by Michael Brunnbauer, 2023-07-21

Check out Exploring
Black Holes, Second Edition by Edwin F. Taylor, John Archibald Wheeler
and Edmund Bertschinger for an accessible introduction to general relativity
using only calculus and algebra. I made
this cheatsheet,
which covers chapters 1-8.

This simulation uses a fourth order
Runge-Kutta
method to solve the equation for \( \frac{dr}{d\tau} \) derived from the
Doran
metric of a spinning black hole:
$$ \frac{dr}{d\tau} = \pm \frac{R}{r} \cdot \sqrt{ \left( \frac{E}{m} - \omega \cdot \frac{L}{m} \right) ^2 - \frac{r^2 \cdot H^2}{R^2} \cdot \left( 1 + \frac{L^2}{m^2 \cdot R^2} \right) } $$
The equation for \( \frac{d\Phi}{d\tau} \) is solved along the way using the
calculated interval points \( r_1 \), \( r_2 \), \( r_3 \), \( r_4 \) used for
\( \frac{dr}{d\tau} \):
$$ \frac{d\Phi}{d\tau} = \frac{1}{r^2 \cdot H^2} \cdot \left( \left( 1 - \frac{2 \cdot M}{r} \right) \cdot \frac{L}{m} + \frac{2 \cdot M \cdot a}{r} \cdot \frac{E}{m} + a \cdot \sqrt{ \frac{2 \cdot M \cdot r}{r^2 + a^2} } \cdot \frac{dr}{d\tau} \right) $$
The result is a plot of global coordinates \( r \) and \( \Phi \) of an object
with rest mass \( m \) inside the equator plane as its
wristwatch time
\( \tau \) advances (\( m \ll M \)). Please note that the equations use a
geometrized
unit system.
$$ M \quad \text{Mass of the black hole in meters (1kg = 7.42e-28 meters)} $$
$$ 0 \le a \le M \quad \text{Spin parameter of the black hole in meters} $$
$$ \frac{E}{m} \quad \text{Specific map energy of the object } $$
$$ \frac{L}{m} \quad \text{Specific map angular momentum of the object } $$
$$ R^2 \equiv r^2 + a^2 + \frac{2 \cdot M \cdot a^2}{r} \quad \text{Reduced circumference} $$
$$ H^2 \equiv \frac{1}{r^2} \cdot ( r^2 - 2 \cdot M \cdot r + a^2 ) \quad \text{Horizon function} $$
$$ \omega \equiv \frac{2 \cdot M \cdot a}{r \cdot R^2} \quad \text{Ring omega} $$
$$ r_{EH} \equiv M + \sqrt{ M^2 - a^2 } \quad \text{Event horizon} $$
$$ r_{CH} \equiv M - \sqrt{ M^2 - a^2 } \quad \text{Cauchy horizon} $$
It can already be seen that the use of specific map energy and map angular
momentum make the equations independent of the mass of the object. The
equations can be transformed such that \( a \), \( \frac{L}{m} \), \( r \),
\( dr \), and \( d\tau \) are always divided by \( M \). Use of these
specific units makes everything independent of the black hole mass.

The Runge-Kutta step size for \( d\tau \) is adjusted dynamically such that
a configureable resolution of data points per unit length \( M \) is achieved.
The horizon function goes towards zero at both horizons. This can cause
\( \frac{d\Phi}{d\tau} \) to blow up - although the rest of the term may
compensate for that. Whether there is such a
coordinate
singularity at a horizon depends on the specified parameters. The
simulation tries to compensate for that with the following combination of
parameters and heuristics:

- M is fixed at 1477 meters - the mass of our sun.
- The default step size is 1 meter (ca. 3 nanoseconds).
- If the singularity is reached or a horizon crossed using the default step size, the step size is not adjusted (this can show as a sudden jump in a straight line).
- The maximum/minimum adjustment factor is 256 or 1/256 (this can also show as sudden jumps in a straight line).

How to use the simulation:

- You can manipulate the parameters with the sliders. The four physical
parameters can also be set by typing a value in the input field and clicking
the corresponding "set" button. If a value set this way is above the
maximum value of the slider for E/m, L/Mm or r
_{0}/M, the maximum value will be adjusted at the cost of slider sensitivity. Reload the page to reset to initial slider ranges. - Only 6 decimal places are displayed for the four physical parameters but they will be more precise even when adjusted with the slider.
- The left graph shows the trajectory plot with start and end position marked by a red dot (no red dot at the end position if the object fell into the singularity). The event horizon is shown as a black circle and for \( a \gt 0 \), the cauchy horizon is shown as a red circle.
- The right graph shows the two effective potentials in green with \( r \) as the x-axis. The areas enclosed by them are forbidden (the square root in the term for \( \frac{dr}{d\tau} \) becomes imaginary. Whenever this happens, the simulation will change the sign of \( \frac{dr}{d\tau} \) and therefore the direction of movement). The object moves along the red horizontal line marking E/m until it hits a green line and changes direction there. The slope of the green line determines the radial acceleration. The starting position is shown as a vertical black line.
- If the simulation shows only a red dot, you started from within the forbidden
areas enclosed by the potentials - adjust E/m, the starting radius
r
_{0}/M or change the potentials by manipulating L/Mm. - The ± slider can be used to fine tune E/m.
- As the potentials can be very flat, you can zoom in around E/m by clicking "Zoom in".
- The simulation will show a circular orbit by default. You can see the
progression of the orbit by manipulating the points/M or the max points
slider. This is example 1. Click on the button "Example 2" to see the next
example. Here are all of them:
- Circular orbit around a non-spinning black hole at 8M.
- Elliptical orbit around a non-spinning black hole with zoom-whirl behavior
- Transfer from the innermost stable circular orbit at 2.54M to an unstable type 1 orbit inside the cauchy horizon of a spinning black hole.
- Transfer from the innermost stable circular orbit at 2.54M to an unstable type 2 orbit inside the cauchy horizon of a spinning black hole. Coordinate singularity when passing the cauchy horizon from outside.
- Wormhole-behavior: Bouncing off inside the cauchy horizon of a spinning black hole and reemerging "somewhere else". Coordinate singularities when passing both horizons from inside.
- Bouncing off inside the horizon of a black hole with maximum spin and reemerging "somewhere else". Coordinate singularity when passing the combined cauchy/event horizon from inside (pushing this particular simulator to its limits).

- For a spinning black hole (\(a > 0)\), you can click on "Show ring" to actually show the ring singularity. This will transform \( r \) to \( \sqrt{r^2 + a^2} \). The ring singularity will then be shown as a grey disk. There is nothing inside the disk as \( r \) would be imaginary there. We can't see a ring because the equations are confined to the equator plane where time and space end at the edge of the disk.
- The initial direction of movement is inward (negative \( \frac{dr}{d\tau} \)). This can be changed by clicking the "Toggle dir" button.