Notizblock

Line element in Boyer Lindquist coordinates, metric signature (+,-,-,-):



Shorthand terms:



with the spin parameter â=Jc/G/M or in dimensionless units a=â/M, the specific electric charge Ω=⚼·√(K/G) and the dimensionless charge ℧=Ω/M. Here we use the units G=M=c=K=ℏ=1 with lengths in GM/c² and times in GM/c³. The relation between the mass-equivalent of the total energy and the irreducible mass Mirr is



Effective mass:



For testparticles with mass μ=-1, for photons μ=0. The specific charge of the test particle is q. Transformation rule for co- and contravariant indices (superscripted letters are not powers but indices):



Co- and contravariant metric:



Electromagnetic potential:



Covariant electromagnetic tensor:



Contravariant Maxwell-tensor:



Magnetic field lines:



Electric field lines:



with the term



With the Christoffel symbols:



the second proper time derivatives of the coordinates are:



Equations of motion:









Canonical 4-momentum, local 3-velocity and 1st proper time derivatives:



From the line element:



we get the total time dilation of a neutral particle:



Total time dilation of a charged particle:



Relation between the first time derivatives and the covariant momentum components:





Relation between the first time derivatives and the local three-velocity components:

   



with the contracted electromagnetic potential



The radial effective potential which defines the turning points at its zero roots is



and the latitudinal potential



with the parameter



For the 3-velocity relative to a local ZAMO we take E and solve for v:



or divide the gravitational time dilation by the total time dilation to get the inverse of the Gamma factor:



Radial escape velocity for a neutral particle:



For the escape velocity of a charged particle with zero orbital angular momentum we set E=1 and solve for v:



1. Constant of motion: Total energy E=-pt



2. Constant of motion: axial angular momentum Lz=+pφ



3. Constant of motion: Carter's constant



with the coaxial component of the angular momentum, which itself is not a constant:



Radial momentum component:



The azimuthal and latitudinal impact parameters are



Gravitative time dilatation of a corotating neutral ZAMO, infinite at the horizon:



Time dilation of a stationary particle, infinite at the ergosphere:



Frame-dragging angular velocity observed at infinity:



Local frame-dragging velocity relative to the fixed stars (c at the ergosphere):



with the relation



Axial and coaxial radius of gyration:



Axial and coaxial circumference:



The radii of the equatorial photon orbits are given implicitly by:



The innermost stable orbit (ISCO) of a neutral particle is given by:



Radial coordinates of the horizons and ergospheres:



Cartesian projection:



r in relation to x,y,z:



Cartesian radius:



Cartesian latitude:



Hawking temperature (with surface gravity κ⁺):

Transformation rule from Boyer Lindquist to Doran Raindrop:



Relative to a local ZAMO the river of space has the negative radial escape velocity:



Metric tensor in Doran Raindrop coordinates, covariant:



Contravariant metric tensor:



Electromagnetic vector potential:



Covariant Maxwell-tensor:



Contravariant electromagnetic tensor:



Total velocity relative to a local raindrop:



Radial local velocity:



Latitudinal local velocity:



Longitudinal local velocity:



Coordinate time differential:







More details: this way, comparison Boyer Lindquist with Raindrop Doran (animation and plots): click, other coordinates: geodesics.yukterez.net, view from the inside of a black hole: click

Horizons and ergospheres in pseudospherical (r,θ,φ) and cartesian (x,y,z) coordinates:



Simulator code: click here, other coordinates: click here