## Kerr Newman Metric

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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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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+.

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Kretschmann scalar, cartesian projektion. The areas around the poles are curved negatively, and those around the equator positively.

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Magnetic (left) and electric (right) field lines, cartesian projektion, view=90° (edge on), plot range=±5.

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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

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

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

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

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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)

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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.

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

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

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

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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.

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

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

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:

Electromagnetic potential:

Covariant electromagnetic tensor:

Contravariant Maxwell-tensor:

Magnetic field lines:

Electric field lines:

with the term

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:

Cartesian latitude:

Transformation rule from Boyer Lindquist to Doran Raindrop:

Relative to a local ZAMO the river of space has the negative radial escape velocity:

Metric tensor in Doran Raindrop coordinates, covariant:

Contravariant metric tensor:

Electromagnetic vector potential:

Covariant Maxwell-tensor:

Contravariant electromagnetic tensor:

Total velocity relative to a local raindrop:

Latitudinal local velocity:

Longitudinal local velocity:

Coordinate time differential:

More details: this way, comparison Boyer Lindquist with Raindrop Doran (animation and plots): click, other coordinates: geodesics.yukterez.net, view from the inside of a black hole: click

Horizons and ergospheres in pseudospherical (r,θ,φ) and cartesian (x,y,z) coordinates: