Notizblock

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




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