Kerr–Newman–de–Sitter metric explained

Λ .

In signature and in natural units of

\rm G = M = c = k _{e=1} the KNdS metric is^{[3] [4] [5] [6]} g _{\rm} = \rm - (3 [ a ^{2} \sin ^{2} \theta \left(a ^{2} Λ \cos ^{2} \theta +3 \right) + a ^{2} \left( Λ r ^{2-3\right)+Λ} r ^{4-3} r ^{2+6} r -3 \mho ^{2} ])/(\left(a ^{2} Λ+3 \right) ^{2} \left(a ^{2} \cos ^{2} \theta + r ^{2\right)}) g _{\rm} = \rm - (a ^{2} \cos ^{2} \theta + r ^{2})/(\left(a ^{2+ r} ^{2\right)} \left( 1- (Λ r ^{2})/(3) \right) -2 r + \mho ^{2}) g _{\rm} = \rm - (3 \left(a ^{2} \cos ^{2} \theta + r ^{2\right)})/(a ^{2} Λ \cos ^{2} \theta +3) g _{\rm} = \rm (9 \{ (1)/(3) \left(a ^{2+r} ^{2\right)} ^{2} \sin ^{2} \theta \left(a ^{2} Λ \cos ^{2} \theta +3 \right) - a ^{2} \sin ^{4} \theta [ \left(a ^{2+r} ^{2\right)} \left( 1-Λ r ^{2/3\right)-2} r + \mho ^{2} ] \)/(}{ - \left(a) ^{2} Λ +3 \right) ^{2} \left(a ^{2} \cos ^{2} \theta + r ^{2\right)}} g _{\rm} = \rm (3 a \sin ^{2} \theta [ a ^{2} Λ \left(a ^{2+r} ^{2\right)} \cos ^{2} \theta + a ^{2} Λ r ^{2+Λ} r ^{4+6} r -3 \mho ^{2} ])/(\left(a ^{2} Λ+3 \right) ^{2} \left(a ^{2} \cos ^{2} \theta + r ^{2\right)})

with all the other metric tensor components

g _{\mu} =0 , where \rm a is the black hole's spin parameter, \rm \mho its electric charge, and \rm Λ=3 H ^{2} ^[7] the cosmological constant with \rm H as the time-independent Hubble parameter. The electromagnetic 4-potential is\rm A _{\mu} = \left\{ (3 r \mho)/(\left(a ^{2} Λ+3 \right) \left(a ^{2} \cos ^{2} \theta + r ^{2\right)}) , 0 , 0 , - (3 a r \mho \sin ^{2} \theta)/(\left(a ^{2} Λ+3 \right) \left(a ^{2} \cos ^{2} \theta + r ^{2\right)}) \right\}

The frame-dragging angular velocity is

\omega = (\rm d\phi)/(\rm d t) =- (g _{\rm})/(g _{\rm}) = \rm (a [ a ^{2} Λ \left(a ^{2+r} ^{2\right)} \cos ^{2} \theta + a ^{2} Λ r ^{2+6} r +Λ r ^{4-3} \mho ^{2} ])/(a ^{2} \sin ^{2} \theta [ a ^{2} \left( Λ r ^{2-3\right)+6} r +Λ r ^{4-3} r ^{2-3} \mho ^{2} ] + a ^{2} Λ \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

\rm \{r, \theta, \phi \} positions (the speed of light at the ergosphere)\nu = \sqrt{g _{\rm} g ^{\rm} } = \rm \sqrt

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

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

The conserved quantities in the equations of motion

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

where

\rm • x is the four velocity, \rm q is the test particle's specific charge and \rm F the Maxwell–Faraday tensor\rm { F} _{\mu} = (\partial A _{\mu})/(- \partial x ^{\nu}) (\partial A _{\nu})/(\partial x ^{\mu})

are the total energy

{\rm E = - p _{t}=g} _{\rm} {\rm • t }+g_ + \rm q \ A_

and the covariant axial angular momentum

{\rm L _{z} = p _{\phi} }=-g_ -g_ - \rm q \ A_\tau or the photon's affine parameter, so \rm • x = dx/d\tau , \ddot{x} = d ^{2x/d\tau} ^{2} .

To get

g _{\rm} =0 coordinates we apply the transformation\rm dt = du - (dr \left(a ^{2} Λ / 3+1 \right) \left(a ^{2+r} ^{2\right)})/(\left(a ^{2+r} ^{2\right)} \left( 1-Λ r ^{2} / 3 \right) -2 r + \mho ^{2}) \rm d \phi = d \varphi - (a dr \left(a ^{2} Λ / 3+1 \right))/(\left(a ^{2+r} ^{2\right)} \left( 1-Λ r ^{2} / 3 \right) -2 r + \mho ^{2})

and get the metric coefficients

g _{\rm} = \rm - (3)/(a ^{2} Λ+3) g _{\rm} = \rm (3 a \sin ^{2} \theta)/(a ^{2} Λ+3) g _{\rm} = g _{\rm} , g _{\theta} = g _{\theta} , g _{\rm} = g _{\rm} , g _{\rm} = g _{\rm}

and all the other

g _{\mu} =0 , with the electromagnetic vector potential\rm A _{\mu} = \left\{ (3 r \mho)/(, \left(a ^{2} Λ+3 \right) \left(a ^{2} \cos ^{2} \theta + r ^{2\right)}) (3 r \mho)/(a ^{2} \left( Λ r ^{2-3\right)+6} r +Λ r ^{4-3} \left(r ^{2+\mho} ^{2\right)}) , 0 , - (3 a r \mho \sin ^{2} \theta)/(\left(a ^{2} Λ+3 \right) \left(a ^{2} \cos ^{2} \theta + r ^{2\right)}) \right\}

Defining

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

The horizons are at

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

\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

Λ (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

\rm a = \mho =0 that would be the case when Λ=1 / 9 ).

The Ricci scalar for the KNdS metric is

\rm R =-4 Λ , and the Kretschmann scalar is\rm K = \{ 220 a ^{12} Λ ^{2} \cos ( 6 \theta ) +66 a ^{12} Λ ^{2} \cos ( 8 \theta ) +12 a ^{12} Λ ^{2} \cos ( 10 \theta ) + a ^{12} Λ ^{2} \cos ( 12 \theta ) + \rm 462 a ^{12} Λ ^{2+1080} a ^{10} Λ ^{2} r ^{2} \cos ( 6 \theta ) +240 a ^{10} Λ ^{2} r ^{2} \cos ( 8 \theta ) +24 a ^{10} Λ ^{2} r ^{2} \cos ( 10 \theta ) + \rm 3024 a ^{10} Λ ^{2} r ^{2+1920} a ^{8} Λ ^{2} r ^{4} \cos ( 6 \theta ) + 240 a ^{8} Λ ^{2} r ^{4} \cos ( 8 \theta ) +8400 a ^{8} Λ ^{2} r ^{4-} \rm 1152 a ^{6} \cos ( 6 \theta ) -11520 a ^{6} +1280 a ^{6} Λ ^{2} r ^{6} \cos ( 6 \theta ) +12800 a ^{6} Λ ^{2} r ^{6+207360} a ^{4} r ^{2-} \rm 138240 a ^{4} r \mho ^{2+11520} a ^{4} Λ ^{2} r ^{8+16128} a ^{4} \mho ^{4-276480} a ^{2} r ^{4+368640} a ^{2} r ^{3} \mho ^{2+} \rm 6144 a ^{2} Λ ^{2} r ^{10} -104448 a ^{2} r ^{2} \mho ^{4+3} a ^{4} \cos ( 4 \theta ) [ 165 a ^{8} Λ ^{2+960} a ^{6} Λ ^{2} r ^{2+2240} a ^{4} Λ ^{2} r ^{4-} \rm 256 a ^{2} ( 9-10 Λ ^{2} r ^{6)+256} ( 90 r ^{2-60} r \mho ^{2+5} Λ ^{2} r ^{8+7} \mho ^{4)} ] +24 a ^{2} \cos ( 2 \theta ) [ 33 a ^{10} Λ ^{2+} \rm 210 a ^{8} Λ ^{2} r ^{2+560} a ^{6} Λ ^{2} r ^{4-80} a ^{4} ( 9-10 Λ ^{2} r ^{6)+128} a ^{2} ( 90 r ^{2-60} r \mho ^{2+5} Λ ^{2} r ^{8+} \rm 7 \mho ^{4)+256} r ^{2} ( -45 r ^{2+60} r \mho ^{2+Λ} ^{2} r ^{8-17} \mho ^{4)} ] +36864 r ^{6-73728} r ^{5} \mho ^{2+} \rm 2048 Λ ^{2} r ^{12} +43008 r ^{4} \mho ^{4} \} ÷ \{ 12 [ a ^{2} \cos ( 2 \theta ) + a ^{2+2} r ^{2} ] ^{6} \} .

See also

• Kerr–Newman metric
• De Sitter–Schwarzschild metric
• de Sitter space
• de Sitter universe
• Anti-de Sitter space
• AdS/CFT correspondence

Notes and References

1. Stuchlik. Bao. Østgaard . Hledik. Kerr-Newman-de Sitter black holes with a restricted repulsive barrier of equatorial photon motion. Physical Review D. 2008 . 58 . 084003. 10.1088/0264-9381/17/21/312. 0803.2539 . 250888923 .
2. Griffiths. Podolsky. Exact spacetimes in Einstein's General Relativity. Cambridge University Press, Cambridge Monographs in Mathematical Physics. 2009 . 10.1017/CBO9780511635397. 9780521889278 .
3. Garnier. Arthur. Motion equations in a Kerr-Newman-de Sitter spacetime. Classical and Quantum Gravity. 2023 . 40 . 13 . 10.1088/1361-6382/accbfe. 2307.04073. 258085066 .
4. Kraniotis. 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. 2014 . 46 . 11 . 1818 . 10.1007/s10714-014-1818-8. 1401.7118 . 2014GReGr..46.1818K . 125791608 .
5. Bhattacharya. Kerr-de Sitter spacetime, Penrose process and the generalized area theorem. Physical Review D. 2018 . 97 . 8 . 084049 . 10.1103/PhysRevD.97.084049. 1710.00997 . 2018PhRvD..97h4049B . 119187422 .
6. Stuchlik. Bao. Østgaard. Null Hypersurfaces in Kerr-Newman-AdS Black Hole and Super-Entropic Black Hole Spacetimes. Classical and Quantum Gravity. 2021 . 38 . 4 . 045018 . 10.1088/1361-6382/abd3e0. 2007.04354 . 2021CQGra..38d5018I . 220424477 .
7. Gaur. Visser. Black holes embedded in FLRW cosmologies. 2023 . gr-qc . 2308.07374.
8. Andrew Hamilton: Black hole Penrose diagrams (JILA Colorado)
9. https://www.researchgate.net/figure/The-Penrose-Carter-diagram-of-the-maximal-Kerr-de-Sitter-spacetime-along-the-symmetry_fig2_338904795 Figure 2
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