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Orbit of a negatively charged particle around a positively charged and rotating black hole


English: Orbit of a negatively charged test particle with charge q/m·√(K/G)=-0.25 around a positively charged and rotating Kerr-Newman black hole with charge ℧/M·√(K/G)=+0.4 and spin Jc/G/M²=+0.9. Initial local velocity and equatorial inclination: v0=0.4c, i0=arctan(5/6)rad=39.8056°. Red: test particle, dashed magenta: locally stationary ZAMO. For the same initial local velocity and inclination, but an electrically neutral testparticle see here.
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 (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)          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)


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

with the parameter

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

where a is in units of M.


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File history

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current18:12, 14 March 2019Thumbnail for version as of 18:12, 14 March 2019758 × 544 (7.02 MB)Yukterez (talk | contribs)adding the 3 components of the local 3-velocity to the numeric display
09:29, 10 March 2019Thumbnail for version as of 09:29, 10 March 2019758 × 544 (6.91 MB)Yukterez (talk | contribs)using a version where the testparticle is also charged
04:18, 10 March 2019Thumbnail for version as of 04:18, 10 March 2019758 × 544 (4.28 MB)Yukterez (talk | contribs)extended numeric display now showing 1st and 2nd derivatives
10:22, 22 August 2017Thumbnail for version as of 10:22, 22 August 2017758 × 500 (4.09 MB)Yukterez (talk | contribs)User created page with UploadWizard
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