## Kerr Metric

English Version
Yukterez
Beiträge: 254
Registriert: Mi 21. Okt 2015, 02:16

### Kerr Metric  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):   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:   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. Simon Tyran aka Симон Тыран @ vk || minds || parler || gab || wikipedia || stackexchange || wolfram 