This is the english version. 
Deutschsprachige Version auf kerr.yukterez.net und Yukipedia.
Shadow and surfaces of a spinning black hole (a=1), click to enlarge (png). Zoom out: [-], Contours: ƒ, Raytracing Code: ▤
Shadow and surfaces of a spinning black hole (a=0.99), Animation parameter: polar angle (θ=1°..90°). Slower: ⎆
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°
Retrograde orbit of a particle around a spinning black hole (a=0.95), coordinates: cartesian
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:
Shorthand terms:
Covariant metric coeffizients:
Contravariant components (superscripted letters are not powers, but indices):
The dimensionless spin parameter is a=Jc/G/M². Transformation into cartesian coordinates:
Line element in Boyer Lindquist coordinates:
Metric tensor (t,r,θ,Ф):
With a=0 Boyer Lindquist coordinates reduce to classical Schwarzschild coordinates.
With the transformation:
where T is a finkelsteinlike time coordinate (radially infalling photons move with dr/dt=1) and ψ the flattened azimuthal angle:
the metric in Kerr Schild coordinates (T,r,θ,ψ) is:
With a=0 Kerr Schild coordinates reduce to Eddington Finkelstein coordinates.
Equations of motion in Boyer Lindquist coordinates
Canonical four-momentum components:
Coordinate time by proper time (dt/dτ):
First proper time derivative of the radial coordinate (dr/dτ):
Radial momentum derivative:
Radial momentum:
Derivative of the poloidial (longitudinal) component of motion (dθ/dτ):
Derivative of the poloidial angular momentum (dpθ/dτ):
Axial (latitudinal) angular momentum:
Derivative of the axial component of motion (dФ/dτ):
Axial angular momentum derivative (pФ/dτ):
Axial component of the angular momentum:
Constant of motion, Carter's constant:
Constant of motion, Carter k:
Constant of motion, total energy:
Constant of motion, axial angular momentum:
Local 3-velocity component along the r-axis:
Local 3-velocity component along the θ-axis:
Local 3-velocity component along the Ф-axis:
Local 3-velocity, total:
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
Shapirodelayed and frame dragged velocity as observed at infinity:
The radial effective potential which defines the turning points is:
Radial escape velocity:
Frame-Dragging angular velocity oberserved at infinity (dФ/dt):
Delayed Frame-Dragging transverse velocity at the equator of the outer horizon:
with the horizons and ergospheres (solution for r at Δ=0 and gtt=0):
r and θ dependend delayed Frame-Dragging transverse velocities:
at the equatorialen plane at θ=π/2:
r und θ dependend local Frame-Dragging transverse velocities (greater than c inside of the ergosphere):
at the equatorialen plane at θ=π/2:
Cartesian projection of the Frame-Dragging transverse velocity:
at the equatorialen plane at θ=π/2:
Gravitational time dilation component relative to a ZAMO (dt/dτ):
Axial and coaxial radius of gyration:
Axial and coaxial circumference:
The innermost stable orbit (ISCO) is at
with the shorthand terms
For images and animations see the german version of this site.
