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English: Kerr-Newman Orbit. Spin: Jc/G/M²=0.9, Charge: Q/M·√(K/G)=0.4. Red: neutral testparticle, dashed magenta: ZAMO
Source Own work (Code)
Author Yukterez (Simon Tyran, Vienna)
Other versions
A test particle approaching the ergosphere in the retrograde direction is forced to change its direction of motion


01) Coordinate time              08) Black hole angular momentum  15) Axial angular momentum  22) Framedragging delayed angular velocity
02) Proper time                  09) Black hole charge            16) Polar angular momentum  23) Framedragging local velocity
03) Total time dilation          10) Axial radius of gyration     17) Radial momentum         24) Framedragging observed velocity
04) Gravitational time dilation  11) kinetic energy               18) Cartesian radius        25) Observed particle velocity
05) Boyer Lindquist radius       12) Potential energy             19) Cartesian x-axis        26) Local escape velocity
06) BL Longitude in radians      13) Total particle energy        20) Cartesian y-axis        27) Delayed particle velocity
07) BL Latitude in radians       14) Carter Constant              21) Cartesian z-axis        28) Local particle velocity


Line-element in Boyer-Lindquist-coordinates:

Shorthand terms:

with the dimensionless spin parameter a=Jc/G/M² and the dimensionless electric charge parameter ℧=Qₑ/M·√(K/G). Here G=M=c=K=1 so that a=J und ℧=Qₑ, with lengths in GM/c² and times in GM/c³.

Co- and contravariant metric:

Contravariant Maxwell tensor:

The coordinate acceleration of a test-particle with the specific charge q is given by

with the Christoffel-symbols

So the second proper time derivatives are

for the time component,

for the radial component,

the poloidial component and

for the axial component of the 4-acceleration. The relation of the local 3-velocity and the first proper time derivatives is

so we get

for the radial,

for the poloidial,

for the axial and

for the total velocity, which also give the initial conditions. The total time dilation is

where the differentiation goes by the proper time τ for charged (q≠0) and neutral (q=0) particles (μ=-1, v<1), and for massless particles (μ=0, v=1) by the spatial affine parameter ŝ. The 3 conserved quantities in relation to the local 3-velocity v=√(vr²+vθ²+vφ²)=√(vx²+vy²+vz²) are

1) the Carter constant:

2) the total energy:

3) the axial angular momentum:

ω is the Frame-Dragging angular velocity

The ergospheres and horizons have the Boyer-Lindquist-radius

In this article the total mass equivalent M, which also contains the rotational and the electrical field energy, is set to 1; the relation of M with the irreducible mass is



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