![Bild](http://org.yukterez.net/strich.gif)
![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/relativistic.raytracer/kerr,shadow.vs.surfaces.gif)
Shadow and surfaces of a spinning black hole (a=1), click to enlarge (png). Zoom out: [-], Contours: ƒ, Raytracing Code: ▤
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://cloud.yukterez.net/relativistic.raytracer/theta-kerr.gif)
Shadow and surfaces of a spinning black hole (a=0.99), Animation parameter: polar angle (θ=1°..90°). Slower: ⎆
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/bh/kerr.accretion.disk.vs.shadow.raytracing.4.gif)
Accretion disk with inner radius ri=isco and outer radius ra=7 around a BH with a=0.95, observer at r=100, θ=70°
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://yukterez.net/org/kerr19.gif)
Retrograde orbit of a particle around a spinning black hole (a=0.95), coordinates: cartesian
![Bild](http://org.yukterez.net/strich.gif)
Here we use natural units of G=M=c=1, so lengths are in GM/c² and times in GM/c³. The metric signature is time-positive (+,-,-,-). a is the spin parameter (for black holes 0≤a≤M), M the mass equivalent of the total energy of the black hole, and Mirr its irreducible mass:
![Bild](http://org.yukterez.net/kerr.latex/217a.gif)
![Bild](http://org.yukterez.net/strich.gif)
Shorthand terms:
![Bild](http://org.yukterez.net/kerr.latex/4.gif)
Covariant metric coeffizients:
![Bild](http://org.yukterez.net/kerr.latex/5a.gif)
Contravariant components (superscripted letters are not powers, but indices):
![Bild](http://org.yukterez.net/kerr.latex/6.gif)
The dimensionless spin parameter is a=Jc/G/M². Transformation into cartesian coordinates:
![Bild](http://org.yukterez.net/kerr.latex/7.gif)
Line element in Boyer Lindquist coordinates:
![Bild](http://org.yukterez.net/kerr.latex/8a.gif)
Metric tensor (t,r,θ,Ф):
![Bild](http://org.yukterez.net/kerr.latex/406.gif)
With a=0 Boyer Lindquist coordinates reduce to classical Schwarzschild coordinates.
![Bild](http://org.yukterez.net/strich.gif)
With the transformation:
![Bild](http://org.yukterez.net/kerr.latex/140.gif)
where T is a finkelsteinlike time coordinate (radially infalling photons move with dr/dt=1) and ψ the flattened azimuthal angle:
![Bild](http://org.yukterez.net/kerr.latex/10.gif)
the metric in Kerr Schild coordinates (T,r,θ,ψ) is:
![Bild](http://org.yukterez.net/kerr.latex/36.gif)
With a=0 Kerr Schild coordinates reduce to Eddington Finkelstein coordinates.
![Bild](http://org.yukterez.net/strich.gif)
Equations of motion in Boyer Lindquist coordinates
![Bild](http://org.yukterez.net/strich.gif)
Canonical four-momentum components:
![Bild](http://yukterez.net/org/kerr.latex/310.gif)
Coordinate time by proper time (dt/dτ):
![Bild](http://org.yukterez.net/kerr.latex/12.gif)
First proper time derivative of the radial coordinate (dr/dτ):
![Bild](http://org.yukterez.net/kerr.latex/13.gif)
Radial momentum derivative:
![Bild](http://org.yukterez.net/kerr.latex/14.gif)
Radial momentum:
![Bild](http://org.yukterez.net/kerr.latex/15.gif)
Derivative of the poloidial (longitudinal) component of motion (dθ/dτ):
![Bild](http://org.yukterez.net/kerr.latex/16.gif)
Derivative of the poloidial angular momentum (dpθ/dτ):
![Bild](http://org.yukterez.net/kerr.latex/17.gif)
Axial (latitudinal) angular momentum:
![Bild](http://org.yukterez.net/kerr.latex/312.gif)
Derivative of the axial component of motion (dФ/dτ):
![Bild](http://org.yukterez.net/kerr.latex/19.gif)
Axial angular momentum derivative (pФ/dτ):
![Bild](http://org.yukterez.net/kerr.latex/20.gif)
Axial component of the angular momentum:
![Bild](http://org.yukterez.net/kerr.latex/21.gif)
Constant of motion, Carter's constant:
![Bild](http://org.yukterez.net/kerr.latex/22.gif)
Constant of motion, Carter k:
![Bild](http://org.yukterez.net/kerr.latex/23.gif)
Constant of motion, total energy:
![Bild](http://org.yukterez.net/kerr.latex/24.gif)
Constant of motion, axial angular momentum:
![Bild](http://org.yukterez.net/kerr.latex/25.gif)
Local 3-velocity component along the r-axis:
![Bild](http://org.yukterez.net/kerr.latex/30.gif)
Local 3-velocity component along the θ-axis:
![Bild](http://org.yukterez.net/kerr.latex/31.gif)
Local 3-velocity component along the Ф-axis:
![Bild](http://org.yukterez.net/kerr.latex/32.gif)
Local 3-velocity, total:
![Bild](http://org.yukterez.net/kerr.latex/216.gif)
For massive testparticles μ=-1 and for photons μ=-0. δ is the inclination angle. With α as the vertical launch anglel the components of the local velocity (relative to a ZAMO) are
![Bild](http://org.yukterez.net/kerr.latex/33.gif)
Shapirodelayed and frame dragged velocity as observed at infinity:
![Bild](http://org.yukterez.net/kerr.latex/34.gif)
The radial effective potential which defines the turning points is:
![Bild](http://cache.yukterez.net/kerr.newman.latex/119.gif)
Radial escape velocity:
![Bild](http://org.yukterez.net/kerr.latex/35.gif)
![Bild](http://org.yukterez.net/strich.gif)
Frame-Dragging angular velocity oberserved at infinity (dФ/dt):
![Bild](http://org.yukterez.net/kerr.latex/26.gif)
Delayed Frame-Dragging transverse velocity at the equator of the outer horizon:
![Bild](http://org.yukterez.net/kerr.latex/300.gif)
with the horizons and ergospheres (solution for r at Δ=0 and gtt=0):
![Bild](http://org.yukterez.net/kerr.latex/101.gif)
r and θ dependend delayed Frame-Dragging transverse velocities:
![Bild](http://org.yukterez.net/kerr.latex/301.gif)
at the equatorialen plane at θ=π/2:
![Bild](http://org.yukterez.net/kerr.latex/302.gif)
r und θ dependend local Frame-Dragging transverse velocities (greater than c inside of the ergosphere):
![Bild](http://org.yukterez.net/kerr.latex/303.gif)
at the equatorialen plane at θ=π/2:
![Bild](http://org.yukterez.net/kerr.latex/304.gif)
Cartesian projection of the Frame-Dragging transverse velocity:
![Bild](http://org.yukterez.net/kerr.latex/305.gif)
at the equatorialen plane at θ=π/2:
![Bild](http://org.yukterez.net/kerr.latex/306.gif)
Gravitational time dilation component relative to a ZAMO (dt/dτ):
![Bild](http://org.yukterez.net/kerr.latex/27.gif)
Axial and coaxial radius of gyration:
![Bild](http://org.yukterez.net/kerr.latex/28.gif)
Axial and coaxial circumference:
![Bild](http://gif.yukterez.net/kerr.latex/29.gif)
The innermost stable orbit (ISCO) is at
![Bild](http://cloud.yukterez.net/kerr.latex/141.gif)
with the shorthand terms
![Bild](http://cloud.yukterez.net/kerr.latex/142.gif)
![Bild](http://org.yukterez.net/strich.gif)
For images and animations see the german version of this site.