![Bild](http://org.yukterez.net/strich.gif)
Kerr-Newman, second order differential equations of motion for a charged particle and photons. Animations by Simon Tyran, Vienna (Yukterez)
![Bild](http://org.yukterez.net/us+br.png)
![Bild](http://org.yukterez.net/de+at.png)
![Bild](http://org.yukterez.net/strich.gif)
![Bild](http://org.yukterez.net/kerr.newman.accretion.disk.89.degree.AB.gif)
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+.
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kretschmann.invariant.kerr.newman.cartesian.png)
Kretschmann scalar, cartesian projektion. The areas around the poles are curved negatively, and those around the equator positively.
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://png.yukterez.net/kerr.newman.black.hole,magnetic+electric.field.lines.png)
Magnetic (left) and electric (right) field lines, cartesian projektion, view=90° (edge on), plot range=±5.
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.orbit.of.a.charged.particle.png)
Retrograde orbit of a charged particle (q=1) around a BH with spin & charge a=√¾ & ℧=⅓. v0 & i0: local initial velocity & inclination
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.orbit.40deg.A.png)
Prograde orbit of a neutral testparticle around a spinning and electrically charged black hole with spin a=0.9 and charge ℧=0.4
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.orbit.40deg.B.png)
Prograde orbit of a negatively charged testparticle around a spiining and positively charged black hole with the same parameters as above
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr-newman-photon-orbit-1.png)
Nonequatorial and retrograde photon orbit around a spinning (a=½) and charged (℧=½) black hole, constant Boyer Lindquist radius
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.photon.orbit.Lz0.png)
Polar photon orbit around a spinning (a=0.5) and charged (℧=0.75) black hole, constant Boyer Lindquist radius
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.charged.particle.orbit.Lz0.png)
Polar orbit (Lz=0) of a positively charged testpaticle (q=⅓) around a positively charged and spinning black hole (℧=a=0.7)
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.negative.charged.particle.plunge.orbit.Lz0.png)
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.
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.naked.singularity.orbit.2.png)
Free fall of a neutral testparticle around a rotating and charged naked singularity with spin a=1.5 and electric charge ℧=0.4
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman,naked.singularity.geodesic.png)
Geodesic orbit around a naked Kerr Newman ringsingularity with the same spin and charge parameters as in the last example
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr-newman-photon-orbit-2.png)
Nonequatorial and retrograde photon orbit around a naked singularity spinning with a=0.9 and charged with ℧=0.9
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr-newman,naked-singularity,photon-orbit-3.png)
Retrograde photon orbit around a naked singularity (a=0.99, ℧=0.99). Local equatorial inclination angle: -2.5rad=-143.23945°
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.naked.ring.singularity.zero.energy.retrograde.stationary.photon.orbit.png)
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.
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.newman.photon.orbit.naked.singularity.png)
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
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/orbit.of.the.3rd.kind.reissner.nordstroem.A.png)
Orbit of a negatively charged particle inside a positively charged Reissner Nordström black hole (also see Dokuchaev, Fig. 1)
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/strich.gif)
![Bild](http://org.yukterez.net/strich.gif)
Line element in Boyer Lindquist coordinates, metric signature (+,-,-,-):
![Bild](http://org.yukterez.net/kerr.newman.latex/1.gif)
Shorthand terms:
![Bild](http://org.yukterez.net/kerr.newman.latex/2.gif)
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=ℏ=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
![Bild](http://org.yukterez.net/kerr.newman.latex/61.gif)
Effective mass:
![Bild](http://org.yukterez.net/kerr.newman.latex/141.gif)
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):
![Bild](http://org.yukterez.net/kerr.newman.latex/60.gif)
Co- and contravariant metric:
![Bild](http://org.yukterez.net/kerr.newman.latex/3.gif)
Electromagnetic potential:
![Bild](http://org.yukterez.net/kerr.newman.latex/36.gif)
Covariant electromagnetic tensor:
![Bild](http://org.yukterez.net/kerr.newman.latex/403.gif)
Contravariant Maxwell-tensor:
![Bild](http://org.yukterez.net/kerr.newman.latex/402.gif)
Magnetic field lines:
![Bild](http://org.yukterez.net/kerr.newman.latex/811.gif)
Electric field lines:
![Bild](http://org.yukterez.net/kerr.newman.latex/812.gif)
with the term
![Bild](http://org.yukterez.net/kerr.newman.latex/813.gif)
With the Christoffel symbols:
![Bild](http://org.yukterez.net/geodesics.latex/63.gif)
the second proper time derivatives of the coordinates are:
![Bild](http://org.yukterez.net/kerr.newman.latex/37.png)
Equations of motion:
![Bild](http://org.yukterez.net/kerr.newman.latex/43.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/44.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/45.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/46.gif)
Canonical 4-momentum, local 3-velocity and 1st proper time derivatives:
![Bild](http://org.yukterez.net/kerr.newman.latex/164.gif)
From the line element:
![Bild](http://org.yukterez.net/geodesics.latex/68.png)
we get the total time dilation of a neutral particle:
![Bild](http://org.yukterez.net/kerr.newman.latex/12.gif)
Total time dilation of a charged particle:
![Bild](http://org.yukterez.net/kerr.newman.latex/42.gif)
Relation between the first time derivatives and the covariant momentum components:
![Bild](http://org.yukterez.net/kerr.newman.latex/705.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/304.gif)
Relation between the first time derivatives and the local three-velocity components:
![Bild](http://org.yukterez.net/kerr.newman.latex/201.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/202.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/173.gif)
with the contracted electromagnetic potential
![Bild](http://org.yukterez.net/kerr.newman.latex/153.gif)
The radial effective potential which defines the turning points at its zero roots is
![Bild](http://org.yukterez.net/kerr.newman.latex/121.gif)
and the latitudinal potential
![Bild](http://org.yukterez.net/kerr.newman.latex/702.gif)
with the parameter
![Bild](http://org.yukterez.net/kerr.newman.latex/703.gif)
For the 3-velocity relative to a local ZAMO we take E and solve for v:
![Bild](http://org.yukterez.net/kerr.newman.latex/104.gif)
or divide the gravitational time dilation by the total time dilation to get the inverse of the Gamma factor:
![Bild](http://org.yukterez.net/kerr.newman.latex/166.gif)
Radial escape velocity for a neutral particle:
![Bild](http://org.yukterez.net/kerr.newman.latex/706.gif)
For the escape velocity of a charged particle with zero orbital angular momentum we set E=1 and solve for v:
![Bild](http://org.yukterez.net/kerr.newman.latex/105.gif)
1. Constant of motion: Total energy E=-pt
![Bild](http://www.yukterez.net/org/kerr.newman.latex/100.gif)
2. Constant of motion: axial angular momentum Lz=+pφ
![Bild](http://org.yukterez.net/kerr.newman.latex/155.gif)
3. Constant of motion: Carter's constant
![Bild](http://org.yukterez.net/kerr.newman.latex/174.gif)
with the coaxial component of the angular momentum, which itself is not a constant:
![Bild](http://org.yukterez.net/kerr.newman.latex/171.gif)
Radial momentum component:
![Bild](http://org.yukterez.net/kerr.newman.latex/162.gif)
The azimuthal and latitudinal impact parameters are
![Bild](http://org.yukterez.net/kerr.newman.latex/125.gif)
Gravitative time dilatation of a corotating neutral ZAMO, infinite at the horizon:
![Bild](http://org.yukterez.net/kerr.newman.latex/27.gif)
Time dilation of a stationary particle, infinite at the ergosphere:
![Bild](http://org.yukterez.net/kerr.newman.latex/65.gif)
Frame-dragging angular velocity observed at infinity:
![Bild](http://org.yukterez.net/kerr.newman.latex/11.gif)
Local frame-dragging velocity relative to the fixed stars (c at the ergosphere):
![Bild](http://org.yukterez.net/kerr.newman.latex/130.gif)
with the relation
![Bild](http://org.yukterez.net/kerr.newman.latex/601.gif)
Axial and coaxial radius of gyration:
![Bild](http://org.yukterez.net/kerr.latex/28.gif)
Axial and coaxial circumference:
![Bild](http://org.yukterez.net/kerr.latex/29.gif)
The radii of the equatorial photon orbits are given implicitly by:
![Bild](http://org.yukterez.net/kerr.newman.latex/124.gif)
The innermost stable orbit (ISCO) of a neutral particle is given by:
![Bild](http://org.yukterez.net/kerr.newman.latex/157.gif)
Radial coordinates of the horizons and ergospheres:
![Bild](http://org.yukterez.net/kerr.newman.latex/16.gif)
Cartesian projection:
![Bild](http://org.yukterez.net/kerr.newman.latex/67.gif)
r in relation to x,y,z:
![Bild](http://org.yukterez.net/kerr.newman.latex/167.gif)
Cartesian radius:
![Bild](http://org.yukterez.net/kerr.newman.latex/r,cartesian.gif)
Cartesian latitude:
![Bild](http://org.yukterez.net/kerr.newman.latex/theta,cartesian.gif)
Hawking temperature (with surface gravity κ⁺):
![Bild](https://org.yukterez.net//kerr.newman.latex/176.gif)
![Bild](http://org.yukterez.net/strich.gif)
![Bild](http://org.yukterez.net/strich.gif)
Transformation rule from Boyer Lindquist to Doran Raindrop:
![Bild](http://gif.yukterez.net/kerr.newman.latex/bl.to.dr.gif)
Relative to a local ZAMO the river of space has the negative radial escape velocity:
![Bild](http://gif.yukterez.net/kerr.newman.latex/311.png)
Metric tensor in Doran Raindrop coordinates, covariant:
![Bild](http://org.yukterez.net/kerr.newman.latex/bh1.gif)
Contravariant metric tensor:
![Bild](http://org.yukterez.net/kerr.newman.latex/bh2.gif)
Electromagnetic vector potential:
![Bild](http://org.yukterez.net/kerr.newman.latex/bh3.gif)
Covariant Maxwell-tensor:
![Bild](http://org.yukterez.net/kerr.newman.latex/bh4.gif)
Contravariant electromagnetic tensor:
![Bild](http://org.yukterez.net/kerr.newman.latex/805.gif)
Total velocity relative to a local raindrop:
![Bild](http://org.yukterez.net/kerr.newman.latex/104.png)
Radial local velocity:
![Bild](http://org.yukterez.net/kerr.newman.latex/vrdoran.gif)
Latitudinal local velocity:
![Bild](http://org.yukterez.net/kerr.newman.latex/vthetadoran.gif)
Longitudinal local velocity:
![Bild](http://png.yukterez.net/kerr.newman.latex/vphi,doran.png)
Coordinate time differential:
![Bild](http://org.yukterez.net/kerr.newman.latex/dt1doran.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/dt2doran.gif)
![Bild](http://org.yukterez.net/kerr.newman.latex/dt3doran.gif)
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
![Bild](http://org.yukterez.net/strich.gif)
![Bild](http://org.yukterez.net/strich.gif)
Horizons and ergospheres in pseudospherical (r,θ,φ) and cartesian (x,y,z) coordinates:
![Bild](http://org.yukterez.net/ksf860.gif)
Simulator code: click here, other coordinates: click here
![Bild](http://org.yukterez.net/strich.gif)
images and animations by Simon Tyran, Vienna (Yukterez) - reuse permitted under the Creative Commons License CC BY-SA 4.0