## Kerr Newman Metric

English Version
Yukterez
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### Kerr Newman Metric Kerr-Newman, second order differential equations of motion for a charged particle and photons. Animations by Simon Tyran, Vienna (Yukterez) This is the english version. Deutschsprachige Version auf kerr.newman.yukterez.net und Yukipedia.  ←
Shadow of an extremal Kerr-Newman BH with a²+℧²=M², Angle of view: edge on. For other parameters see here. Raw material: Commons.  ←
Zoom with overlayed ergoshphere and horizon surfaces. Comparison with an uncharged Kerr black hole: click here  ←
Accretion disk around a spinning and charged BH with a=0.95, ℧=0.3, ri=isco, ra=10, viewpoint=89°. Earth surface at r=1.01r+.  ←
Retrograde orbit of a charged particle (q=1) around a BH with spin & charge a=√¾ & ℧=⅓. v0 & i0: local initial velocity & inclination  ←
Prograde orbit of a neutral testparticle around a spinning and electrically charged black hole with spin a=0.9 and charge ℧=0.4  ←
Prograde orbit of a negatively charged testparticle around a spiining and positively charged black hole with the same parameters as above  ←
Nonequatorial and retrograde photon orbit around a spinning (a=½) and charged (℧=½) black hole, constant Boyer Lindquist radius  ←
Polar photon orbit around a spinning (a=0.5) and charged (℧=0.75) naked singularity, constant Boyer Lindquist radius  ←
Polar orbit (Lz=0) of a positively charged testpaticle (q=⅓) around a positively charged and spinning black hole (℧=a=0.7)  ←
Plunge orbit of a negative particle (q=-⅓), BH like above. The nonpolar axial velocity for q<0 is positive for Lz=0 due to electric force.  ←
Free fall of a neutral testparticle around a rotating and charged naked singularity with spin a=1.5 and electric charge ℧=0.4  ←
Geodesic orbit around a naked Kerr Newman ringsingularity with the same spin and charge parameters as in the last example  ←
Nonequatorial and retrograde photon orbit around a naked singularity spinning with a=0.9 and charged with ℧=0.9  ←
Retrograde photon orbit around a naked singularity (a=0.99, ℧=0.99). Local equatorial inclination angle: -2.5rad=-143.23945°  ←
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.  ←
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  ←
Retrograde photon orbit of the third kind insinde a spinning (a=0.75) and charged (℧=0.5) black hole, constant BL radius  ←
Retrograde (but due to superluminal frame dragging apparently prograde) photon orbit inside a BH with a=℧=0.707 at r=0.8  ←
Orbit of a negatively charged particle inside a positively charged Reissner Nordström black hole (also see Dokuchaev, Fig. 1)  Simon Tyran aka Симон Тыран @ minds || vk || wikipedia || stackexchange || wolfram Yukterez
Beiträge: 221
Registriert: Mi 21. Okt 2015, 02:16

### Kerr Newman Metric Line element in Boyer Lindquist coordinates, metric signature (+,-,-,-): Shorthand terms: 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 Effective mass: 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): Co- and contravariant metric: Elektromagnetic potential: Covariant elektromagnetic tensor: Contravariant Maxwell-tensor: with the Christoffel symbols: the second proper time derivatives of the coordinates are: Equations of motion:    Canonical 4-momentum, local 3-velocity and 1st proper time derivatives: From the line element: we get the total time dilation of a neutral particle: Total time dilation of a charged particle: Relation between the first time derivatives and the covariant momentum components:  Relation between the first time derivatives and the local three-velocity components:   with the contracted electromagnetic potential The radial effective potential which defines the turning points at its zero roots is and the latitudinal potential with the parameter For the 3-velocity relative to a local ZAMO we take E and solve for v: or divide the gravitational time dilation by the total time dilation to get the inverse of the Gamma factor: Radial escape velocity for a neutral particle: For the escape velocity of a charged particle with zero orbital angular momentum we set E=1 and solve for v: 1. Constant of motion: Total energy E=-pt 2. Constant of motion: axial angular momentum Lz=+pφ 3. Constant of motion: Carter's constant with the coaxial component of the angular momentum, which itself is not a constant:  The azimuthal and latitudinal impact parameters are Gravitative time dilatation of a corotating neutral ZAMO, infinite at the horizon: Time dilation of a stationary particle, infinite at the ergosphere: Frame-dragging angular velocity observed at infinity: Local frame-dragging velocity relative to the fixed stars (c at the ergosphere): with the relation Axial and coaxial radius of gyration: Axial and coaxial circumference: The radii of the equatorial photon orbits are given implicitly by: The innermost stable orbit (ISCO) of a neutral particle is given by: Radial coordinates of the horizons and ergospheres: Cartesian projection: r in relation to x,y,z:    Simon Tyran aka Симон Тыран @ minds || vk || wikipedia || stackexchange || wolfram 