English: Orbit of a negatively charged test particle (q/m·√(K/G)=-0.25) around a positively charged and rotating Kerr-Newman black hole (charge ℧/M·√(K/G)=+0.4 and spin Jc/G/M²=+0.9). Initial local velocity and equatorial inclination: v0=0.4c, i0=40°. Red: test particle, dashed magenta: locally stationary ZAMO. For an other example with different initial conditions see here.
01) Coordinate time (GM/c^3) 11) BL r coordinate (GM/c^2) 21) BH central charge (M/√(K/G)) 31) Observed framedragging rate (c^3/G/M)
02) Proper time (GM/c^3) 12) BL φ coordinate (radians) 22) Particle charge (m/√(K/G)) 32) Local framedragging velocity (c)
03) Total time dilation (dt/dτ) 13) BL θ coordinate (radians) 23) BH Irreducible mass (M) 33) Cartesian framedragging velocity (c)
04) Grav. time dilation (dt/dτ) 14) dr/dτ (c) 24) Kinetic energy (mc^2) 34) Proper velocity (c, dl/dτ)
05) Local energy (dt/dτ, mc^2) 15) dφ/dτ (c^3/G/M) 25) Potential energy (mc^2) 35) Observed velocity (c, d{x,y,z}/dt)
06) Cartesian radius (GM/c^2) 16) dθ/dτ (c^3/G/M) 26) Total energy (mc^2) 36) Escape velocity (c)
07) x Axis (GM/c^2) 17) d^2r/dτ^2 (c^6/G/M) 27) Carter constant (GMm/c)^2 37) Local r velocity (c)
08) y Axis (GM/c^2) 18) d^2φ/dτ^2 (c^6/G^2/M^2) 28) φ angular momentum (GMm/c) 38) Local θ velocity (c)
09) z Axis (GM/c^2) 19) d^2θ/dτ^2 (c^6/G^2/M^2) 29) θ angular momentum (GMm/c) 39) Local φ velocity (c)
10) travelled distance (GM/c^2) 20) Spin parameter (GM^2/c) 30) Radial momentum (mc) 40) Total local velocity (c)
Equations
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 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 relation between the first proper time derivatives and the local three-velocity components relative to a ZAMO is
The local three-velocity in terms of the position and the constants of motion is
which reduces to
if the charge of the test particle is q=0. The escape velocity of a charged particle with zero orbital angular momentum is
which for a neutral test particle with q=0 reduces to
with the gravitational time dilation of a locally stationary ZAMO
which is infinite at the horizon. The time dilation of a globally stationary particle (with respect to the fixed stars) is
which is infinite at the ergosphere. The Frame-Dragging angular velocity observed at infinity is
The local frame dragging velocity with respect to the fixed stars is therefore
which is c at the ergosphere. The axial radius of gyration is
The 3 conserved quantities are 1) the total energy:
2) the axial angular momentum:
3) the Carter constant:
The effective radial potential whose zero roots define the turning points is
and the poloidial potential
with the parameter
The azimutal and latitudinal impact parameters are
The horizons and ergospheres 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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