Notizblock

This thread shows the perspective of an observer that falls freely from infinity (with the negative escape velocity) into a
rotating black hole with spin a=0.7 and electric charge ℧=0.7, and compares it to the perspective of an observer in a radially
stationary LNRF at the same position. The black hole has an accretion disk rotating with the local circular orbit velocity, an
inner disk radius rᵢ=ISCO=1.63678 and outer disk radius rₐ=7; the raytraced images consist of 4 parts:



The first part shows the black hole with an accretion disk, and the second one without. In the third part a hollow shell with
a world map pattern is hovering slightly outside of the outer event horizon at 1.0001r₊ and corotates with the local frame
dragging velocity, see left for a cartesian x,y,z and right for a pseudospherical r,θ,φ projection:



In the cartesian representation it is shown that the singularity is actually a ring with the physical radius R=a, while in the
pseudospherical representation it is depicted at r=0. The relation of the cartesian radius R and angle Θ to the pseudospherical
coordinates r,θ,φ is R=√(r²+a² Sin[θ]), Θ=ArcCos[(r Cos[θ])/√(r²+a² Sin[θ]²)], Φ=φ. A freefaller from infinity (E=1) has a
constant θ in BL coordinates and also constant φ in Raindrop coordinates (but in both coordinates no constant Θ). The fourth
part shows the frequency shift of the background sky and the disk, color code in the range from fe/f0=0..2 (white for fe/f0=1):



In the ZAMO perspective the observer is not at the horizons, but slightly above the outer and below the inner horizon where the
escape/freefall-velocity (and therefore the local velocity relative to a raindrop) is v=0.9999c. Profile of the accretion disc's
rotation: local circular orbit velocity relative to a ZAMO and shapiro delayed angular velocity as a funktion of r (the disc spins
counterclockwise with its left side toward the observer):



The perspectives from r=12, r=4, r=r₊ (outer horizon), r=r₋ (inner horizon) & r=0.5 for latitudes of θ=80°, θ=45° & θ=10° in
360°×180° full panorama in equirectangular projection and as stereographic projektion with front and rear view are shown, the
numeric display shows the velocities relative to a local ZAMO (at the horizons the only possible ZAMO is a photon, therefore v=c):

r=12, θ=80°  →  R=12.01979, Θ=80.01663°











r=4, θ=80°  →  R=4.058968, Θ=80.14674°











r=r₊=1.141421 (outer event horizon), θ=80°  →  R=1.333442, Θ=81.45175°











r=r₋=0.858579 (inner horizon), θ=80°  →  R=1.101082, Θ=82.21803°











r=0.5, θ=80°  →  R=0.851601, Θ=84.14830°

r=12, θ=80°  →  R=12.01979, Θ=80.01663°











r=4, θ=80°  →  R=4.058968, Θ=80.14674°











r=1.143138 (slightly above the outer event horizon), θ=80°  →  R=1.334912, Θ=81.448296°











r=0.857171 (slightly below the inner horizon), θ=80°  →  R=1.099985, Θ=82.223075°











r=0.5, θ=80°  →  R=0.851601, Θ=84.14830°

r=12, θ=45°  →  R=12.01020, Θ=45.04866°











r=4, θ=45°  →  R=4.030509, Θ=45.43207°











r=r₊=1.141421 (outer event horizon), θ=45°  →  R=1.244123, Θ=49.55368°











r=r₋=0.858579 (inner horizon), θ=45°  →  R=0.991039, Θ=52.22251°











r=0.5, θ=45°  →  R=0.703562, Θ=59.83321°

r=12, θ=45°  →  R=12.01020, Θ=45.04866°











r=4, θ=45°  →  R=4.030509, Θ=45.43207°











r=1.1428844 (slightly above the outer event horizon), θ=45°  →  R=1.2454656, Θ=49.543783°











r=0.8574999 (slightly below the inner horizon), θ=45°  →  R=0.990104, Θ=52.236446°











r=0.5, θ=45°  →  R=0.703562, Θ=59.83321°

r=12, θ=10°  →  R=12.00062, Θ=10.01665°











r=4, θ=10°  →  R=4.001847, Θ=10.14883°











r=r₊=1.141421 (outer event horizon), θ=10°  →  R=1.147875, Θ=11.68651°











r=r₋=0.858579 (inner horizon), θ=10°  →  R=0.867141, Θ=12.81686°











r=0.5, θ=10°  →  R=0.514563, Θ=16.87593°

r=12, θ=10°  →  R=12.00062, Θ=10.01665°











r=4, θ=10°  →  R=4.001847, Θ=10.14883°











r=1.1426959 (slightly above the outer event horizon), θ=10°  →  R=1.1491428, Θ=11.683043°











r=0.857704 (slightly below the inner horizon), θ=10°  →  R=0.8662746, Θ=12.821906°











r=0.5, θ=10°  →  R=0.514563, Θ=16.87593°

The black noise observed inside the Cauchy horizon is numeric and occours around the ring singularity. Zoom:





















Between the ring you have the window into negative space (the Antiverse), which (if it isn't just a mathematical artefact) is
propably empty. In the pictures on this site a mirror is spanned between the ring, but it is basically unknown what you would
see at this spot; it is even unclear if the region inside the Cauchy horizon can be considered as physical.

Undistorted view on the black hole's disk and surfaces from an angle of Θ=80°:



This is what an observer would see if neither gravitational lensing nor gravitational and kinematic frequency shift would occur
and the BH surfaces and singularity were visible. The BL inner/outer radii of the accretion disk rᵢ=1.63678, rₐ=7 project with
Rᵢ=1.780184, Rₐ= 7.034913 into the cartesian flat space. The outer horizon is at r₊=1.1414214 → R₊=1.3389708 and the ring
singularity at rᵣ=0 → Rᵣ=a=0.7

In these images the celestial sphere is spanned at rₛₚₕ=500 (in natural units of G=M=c=k=1), and the affine parameter's range of
integration for the photon of interest to hit the sphere tᵢ=-100rₛₚₕ. If the stepsize of integration is smaller than 10⁻⁵ or the
maximum affine parameter is reached the integration for the pixel is aborted and displayed black.

The coordinates used are Raindrop Doran coordinates (for more details see here). Outside of the horizon, Boyer Lindquist
coordinates are recommended, since the BL Code renders faster and needs less memory. In both cases the output is a full
panorama in equirectangular projection and can be transformed into other formats using the code at kartographie.yukterez.net,
like in the examples in stereographic projection.

Inside the horizon Raindrop or Kerr Schild coordinates are needed, even if the perspective of a ZAMO is rendered, the
conversion for the local velocities from one system to the other can be found here (v=0 in Raindrop coordinates means free fall
from infinity, and v=0 in BL coordinates the system of a ZAMO). In the numeric display, the velocity is given relative to a ZAMO.

The field equations and diverse coordinate systems can be found at geodesics.yukterez.net, the code for the simulator at
kerr.newman.yukterez.net and the relativististic raytracer at raytracing.yukterez.net; for the free fall into a nonrotating and
uncharged black hole without an accretion disc see schwarzschild.yukterez.net

Alternative layout with the raindrop perspective in the left column and the ZAMO perspective in the right column: click here