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Black.Hole,Extremal.Kerr.Newman,Raytracing.png (800 × 430 pixels, file size: 262 KB, MIME type: image/png)

Summary

Description
English: Extremal Kerr-Newman black hole with spin a/M=√½ and charge Q/M=√½, in natural units of G=K=c=1. Therefore a²+Q²=M². The observer is at r=50M and views the black hole from the equatorial plane (edge on). FOV: 77.4°×38.7°. The equations that were used to raytrace the image can be found here.
Date
Source Own work
Author Yukterez (Simon Tyran, Vienna). Source material for the Milky way background (also available on Commons): ESO/S.Brunier. Code for the relativistic raytracer: yukterez.net
Other versions
Naked singularity viewed from the equatorial plane
The same naked singularity viewed from 45°
The same naked singularity viewed from the top

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

where a is in units of M.

Licensing

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Captions

spinning and charged black hole

Items portrayed in this file

depicts

10 April 2018

image/png

a203269c7f3ed97e2242933c91863991d72e80c6

268,527 byte

430 pixel

800 pixel

File history

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Date/TimeThumbnailDimensionsUserComment
current02:00, 21 April 2018Thumbnail for version as of 02:00, 21 April 2018800 × 430 (262 KB)Yukterezdisplay the distance and angle of view in the right upper corner
20:13, 11 April 2018Thumbnail for version as of 20:13, 11 April 2018800 × 430 (262 KB)Yukterezreduced width to 800px to avoid rescaling
19:59, 11 April 2018Thumbnail for version as of 19:59, 11 April 2018860 × 430 (285 KB)YukterezUser created page with UploadWizard

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