DescriptionOrbit in the Kerr Newman De Sitter Spacetime.png	English: Unstable orbit at constant Boyer Lindquist r in the vicinity of a Kerr Newman De Sitter (KNdS) black hole. The local velocities on the bottom right of the numeric display are relative to local ZAMOs, while the ZAMOs' frame dragging velocities (larger than the speed of light at this point) are relative to stationary [r,θ,φ]-positions.
Date	23 August 2023
Source	Own work, Code: knds.yukterez.net, Equations of motion: relativity.yukterez.net/74.html#11
Author	Yukterez (Simon Tyran, Vienna)
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The Kerr–Newman–de–Sitter metric (KNdS) ^[1][2] is the one of the most general stationary solutions of the Einstein–Maxwell equations in [1] that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass embedded in an expanding universe. It generalizes the Kerr–Newman metric by taking into account the cosmological constant ${\displaystyle \Lambda }$.

In (+, −, −, −) signature and in natural units of ${\displaystyle {\rm {G=M=c=k_{e}=1}}}$ the KNdS metric is^[3][4][5][6]

${\displaystyle g_{\rm {tt}}={\rm {-{\frac {3\ [a^{2}\ \sin ^{2}\theta \left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)+a^{2}\left(\Lambda \ r^{2}-3\right)+\Lambda \ r^{4}-3\ r^{2}+6\ r-3\mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}}$

${\displaystyle g_{\rm {rr}}={\rm {-{\frac {a^{2}\ \cos ^{2}\theta +r^{2}}{\left(a^{2}+r^{2}\right)\left(1-{\frac {\Lambda \ r^{2}}{3}}\right)-2\ r+\mho ^{2}}}}}}$

${\displaystyle g_{\rm {\theta \theta }}={\rm {-{\frac {3\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}{a^{2}\ \Lambda \ \cos ^{2}\theta +3}}}}}$

${\displaystyle g_{\rm {\phi \phi }}={\rm {\frac {9\ \{{\frac {1}{3}}\left(a^{2}+r^{2}\right)^{2}\sin ^{2}\theta \left(a^{2}\ \Lambda \cos ^{2}\theta +3\right)-a^{2}\sin ^{4}\theta \ [\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}]\}}{-\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}$

${\displaystyle g_{\rm {t\phi }}={\rm {\frac {3\ a\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+\Lambda \ r^{4}+6\ r-3\ \mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}}}$

with all the other ${\displaystyle g_{\mu \nu }=0}$, where ${\displaystyle {\rm {a}}}$ is the black hole's spin parameter, ${\displaystyle {\rm {\mho }}}$ its electric charge and ${\displaystyle {\rm {\Lambda =3H^{2}}}}$^[7] the cosmological constant with ${\displaystyle {\rm {H}}}$ as the time-independent Sitter universe#Mathematical expression Hubble parameter. The electromagnetic 4-potential is

${\displaystyle {\rm {A_{\mu }=\left\{{\frac {3\ r\ \mho }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}},\ 0,\ 0,\ -{\frac {3\ a\ r\ \mho \ \sin ^{2}\theta }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}\right\}}}}$

The frame-dragging angular velocity is

${\displaystyle \omega ={\frac {\rm {d\phi }}{\rm {dt}}}=-{\frac {g_{\rm {t\phi }}}{g_{\rm {\phi \phi }}}}={\rm {\frac {a\ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]}{a^{2}\ \sin ^{2}\theta \ [a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]+a^{2}\ \Lambda \ \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\ \left(a^{2}+r^{2}\right)^{2}}}}}$

and the local frame-dragging velocity relative to constant ${\displaystyle {\rm {\{r,\theta ,\phi \}}}}$ positions (the speed of light at the ergosphere)

${\displaystyle \nu ={\sqrt {g_{\rm {t\phi }}\ g^{\rm {t\phi }}}}={\rm {\sqrt {-{\frac {a^{2}\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]^{2}}{\left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}[a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]}}}}}}$

The escape velocity (the speed of light at the horizons) relative to the local corotating ZAMO (zero angular momentum observer) is

${\displaystyle {\rm {v}}={\sqrt {1-1/g^{\rm {tt}}}}={\rm {\sqrt {{\frac {3\left(a^{2}\Lambda \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}\left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]}{\left(a^{2}\Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)\{a^{2}\Lambda \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\left(a^{2}+r^{2}\right)^{2}+a^{2}\sin ^{2}\theta \left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]\}}}+1}}}}$

The conserved quantities in the equations of motion

${\displaystyle {\rm {{\ddot {x}}^{\mu }=-\sum _{\alpha ,\beta }\ (\Gamma _{\alpha \beta }^{\mu }\ {\dot {x}}^{\alpha }\ {\dot {x}}^{\beta }+q\ {\rm {F}}^{\mu \beta }\ {\rm {\dot {x}}}^{\alpha }}}\ g_{\alpha \beta })}$

where ${\displaystyle {\rm {\dot {x}}}}$ is the four velocity, ${\displaystyle {\rm {q}}}$ is the test particle's specific charge and ${\displaystyle {\rm {F}}}$ the Maxwell–Faraday tensor

${\displaystyle {\rm {{\ F}_{\mu \nu }={\frac {\partial A_{\mu }}{\partial x^{\nu }}}-{\frac {\partial A_{\nu }}{\partial x^{\mu }}}}}}$

are the total energy

${\displaystyle {\rm {E=-p_{t}}}=g_{\rm {tt}}{\rm {\dot {t}}}+g_{\rm {t\phi }}{\rm {\dot {\phi }}}+{\rm {q\ A_{t}}}}$

and the covariant axial angular momentum

${\displaystyle {\rm {L_{z}=p_{\phi }}}=-g_{\rm {\phi \phi }}{\rm {\dot {\phi }}}-g_{\rm {t\phi }}{\rm {\dot {t}}}-{\rm {q\ A_{\phi }}}}$

The for differentiation overdot stands for differentiation by the testparticle's proper time ${\displaystyle \tau }$ or the photon's affine parameter, so ${\displaystyle {\rm {{\dot {x}}=dx/d\tau ,\ {\ddot {x}}=d^{2}x/d\tau ^{2}}}}$.

To get ${\displaystyle g_{\rm {rr}}=0}$ coordinates we apply the transformation

${\displaystyle {\rm {dt=du-{\frac {dr\left(a^{2}\ \Lambda /3+1\right)\left(a^{2}+r^{2}\right)}{\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}}}}}}$

${\displaystyle {\rm {d\phi =d\varphi -{\frac {a\ dr\left(a^{2}\ \Lambda /3+1\right)}{\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}}}}}}$

and get the metric coefficients

${\displaystyle g_{\rm {ur}}={\rm {-{\frac {3}{a^{2}\ \Lambda +3}}}}}$

${\displaystyle g_{\rm {r\varphi }}={\rm {\frac {3\ a\sin ^{2}\theta }{a^{2}\ \Lambda +3}}}}$

${\displaystyle g_{\rm {uu}}=g_{\rm {tt}}\ ,\ \ g_{\theta \theta }=g_{\theta \theta }\ ,\ \ g_{\rm {\varphi \varphi }}=g_{\rm {\phi \phi }}\ ,\ \ g_{\rm {u\varphi }}=g_{\rm {t\phi }}}$

and all the other ${\displaystyle g_{\mu \nu }=0}$, with the electromagnetic vector potential

${\displaystyle {\rm {A_{\mu }=\left\{{\frac {3\ r\ \mho }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\cos ^{2}\theta +r^{2}\right)}},{\frac {3\ r\ \mho }{a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\left(r^{2}+\mho ^{2}\right)}},\ 0,\ -{\frac {3\ a\ r\ \mho \sin ^{2}\theta }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}\right\}}}}$

Defining ${\displaystyle {\rm {{\bar {t}}=u-r}}}$ ingoing lightlike worldlines give a ${\displaystyle 45^{\circ }}$ light cone on a ${\displaystyle \{{\rm {{\bar {t}},\ r\}}}}$ spacetime diagram.

The horizons are at ${\displaystyle g^{\rm {rr}}=0}$ and the ergospheres at ${\displaystyle g_{\rm {tt}}||g_{\rm {uu}}=0}$. This can be solved numerically or analytically. Like in the Kerr and Kerr–Newman metrics the horizons have constant Boyer-Lindquist ${\displaystyle {\rm {r}}}$, while the ergospheres' radii also depend on the polar angle ${\displaystyle \theta }$.

This gives 3 positive solutions each (including the black hole's inner and outer horizons and ergospheres as well as the cosmic ones) and a negative solution for the space at ${\displaystyle {\rm {r<0}}}$ in the antiverse^[8][9] behind the ring singularity, which is part of the probably unphysical extended solution of the metric.

With a negative ${\displaystyle \Lambda }$ (the Anti–de–Sitter variant with an attractive cosmological constant) there are no cosmic horizon and ergosphere, only the black hole related ones.

In the Nariai limit^[10] the black hole's outer horizon and ergosphere coincide with the cosmic ones (in the Schwarzschild–de–Sitter metric to which the KNdS reduces with ${\displaystyle {\rm {a=\mho =0}}}$ that would be the case when ${\displaystyle \Lambda =1/9}$).

The Ricci scalar for the KNdS metric is ${\displaystyle {\rm {R=-4\Lambda }}}$, and the Kretschmann scalar

${\displaystyle {\rm {K=\{220a^{12}\Lambda ^{2}\cos(6\theta )+66a^{12}\Lambda ^{2}\cos(8\theta )+12a^{12}\Lambda ^{2}\cos(10\theta )+a^{12}\Lambda ^{2}\cos(12\theta )+}}}$

${\displaystyle {\rm {462a^{12}\Lambda ^{2}+1080a^{10}\Lambda ^{2}r^{2}\cos(6\theta )+240a^{10}\Lambda ^{2}r^{2}\cos(8\theta )+24a^{10}\Lambda ^{2}r^{2}\cos(10\theta )+}}}$

${\displaystyle {\rm {3024a^{10}\Lambda ^{2}r^{2}+1920a^{8}\Lambda ^{2}r^{4}\cos(6\theta )+240a^{8}\Lambda ^{2}r^{4}\cos(8\theta )+8400a^{8}\Lambda ^{2}r^{4}-}}}$

${\displaystyle {\rm {1152a^{6}\cos(6\theta )-11520a^{6}+1280a^{6}\Lambda ^{2}r^{6}\cos(6\theta )+12800a^{6}\Lambda ^{2}r^{6}+207360a^{4}r^{2}-}}}$

${\displaystyle {\rm {138240a^{4}r\mho ^{2}+11520a^{4}\Lambda ^{2}r^{8}+16128a^{4}\mho ^{4}-276480a^{2}r^{4}+368640a^{2}r^{3}\mho ^{2}+}}}$

${\displaystyle {\rm {6144a^{2}\Lambda ^{2}r^{10}-104448a^{2}r^{2}\mho ^{4}+3a^{4}\cos(4\theta )[165a^{8}\Lambda ^{2}+960a^{6}\Lambda ^{2}r^{2}+2240a^{4}\Lambda ^{2}r^{4}-}}}$

${\displaystyle {\rm {256a^{2}(9-10\Lambda ^{2}r^{6})+256(90r^{2}-60r\mho ^{2}+5\Lambda ^{2}r^{8}+7\mho ^{4})]+24a^{2}\cos(2\theta )[33a^{10}\Lambda ^{2}+}}}$

${\displaystyle {\rm {210a^{8}\Lambda ^{2}r^{2}+560a^{6}\Lambda ^{2}r^{4}-80a^{4}(9-10\Lambda ^{2}r^{6})+128a^{2}(90r^{2}-60r\mho ^{2}+5\Lambda ^{2}r^{8}+}}}$

${\displaystyle {\rm {7\mho ^{4})+256r^{2}(-45r^{2}+60r\mho ^{2}+\Lambda ^{2}r^{8}-17\mho ^{4})]+36864r^{6}-73728r^{5}\mho ^{2}+}}}$

${\displaystyle {\rm {2048\Lambda ^{2}r^{12}+43008r^{4}\mho ^{4}\}\div \{12[a^{2}\cos(2\theta )+a^{2}+2r^{2}]^{6}\}}}}$

For the transformation see here and the links therein. More tensors and scalars for the KNdS metric: in Boyer Lindquist and Null coordinates, higher resolution: video, advised references: arxiv:1710.00997 & arxiv:2007.04354. More snapshots of this series can be found here, those are also under the creative commons license.

1. ↑ (2008). "Kerr-Newman-de Sitter black holes with a restricted repulsive barrier of equatorial photon motion". Physical Review D 58: 084003. DOI:10.1088/0264-9381/17/21/312.
2. ↑ (2009). "Exact spacetimes in Einstein's General Relativity". Cambridge University Press, Cambridge Monographs in Mathematical Physics. DOI:10.1017/CBO9780511635397.
3. ↑ (2023). "Motion equations in a Kerr-Newman-de Sitter spacetime". Classical and Quantum Gravity 40 (13). DOI:10.1088/1361-6382/accbfe.
4. ↑ (2014). "Gravitational lensing and frame-dragging of light in the Kerr–Newman and the Kerr–Newman (anti) de Sitter black hole spacetimes". General Relativity and Gravitation 46 (11): 1818. DOI:10.1007/s10714-014-1818-8.
5. ↑ (2018). "Kerr-de Sitter spacetime, Penrose process and the generalized area theorem". Physical Review D 97 (8): 084049. DOI:10.1103/PhysRevD.97.084049.
6. ↑ (2021). "Null Hypersurfaces in Kerr-Newman-AdS Black Hole and Super-Entropic Black Hole Spacetimes". Classical and Quantum Gravity 38 (4): 045018. DOI:10.1088/1361-6382/abd3e0.
7. ↑ Gaur & Visser: Black holes embedded in FLRW cosmologies (2023) class=gr-qc, arxiv eprint=2308.07374
8. ↑ Andrew Hamilton: Black hole Penrose diagrams (JILA Colorado)
9. ↑ Figure 2 in (2020). "Influence of Cosmic Repulsion and Magnetic Fields on Accretion Disks Rotating around Kerr Black Holes". Universe. DOI:10.3390/universe6020026.
10. ↑ Leonard Susskind: Aspects of de Sitter Holography, timestamp 38:27: video of the online seminar on de Sitter space and Holography, Sept 14, 2021