Kerr Newman Metric

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Yukterez
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Kerr Newman Metric

Beitragvon Yukterez » Do 12. Apr 2018, 06:28

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Kerr-Newman, second order differential equations of motion for a charged particle and photons. Animations by Simon Tyran, Vienna (Yukterez)
Bild This is the english version.   Bild Deutschsprachige Version auf kerr.newman.yukterez.net und Yukipedia.
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Shadow of an extremal Kerr-Newman BH with a²+℧²=M², Angle of view: edge on. For other parameters see here. Raw material: Commons.Bild

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Zoom with overlayed ergoshphere and horizon surfaces. Comparison with an uncharged Kerr black hole: click hereBild

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Accretion disk around a spinning and charged BH with a=0.95, ℧=0.3, inner radius ri=isco, outer radius ra=10, viewpoint=89°    
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Gravitylensed surface of a charged and rotating body with the same surface as the earth. a=0.95, ℧=0.3, r=1.01r+, viewpoint=89°    
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Retrograde orbit of a charged particle (q=1) around a BH with spin & charge a=√¾ & ℧=⅓. v0 & i0: local initial velocity & inclination  Bild

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Prograde orbit of a neutral testparticle around a spinning and electrically charged black hole with spin a=0.9 and charge ℧=0.4  Bild

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Prograde orbit of a negatively charged testparticle around a spiining and positively charged black hole with the same parameters as above  Bild

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Nonequatorial and retrograde photon orbit around a spinning (a=½) and charged (℧=½) black hole, constant Boyer Lindquist radius    Bild

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Polar photon orbit around a spinning (a=0.5) and charged (℧=0.75) naked singularity, constant Boyer Lindquist radius    Bild

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Polar orbit (Lz=0) of a positively charged testpaticle (q=⅓) around a positively charged and spinning black hole (℧=a=0.7)   Bild

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Free fall of a neutral testparticle around a rotating and charged naked singularity with spin a=1.5 and electric charge ℧=0.4  Bild

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Geodesic orbit around a naked Kerr Newman ringsingularity with the same spin and charge parameters as in the last example    Bild

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Nonequatorial and retrograde photon orbit around a naked singularity spinning with a=0.9 and charged with ℧=0.9    Bild

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Retrograde photon orbit around a naked singularity (a=0.99, ℧=0.99). Local equatorial inclination angle: -2.5rad=-143.23945°    Bild

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Stationary photon orbit (E=0) around a ringsingularity (a=½, ℧=1). Except at r=1, θ=90° v framedrag is <c everywhere, therefore no ergospheres.    Bild

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Equatorial retrograde photon orbit, singularity at r=0→R=√(r²+a²)=a=½. Ergoring (green) at r=1, turning points at r=0.8 and r=1.3484    Bild

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Retrograde photon orbit of the third kind insinde a spinning (a=0.75) and charged (℧=0.5) black hole, constant BL radius    Bild

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Orbit of a negatively charged particle inside a positively charged Reissner Nordström black hole (also see Dokuchaev, Fig. 1)    Bild
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Simon Tyran aka Симон Тыран @ minds || vk || 1984 || wikipedia || stackexchange || wolframBild

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Yukterez
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Beiträge: 207
Registriert: Mi 21. Okt 2015, 02:16

Kerr Newman Metric

Beitragvon Yukterez » Do 12. Apr 2018, 21:02

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Line element in Boyer Lindquist coordinates, metric signature (+,-,-,-):

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Shorthand terms:

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with the spin parameter â=Jc/G/M or in dimensionless units a=â/M, the specific electric charge Ω=·√(K/G) and the dimensionless charge ℧=Ω/M. Here we use the units G=M=c=K=1 with lengths in GM/c² and times in GM/c³. The relation between the mass-equivalent of the total energy and the irreducible mass Mirr is

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Effective mass:

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For testparticles with mass μ=-1, for photons μ=0. The specific charge of the test particle is q. Transformation rule for co- and contravariant indices (superscripted letters are not powers but indices):

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Co- and contravariant metric:

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Elektromagnetic potential:

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Covariant elektromagnetic tensor:

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Contravariant Maxwell-tensor:

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Proper time derivatives of the coordinates:

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Equations of motion:

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Canonical 4-momentum:

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Total time dilation of a neutral particle:

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Additional time dilation for a charged particle:

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Total time dilation of a charged particle:

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Relation between the first time derivatives and the covariant momentum components:

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Relation between the first time derivatives and the local three-velocity components:

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with the contracted electromagnetic potential

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The radial effective potential which defines the turning points at its zero roots is

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and the latitudinal potential

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with the parameter

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For the 3-velocity relative to a local ZAMO we take E and solve for v:

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Radial escape velocity for a neutral particle:

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For the escape velocity of a charged particle with zero orbital angular momentum we set E=1 and solve for v:

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1. Constant of motion: Total energy E=-pt

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2. Constant of motion: axial angular momentum Lz=+pφ

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3. Constant of motion: Carter's constant

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with the coaxial component of the angular momentum, which itself is not a constant:

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The azimuthal and latitudinal impact parameters are

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Gravitative time dilatation of a corotating neutral ZAMO, infinite at the horizon:

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Time dilation of a stationary particle, infinite at the ergosphere:

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Frame-dragging angular velocity observed at infinity:

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Local frame-dragging velocity relative to the fixed stars (c at the ergosphere):

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with the relation

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Axial and coaxial radius of gyration:

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Axial and coaxial circumference:

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The radii of the equatorial photon orbits are given implicitly by:

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The innermost stable orbit (ISCO) of a neutral particle is given by the polynomial:

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Radial coordinates of the horizons and ergospheres:

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Cartesian projection:

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Code: original / backup

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images and animations by Simon Tyran, Vienna (Yukterez) - reuse permitted under the Creative Commons License CC BY-SA 4.0
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Simon Tyran aka Симон Тыран @ minds || vk || 1984 || wikipedia || stackexchange || wolframBild


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