Kerr Newman Metric (yukterez.net Backup)

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.


  ←
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+.    
 

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

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

  ←
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 (violet) at r=1, turning points at r=0.8 and r=1.3484      

  ←
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:



Radial momentum component:



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 radius:



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:



Radial local velocity:



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:



Simulator code: original / backup, other coordinates: click here

images and animations by Simon Tyran, Vienna (Yukterez) - reuse permitted under the Creative Commons License CC BY-SA 4.0