## Kerr Metric

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### 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=10, θ=70°

The same black hole like in the example above, but with the observer at r=100. For more details click on the images.

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.

In Doran coordinates (т,r,θ,φ) where the coordinate time is the proper time of local free fallers from infinity the metric tensor becomes

With a=0 Doran coordinates reduce to Gullstrand Painlevé 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 component of motion (dθ/dτ):

Derivative of the poloidial angular momentum (dpθ/dτ):

Axial angular momentum:

Derivative of the axial component of motion (dФ/dτ):

Axial angular momentum derivative (pФ/dτ):

Longitudinal 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 @ minds || vk || wikipedia || stackexchange || wolfram