}+10a^{4}r-}}}
5
a
4
℧
2
+
4
a
2
(
a
2
+
2
r
2
)
cos
(
2
θ
)
(
a
2
+
(
r
−
2
)
r
+
℧
2
)
−
8
a
2
℧
2
r
2
+
16
a
2
r
4
+
16
a
2
r
3
+
a
4
cos
(
4
θ
)
(
a
2
+
{\displaystyle {\rm {5a^{4}\mho ^{2}+4a^{2}(a^{2}+2r^{2})\cos(2\theta )(a^{2}+(r-2)r+\mho ^{2})-8a^{2}\mho ^{2}r^{2}+16a^{2}r^{4}+16a^{2}r^{3}+a^{4}\cos(4\theta )(a^{2}+}}}
(
r
−
2
)
r
+
℧
2
)
+
8
r
6
)
)
)
/
(
16
(
a
2
cos
2
θ
+
r
2
)
3
)
{\displaystyle {\rm {(r-2)r+\mho ^{2})+8r^{6})))/(16(a^{2}\cos ^{2}\theta +r^{2})^{3})}}}
the poloidial component and
ϕ
¨
=
−
(
(
r
˙
(
4
a
q
℧
(
a
2
cos
2
θ
−
r
2
)
−
8
a
t
˙
(
a
2
cos
2
θ
+
r
(
℧
2
−
r
)
)
+
ϕ
˙
(
2
a
4
sin
2
(
2
θ
)
+
{\displaystyle {\rm {{\ddot {\phi }}=-(({\dot {r}}(4a\ q\ \mho \ (a^{2}\cos ^{2}\theta -r^{2})-8a\ {\dot {t}}(a^{2}\cos ^{2}\theta +r\ (\mho ^{2}-r))+{\dot {\phi }}\ (2a^{4}\sin ^{2}(2\theta )+}}}
8
r
3
(
a
2
cos
(
2
θ
)
+
a
2
+
℧
2
)
+
a
2
r
(
a
2
(
4
cos
(
2
θ
)
+
cos
(
4
θ
)
)
+
3
a
2
+
8
℧
2
)
−
4
a
2
r
2
(
cos
(
2
θ
)
+
3
)
+
8
r
5
−
16
r
4
)
)
)
/
(
a
2
+
{\displaystyle {\rm {8r^{3}(a^{2}\cos(2\theta )+a^{2}+\mho ^{2})+a^{2}r\ (a^{2}(4\cos(2\theta )+\cos(4\theta ))+3a^{2}+8\mho ^{2})-4a^{2}r^{2}(\cos(2\theta )+3)+8r^{5}-16r^{4})))/(a^{2}+}}}
(
r
−
2
)
r
+
℧
2
)
+
θ
˙
(
ϕ
˙
(
a
4
(
−
sin
(
4
θ
)
)
−
2
a
2
sin
(
2
θ
)
(
3
a
2
+
4
(
r
−
1
)
r
+
2
℧
2
)
+
8
(
a
2
+
r
2
)
2
cot
θ
)
+
{\displaystyle {\rm {(r-2)\ r+\mho ^{2})+{\dot {\theta }}\ ({\dot {\phi }}\ (a^{4}(-\sin(4\theta ))-2a^{2}\sin(2\theta )(3a^{2}+4(r-1)r+2\mho ^{2})+8\ (a^{2}+r^{2})^{2}\cot \theta )+}}}
8
a
cot
θ
(
q
℧
r
+
(
℧
2
−
2
r
)
t
˙
)
)
)
/
(
4
(
a
2
cos
2
θ
+
r
2
)
2
)
{\displaystyle {\rm {8a\cot \theta \ (q\ \mho \ r+(\mho ^{2}-2r)\ {\dot {t}})))/(4(a^{2}\cos ^{2}\theta +r^{2})^{2})}}}
for the axial component of the 4-acceleration. The total time dilation is
t
˙
{\displaystyle {\rm {\dot {t}}}}
=
csc
2
θ
(
L
z
(
a
Δ
sin
2
θ
−
a
(
a
2
+
r
2
)
sin
2
θ
)
−
q
℧
r
(
a
2
+
r
2
)
sin
2
θ
+
E
(
(
a
2
+
r
2
)
2
sin
2
θ
−
a
2
Δ
sin
4
θ
)
)
Δ
Σ
{\displaystyle {\rm {={\frac {\csc ^{2}\theta \ ({L_{z}}(a\ \Delta \sin ^{2}\theta -a\ (a^{2}+r^{2})\sin ^{2}\theta )-q\ \mho \ r\ (a^{2}+r^{2})\sin ^{2}\theta +E((a^{2}+r^{2})^{2}\sin ^{2}\theta -a^{2}\Delta \sin ^{4}\theta ))}{\Delta \Sigma }}}}}
=
a
(
L
z
−
a
E
sin
2
θ
)
+
(
r
2
+
a
2
)
P
/
Δ
Σ
{\displaystyle {\rm {={\frac {a(L_{z}-aE\sin ^{2}\theta )+(r^{2}+a^{2})P/\Delta }{\Sigma }}}}}
where the differentiation goes by the proper time τ for charged (q≠0) and neutral (q=0) particles (μ=-1, v<1), and for massless particles (μ=0, v=1) by the spatial affine parameter ŝ. The relation between the first proper time derivatives and the local three-velocity components relative to a ZAMO is
r
˙
=
v
r
Δ
Σ
(
1
−
μ
2
v
2
)
=
S
i
g
n
(
v
r
)
V
r
Σ
θ
˙
=
v
θ
Σ
(
1
−
μ
2
v
2
)
=
S
i
g
n
(
v
θ
)
V
θ
Σ
{\displaystyle {\rm {{\dot {r}}={\frac {v_{r}{\sqrt {\Delta }}}{\sqrt {\Sigma (1-\mu ^{2}v^{2})}}}}}={\frac {{\rm {Sign}}({\rm {v_{r}){\sqrt {\rm {V_{r}}}}}}}{\Sigma }}\ \ \ \ \ \ \ \ {\rm {{\dot {\theta }}={\frac {v_{\theta }}{\sqrt {\Sigma (1-\mu ^{2}v^{2})}}}={\frac {\rm {Sign(v_{\theta }){\sqrt {\rm {V_{\theta }}}}}}{\Sigma }}}}}
ϕ
˙
=
a
(
a
2
E
−
a
L
z
−
Δ
E
−
q
r
℧
+
E
r
2
)
+
Δ
L
z
csc
2
θ
Δ
Σ
{\displaystyle {\dot {\phi }}{\rm {={\frac {a\left(a^{2}E-aL_{z}-\Delta E-qr\mho +Er^{2}\right)+\Delta L_{z}\csc ^{2}\theta }{\Delta \Sigma }}}}}
The local three-velocity in terms of the position and the constants of motion is
v
=
|
−
a
2
L
z
2
Σ
2
(
℧
2
−
2
r
)
2
+
2
a
L
z
Σ
χ
(
2
r
−
℧
2
)
(
E
Σ
−
q
r
℧
)
+
χ
(
Δ
Σ
3
−
χ
(
E
Σ
−
q
r
℧
)
2
)
a
L
z
Σ
(
℧
2
−
2
r
)
+
χ
(
E
Σ
−
q
r
℧
)
|
{\displaystyle {\rm {v=\left|{\frac {\sqrt {-a^{2}L_{z}^{2}\Sigma ^{2}\left(\mho ^{2}-2r\right)^{2}+2aL_{z}\Sigma \chi \left(2r-\mho ^{2}\right)(E\Sigma -qr\mho )+\chi \left(\Delta \Sigma ^{3}-\chi (E\Sigma -qr\mho )^{2}\right)}}{aL_{z}\Sigma \left(\mho ^{2}-2r\right)+\chi (E\Sigma -qr\mho )}}\right|}}}
which reduces to
v
=
χ
(
E
−
L
z
Ω
)
2
−
Δ
Σ
χ
(
E
−
L
z
Ω
)
2
=
t
˙
2
−
ς
2
t
˙
{\displaystyle {\rm {v={\sqrt {\frac {\chi \ (E-L_{z}\ \Omega )^{2}-\Delta \ \Sigma }{\chi \ (E-L_{z}\ \Omega )^{2}}}}={\frac {\sqrt {{\dot {t}}^{2}-\varsigma ^{2}}}{\dot {t}}}}}}
if the charge of the test particle is q=0. The escape velocity of a charged particle with zero orbital angular momentum is
v
e
s
c
=
|
a
4
cos
4
θ
(
Δ
Σ
−
χ
)
+
2
a
2
r
cos
2
θ
(
q
χ
℧
+
Δ
r
Σ
−
r
χ
)
+
r
2
(
−
q
2
χ
℧
2
+
2
q
r
χ
℧
+
r
2
(
Δ
Σ
−
χ
)
)
χ
(
a
2
cos
2
θ
+
r
(
r
−
q
℧
)
)
|
{\displaystyle {\rm {v_{esc}=\left|{\frac {\sqrt {a^{4}\cos ^{4}\theta (\Delta \Sigma -\chi )+2a^{2}r\cos ^{2}\theta (q\chi \mho +\Delta r\Sigma -r\chi )+r^{2}\left(-q^{2}\chi \mho ^{2}+2qr\chi \mho +r^{2}(\Delta \Sigma -\chi )\right)}}{{\sqrt {\chi }}\left(a^{2}\cos ^{2}\theta +r(r-q\mho )\right)}}\right|}}}
which for a neutral test particle with q=0 reduces to
v
e
s
c
=
ς
2
−
1
ς
{\displaystyle {\rm {v_{esc}}}={\frac {\sqrt {\varsigma ^{2}-1}}{\varsigma }}}
with the gravitational time dilation of a locally stationary ZAMO
ς
=
d
t
d
τ
=
|
g
t
t
|
=
χ
Δ
Σ
{\displaystyle \varsigma ={\frac {\rm {dt}}{\rm {d\tau }}}={\sqrt {|g^{\rm {tt}}|}}={\sqrt {\frac {\chi }{\Delta \ \Sigma }}}}
which is infinite at the horizon. The time dilation of a globally stationary particle (with respect to the fixed stars) is
σ
=
d
t
d
τ
=
|
1
/
g
t
t
|
=
1
1
−
2
r
−
℧
2
Σ
{\displaystyle \sigma ={\frac {\rm {dt}}{\rm {d\tau }}}={\sqrt {|1/g_{\rm {tt}}|}}={\frac {1}{\sqrt {1-{\frac {\rm {2r-\mho ^{2}}}{\Sigma }}}}}}
which is infinite at the ergosphere. The Frame-Dragging angular velocity observed at infinity is
ω
=
|
g
t
ϕ
g
ϕ
ϕ
|
=
a
(
2
r
−
℧
2
)
/
χ
{\displaystyle \omega =\left|{\frac {g_{\rm {t\phi }}}{g_{\phi \phi }}}\right|={\rm {a\left(2r-\mho ^{2}\right)/\chi }}}
The local frame dragging velocity with respect to the fixed stars is therefore
v
ϕ
=
g
t
ϕ
g
t
ϕ
=
1
−
g
t
t
g
t
t
=
|
g
t
ϕ
g
ϕ
ϕ
g
t
t
g
ϕ
ϕ
|
=
ω
R
¯
ϕ
ς
{\displaystyle v_{\phi }={\sqrt {g_{\rm {t\phi }}\ g^{\rm {t\phi }}}}={\sqrt {1-g_{\rm {tt}}\ g^{\rm {tt}}}}=|{\frac {g_{\rm {t\phi }}}{g_{\rm {\phi \phi }}}}{\sqrt {g^{\rm {tt}}}}\ {\sqrt {g_{\rm {\phi \phi }}}}|=\omega {\bar {\rm {R}}}_{\phi }\varsigma }
which is c at the ergosphere. The axial radius of gyration is
R
¯
ϕ
=
|
g
ϕ
ϕ
|
=
χ
Σ
sin
θ
{\displaystyle {\bar {\rm {R}}}_{\phi }={\sqrt {|g_{\phi \phi }|}}={\sqrt {\frac {\chi }{\Sigma }}}\ \sin \theta }
The 3 conserved quantities are 1) the total energy:
E
=
g
t
t
t
˙
+
g
t
ϕ
ϕ
˙
+
q
A
t
=
t
˙
(
1
−
2
r
−
℧
2
Σ
)
+
ϕ
˙
a
sin
2
θ
(
2
r
−
℧
2
)
Σ
+
℧
q
r
Σ
=
Δ
Σ
(
1
−
μ
2
v
2
)
χ
+
ω
L
z
+
℧
q
r
Σ
{\displaystyle {{\rm {E}}=g_{\rm {tt}}\ {\dot {\rm {t}}}+g_{\rm {t\phi }}\ {\rm {{\dot {\phi }}+{\rm {q\ A_{t}={\dot {t}}\left(1-{\frac {2r-\mho ^{2}}{\Sigma }}\right)+{\dot {\phi }}{\frac {a\sin ^{2}\theta \left(2r-\mho ^{2}\right)}{\Sigma }}+{\frac {\mho \ q\ r}{\Sigma }}={\rm {{\sqrt {\frac {\Delta \ \Sigma }{(1-\mu ^{2}v^{2})\ \chi }}}+\omega \ L_{z}+{\frac {\mho \ q\ r}{\Sigma }}}}}}}}}}
2) the axial angular momentum:
L
z
=
−
g
ϕ
ϕ
ϕ
˙
−
g
t
ϕ
t
˙
−
q
A
ϕ
=
ϕ
˙
χ
sin
2
θ
Σ
−
t
˙
a
sin
2
θ
(
2
r
−
Q
2
)
Σ
+
a
r
℧
q
sin
2
θ
Σ
=
v
ϕ
R
¯
ϕ
1
−
μ
2
v
2
+
(
1
−
μ
2
v
2
)
a
r
℧
q
sin
2
θ
Σ
{\displaystyle {\rm {L_{z}}}=-g_{\phi \phi }\ {\dot {\phi }}-g_{\rm {t\phi }}\ {\rm {{\dot {t}}-q\ A_{\phi }={\rm {{\frac {{\dot {\phi }}\ \chi \sin ^{2}\theta }{\Sigma }}-{\frac {{\dot {t}}\ a\ \sin ^{2}\theta \left(2r-Q^{2}\right)}{\Sigma }}+{\frac {a\ r\ \mho \ q\ \sin ^{2}\theta }{\Sigma }}}}={\frac {v_{\phi }\ {\bar {R}}_{\phi }}{\sqrt {1-\mu ^{2}\ v^{2}}}}+{\frac {(1-\mu ^{2}v^{2})\ a\ r\ \mho \ q\ \sin ^{2}\theta }{\Sigma }}}}}
3) the Carter constant:
Q
=
v
θ
2
Σ
1
−
μ
2
v
2
+
cos
2
θ
(
a
2
(
μ
2
−
E
2
)
+
L
z
2
sin
2
θ
)
{\displaystyle {\rm {Q={\frac {{v_{\theta }}^{2}\ \Sigma }{1-\mu ^{2}v^{2}}}+\cos ^{2}\theta \left(a^{2}(\mu ^{2}-E^{2})+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)}}}
The effective radial potential whose zero roots define the turning points is
V
r
=
P
2
−
Δ
(
(
L
z
−
a
E
)
2
+
Q
+
μ
2
r
2
)
{\displaystyle {\rm {V_{r}=P^{2}-\Delta \left((L_{z}-aE)^{2}+Q+\mu ^{2}r^{2}\right)}}}
and the poloidial potential
V
θ
=
v
θ
2
Σ
1
−
μ
2
v
2
=
Q
−
cos
2
θ
(
a
2
(
μ
2
−
E
2
)
+
L
z
2
sin
2
θ
)
{\displaystyle {\rm {V_{\theta }={\frac {{v_{\theta }}^{2}\ \Sigma }{1-\mu ^{2}v^{2}}}=Q-\cos ^{2}\theta \left(a^{2}\left(\mu ^{2}-E^{2}\right)+{\frac {\rm {L_{z}^{2}}}{\sin ^{2}\theta }}\right)}}}
with the parameter
P
=
E
(
a
2
+
r
2
)
−
a
L
z
+
q
r
℧
{\displaystyle {\rm {P=E\left(a^{2}+r^{2}\right)-aL_{z}+qr\mho }}}
The azimutal and latitudinal impact parameters are
b
ϕ
=
L
z
E
,
b
θ
=
Q
E
2
{\displaystyle {\rm {b_{\phi }={\frac {L_{z}}{E}}\ ,\ \ b_{\theta }={\sqrt {\frac {Q}{E^{2}}}}}}}
The horizons and ergospheres have the Boyer-Lindquist-radius
r
H
±
=
1
±
1
−
a
2
−
℧
2
,
r
E
±
=
1
±
1
−
a
2
cos
2
θ
−
℧
2
{\displaystyle {\rm {r_{H}^{\pm }=1\pm {\sqrt {1-a^{2}-\mho ^{2}}}}}\ ,\ \ {\rm {r_{E}^{\pm }=1\pm {\sqrt {\rm {1-a^{2}\cos ^{2}\theta -\mho ^{2}}}}}}}
In this article the total mass equivalent M, which also contains the rotational and the electrical field energy, is set to 1; the relation of M with the irreducible mass is
M
i
r
r
=
2
M
2
−
℧
2
+
2
M
M
2
−
℧
2
−
a
2
2
→
M
=
16
M
i
r
r
4
+
8
M
i
r
r
2
℧
2
+
℧
4
16
M
i
r
r
2
−
4
a
2
{\displaystyle {\rm {M_{\rm {irr}}={\frac {\sqrt {2M^{2}-\mho ^{2}+2M{\sqrt {M^{2}-\mho ^{2}-a^{2}}}}}{2}}\ \to \ M={\sqrt {\frac {16M_{\rm {irr}}^{4}+8M_{\rm {irr}}^{2}\ \mho ^{2}+\mho ^{4}}{16M_{\rm {irr}}^{2}-4a^{2}}}}}}}
where a is in units of M.
Giấy phép
Tôi, người giữ bản quyền tác phẩm này, từ đây phát hành nó theo giấy phép sau:
Bạn được phép:
chia sẻ – sao chép, phân phối và chuyển giao tác phẩm
pha trộn – để chuyển thể tác phẩm
Theo các điều kiện sau:
ghi công – Bạn phải ghi lại tác giả và nguồn, liên kết đến giấy phép, và các thay đổi đã được thực hiện, nếu có. Bạn có thể làm các điều trên bằng bất kỳ cách hợp lý nào, miễn sao không ám chỉ rằng người cho giấy phép ủng hộ bạn hay việc sử dụng của bạn.
chia sẻ tương tự – Nếu bạn biến tấu, biến đổi, hoặc làm tác phẩm khác dựa trên tác phẩm này, bạn chỉ được phép phân phối tác phẩm mới theo giấy phép y hệt hoặc tương thích với tác phẩm gốc. https://creativecommons.org/licenses/by-sa/4.0 CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 true true Tiếng Việt Ghi một dòng giải thích những gì có trong tập tin này
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