In (+, −, −, −) signature and in natural units of G = M = c = k e = 1 {\displaystyle {\rm {G=M=c=k_{e}=1}}} the KNdS metric is[3] [4] [5] [6]
g t t = − 3 [ a 2 sin 2 θ ( a 2 Λ cos 2 θ + 3 ) + a 2 ( Λ r 2 − 3 ) + Λ r 4 − 3 r 2 + 6 r − 3 ℧ 2 ] ( a 2 Λ + 3 ) 2 ( a 2 cos 2 θ + r 2 ) {\displaystyle g_{\rm {tt}}={\rm {-{\frac {3\ [a^{2}\ \sin ^{2}\theta \left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)+a^{2}\left(\Lambda \ r^{2}-3\right)+\Lambda \ r^{4}-3\ r^{2}+6\ r-3\mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}}
g r r = − a 2 cos 2 θ + r 2 ( a 2 + r 2 ) ( 1 − Λ r 2 3 ) − 2 r + ℧ 2 {\displaystyle g_{\rm {rr}}={\rm {-{\frac {a^{2}\ \cos ^{2}\theta +r^{2}}{\left(a^{2}+r^{2}\right)\left(1-{\frac {\Lambda \ r^{2}}{3}}\right)-2\ r+\mho ^{2}}}}}}
g θ θ = − 3 ( a 2 cos 2 θ + r 2 ) a 2 Λ cos 2 θ + 3 {\displaystyle g_{\rm {\theta \theta }}={\rm {-{\frac {3\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}{a^{2}\ \Lambda \ \cos ^{2}\theta +3}}}}}
g ϕ ϕ = 9 { 1 3 ( a 2 + r 2 ) 2 sin 2 θ ( a 2 Λ cos 2 θ + 3 ) − a 2 sin 4 θ [ ( a 2 + r 2 ) ( 1 − Λ r 2 / 3 ) − 2 r + ℧ 2 ] } − ( a 2 Λ + 3 ) 2 ( a 2 cos 2 θ + r 2 ) {\displaystyle g_{\rm {\phi \phi }}={\rm {\frac {9\ \{{\frac {1}{3}}\left(a^{2}+r^{2}\right)^{2}\sin ^{2}\theta \left(a^{2}\ \Lambda \cos ^{2}\theta +3\right)-a^{2}\sin ^{4}\theta \ [\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}]\}}{-\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}
g t ϕ = 3 a sin 2 θ [ a 2 Λ ( a 2 + r 2 ) cos 2 θ + a 2 Λ r 2 + Λ r 4 + 6 r − 3 ℧ 2 ] ( a 2 Λ + 3 ) 2 ( a 2 cos 2 θ + r 2 ) {\displaystyle g_{\rm {t\phi }}={\rm {\frac {3\ a\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+\Lambda \ r^{4}+6\ r-3\ \mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}}}
with all the other metric tensor components g μ ν = 0 {\displaystyle g_{\mu \nu }=0} , where a {\displaystyle {\rm {a}}} is the black hole's spin parameter, ℧ {\displaystyle {\rm {\mho }}} its electric charge, and Λ = 3 H 2 {\displaystyle {\rm {\Lambda =3H^{2}}}} [7] the cosmological constant with H {\displaystyle {\rm {H}}} as the time-independent Hubble parameter . The electromagnetic 4-potential is
A μ = { 3 r ℧ ( a 2 Λ + 3 ) ( a 2 cos 2 θ + r 2 ) , 0 , 0 , − 3 a r ℧ sin 2 θ ( a 2 Λ + 3 ) ( a 2 cos 2 θ + r 2 ) } {\displaystyle {\rm {A_{\mu }=\left\{{\frac {3\ r\ \mho }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}},\ 0,\ 0,\ -{\frac {3\ a\ r\ \mho \ \sin ^{2}\theta }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}\right\}}}}
The frame-dragging angular velocity is
ω = d ϕ d t = − g t ϕ g ϕ ϕ = a [ a 2 Λ ( a 2 + r 2 ) cos 2 θ + a 2 Λ r 2 + 6 r + Λ r 4 − 3 ℧ 2 ] a 2 sin 2 θ [ a 2 ( Λ r 2 − 3 ) + 6 r + Λ r 4 − 3 r 2 − 3 ℧ 2 ] + a 2 Λ ( a 2 + r 2 ) 2 cos 2 θ + 3 ( a 2 + r 2 ) 2 {\displaystyle \omega ={\frac {\rm {d\phi }}{\rm {dt}}}=-{\frac {g_{\rm {t\phi }}}{g_{\rm {\phi \phi }}}}={\rm {\frac {a\ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]}{a^{2}\ \sin ^{2}\theta \ [a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]+a^{2}\ \Lambda \ \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\ \left(a^{2}+r^{2}\right)^{2}}}}}
and the local frame-dragging velocity relative to constant { r , θ , ϕ } {\displaystyle {\rm {\{r,\theta ,\phi \}}}} positions (the speed of light at the ergosphere )
ν = g t ϕ g t ϕ = − a 2 sin 2 θ [ a 2 Λ ( a 2 + r 2 ) cos 2 θ + a 2 Λ r 2 + 6 r + Λ r 4 − 3 ℧ 2 ] 2 ( a 2 Λ cos 2 θ + 3 ) ( a 2 + r 2 − a 2 sin 2 θ ) 2 [ a 2 ( Λ r 2 − 3 ) + 6 r + Λ r 4 − 3 r 2 − 3 ℧ 2 ] {\displaystyle \nu ={\sqrt {g_{\rm {t\phi }}\ g^{\rm {t\phi }}}}={\rm {\sqrt {-{\frac {a^{2}\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]^{2}}{\left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}[a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]}}}}}}
The escape velocity (the speed of light at the horizons) relative to the local corotating zero-angular momentum observer is
v = 1 − 1 / g t t = 3 ( a 2 Λ cos 2 θ + 3 ) ( a 2 + r 2 − a 2 sin 2 θ ) 2 [ a 2 ( Λ r 2 − 3 ) + Λ r 4 − 3 r 2 + 6 r − 3 ℧ 2 ] ( a 2 Λ + 3 ) 2 ( a 2 cos 2 θ + r 2 ) { a 2 Λ ( a 2 + r 2 ) 2 cos 2 θ + 3 ( a 2 + r 2 ) 2 + a 2 sin 2 θ [ a 2 ( Λ r 2 − 3 ) + Λ r 4 − 3 r 2 + 6 r − 3 ℧ 2 ] } + 1 {\displaystyle {\rm {v}}={\sqrt {1-1/g^{\rm {tt}}}}={\rm {\sqrt {{\frac {3\left(a^{2}\Lambda \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}\left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]}{\left(a^{2}\Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)\{a^{2}\Lambda \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\left(a^{2}+r^{2}\right)^{2}+a^{2}\sin ^{2}\theta \left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]\}}}+1}}}}
The conserved quantities in the equations of motion
x ¨ μ = − ∑ α , β ( Γ α β μ x ˙ α x ˙ β + q F μ β x ˙ α g α β ) {\displaystyle {\rm {{\ddot {x}}^{\mu }=-\sum _{\alpha ,\beta }\ (\Gamma _{\alpha \beta }^{\mu }\ {\dot {x}}^{\alpha }\ {\dot {x}}^{\beta }+q\ {\rm {F}}^{\mu \beta }\ {\rm {\dot {x}}}^{\alpha }}}\ g_{\alpha \beta })}
where x ˙ {\displaystyle {\rm {\dot {x}}}} is the four velocity , q {\displaystyle {\rm {q}}} is the test particle's specific charge and F {\displaystyle {\rm {F}}} the Maxwell–Faraday tensor
F μ ν = ∂ A μ ∂ x ν − ∂ A ν ∂ x μ {\displaystyle {\rm {{\ F}_{\mu \nu }={\frac {\partial A_{\mu }}{\partial x^{\nu }}}-{\frac {\partial A_{\nu }}{\partial x^{\mu }}}}}}
are the total energy
E = − p t = g t t t ˙ + g t ϕ ϕ ˙ + q A t {\displaystyle {\rm {E=-p_{t}}}=g_{\rm {tt}}{\rm {\dot {t}}}+g_{\rm {t\phi }}{\rm {\dot {\phi }}}+{\rm {q\ A_{t}}}}
and the covariant axial angular momentum
L z = p ϕ = − g ϕ ϕ ϕ ˙ − g t ϕ t ˙ − q A ϕ {\displaystyle {\rm {L_{z}=p_{\phi }}}=-g_{\rm {\phi \phi }}{\rm {\dot {\phi }}}-g_{\rm {t\phi }}{\rm {\dot {t}}}-{\rm {q\ A_{\phi }}}}
The overdot stands for differentiation by the testparticle's proper time τ {\displaystyle \tau } or the photon's affine parameter , so x ˙ = d x / d τ , x ¨ = d 2 x / d τ 2 {\displaystyle {\rm {{\dot {x}}=dx/d\tau ,\ {\ddot {x}}=d^{2}x/d\tau ^{2}}}} .
To get g r r = 0 {\displaystyle g_{\rm {rr}}=0} coordinates we apply the transformation
d t = d u − d r ( a 2 Λ / 3 + 1 ) ( a 2 + r 2 ) ( a 2 + r 2 ) ( 1 − Λ r 2 / 3 ) − 2 r + ℧ 2 {\displaystyle {\rm {dt=du-{\frac {dr\left(a^{2}\ \Lambda /3+1\right)\left(a^{2}+r^{2}\right)}{\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}}}}}}
d ϕ = d φ − a d r ( a 2 Λ / 3 + 1 ) ( a 2 + r 2 ) ( 1 − Λ r 2 / 3 ) − 2 r + ℧ 2 {\displaystyle {\rm {d\phi =d\varphi -{\frac {a\ dr\left(a^{2}\ \Lambda /3+1\right)}{\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}}}}}}
and get the metric coefficients
g u r = − 3 a 2 Λ + 3 {\displaystyle g_{\rm {ur}}={\rm {-{\frac {3}{a^{2}\ \Lambda +3}}}}}
g r φ = 3 a sin 2 θ a 2 Λ + 3 {\displaystyle g_{\rm {r\varphi }}={\rm {\frac {3\ a\sin ^{2}\theta }{a^{2}\ \Lambda +3}}}}
g u u = g t t , g θ θ = g θ θ , g φ φ = g ϕ ϕ , g u φ = g t ϕ {\displaystyle g_{\rm {uu}}=g_{\rm {tt}}\ ,\ \ g_{\theta \theta }=g_{\theta \theta }\ ,\ \ g_{\rm {\varphi \varphi }}=g_{\rm {\phi \phi }}\ ,\ \ g_{\rm {u\varphi }}=g_{\rm {t\phi }}}
and all the other g μ ν = 0 {\displaystyle g_{\mu \nu }=0} , with the electromagnetic vector potential
A μ = { 3 r ℧ ( a 2 Λ + 3 ) ( a 2 cos 2 θ + r 2 ) , 3 r ℧ a 2 ( Λ r 2 − 3 ) + 6 r + Λ r 4 − 3 ( r 2 + ℧ 2 ) , 0 , − 3 a r ℧ sin 2 θ ( a 2 Λ + 3 ) ( a 2 cos 2 θ + r 2 ) } {\displaystyle {\rm {A_{\mu }=\left\{{\frac {3\ r\ \mho }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\cos ^{2}\theta +r^{2}\right)}},{\frac {3\ r\ \mho }{a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\left(r^{2}+\mho ^{2}\right)}},\ 0,\ -{\frac {3\ a\ r\ \mho \sin ^{2}\theta }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}\right\}}}}
Defining t ¯ = u − r {\displaystyle {\rm {{\bar {t}}=u-r}}} ingoing lightlike worldlines give a 45 ∘ {\displaystyle 45^{\circ }} light cone on a { t ¯ , r } {\displaystyle \{{\rm {{\bar {t}},\ r\}}}} spacetime diagram .
See also
References