Black hole trajectory plotter

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:

Note that for an object dropped at rest from infinity, the global coordinate \( \Phi \) does not change by definition - although the object will be dragged around a spinning black hole due to frame-dragging.

How to use the simulation: