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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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  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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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)  Bild

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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) Bild

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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)   Bild

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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  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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Nichtäquatorialer und retrograder Photonenorbit um eine mit a=0.9 und ℧=0.9 geladene nackte Singularität, konstanter Boyer Lindquist Radius    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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Escape of neutral and charged particles from inside the ergosphere, startposition above the outer horizon. v local = v escape    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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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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Simon Tyran aka Симон Тыран @ vk || wikipedia || stackexchange || wolfram

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Yukterez
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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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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, local 3-velocity and coordinate celerity:

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

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

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with the parameter

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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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Axial and coaxial radius of gyration:

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Axial and coaxial circumference:

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

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


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