## 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.

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Retrograde orbit of a particle with charge q=1 around a black hole with spin a=√¾ and charge ℧=⅓. v0 & i0: initial local velocity & inclination

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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

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Same initial conditions as above (R0=x0=7GM/c², v0=0.4c, i0=39.8056°=atan(5/6)rad), but with a negatively charged testparticle (q=-¼)

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Orbit around a BH with a=√½ und ℧=√½. Particle: q=⅓, Lz=0, Initial conditions: v0=0.57: vr=0, vθ=√(782759/2409750), vφ=-√(6751/96390000)

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Charged particle orbit; a=0.8 und ℧=½. Particle: q=-½, Lz=0 (note that the local φ-velocity is not 0 despite Lz=0 because q is also not 0)

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Orbit of a charged particle with q=½ around a naked singularity with a=¾ and ℧=⅔; colored surfaces: outer and inner ergosphere

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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

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Escape of neutral and charged particles from inside the ergosphere, startposition above the outer horizon. v local = v escape

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Orbit of a negatively charged particle inside a positively charged Reissner Nordström black hole (also see Dokuchaev, Fig. 1)

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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.

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Zoom with overlayed ergoshphere and horizon surfaces. Comparison with an uncharged Kerr black hole: click here  ←
Orbit of a negatively charged particle inside a positively charged Reissner Nordström black hole (also see Dokuchaev, Fig. 1)

Simon Tyran aka Симон Тыран @ vk || wikipedia || stackexchange || wolfram

Yukterez
Beiträge: 196
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

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:

Proper time derivatives of the coordinates:

Equations of motion:

Canonical 4-momentum, local 3-velocity and coordinate celerity:

Total time dilation of a neutral particle:

Additional time dilation for a charged 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:

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

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:

The required local vertical launch angle is π/2 if q=0, and if q≠0:

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

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

with the parameter

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

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

Axial and coaxial radius of gyration:

Axial and coaxial circumference:

frame-dragging angular velocity observed at infinity:

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

Radial coordinates of the horizons and ergospheres:

Cartesian projection:

Code: original / backup

Simon Tyran aka Симон Тыран @ vk || wikipedia || stackexchange || wolfram