The Mathematics Behind It

The Mathematics behind SAB-GHOOMo

The Schwarzschild black hole is a static, non-rotating and uncharged black hole. It describes the geometry of spacetime around a static, spherical black hole immersed in a vacuum. It is described by the Schwarzschild metric, which in itself is an exact solution to the Einstein Field Equations:

Here, (t, r, θ, φ) are coordinates defining the Schwarzschild spacetime, and M is the mass of the black hole.

This metric has some interesting properties:

• When r approaches infinity, the metric turns into the Minkowski metric. This shows that the Schwarzschild spacetime is asymptotically flat - meaning if you're standing far away enough from the Schwarzschild black hole, you'll notice less curvature and more flatness. 

• r = 2M sketches a spherical boundary around the black hole, called an event horizon. That is what you would call a "point of no return" - once a light ray or massive particle enters this region, it cannot turn around and go back; in technical terms, all future events of any particle crossing the event horizon point towards radial positions r < 2M. The particle's fate is sealed: it has no choice but to fall into the center of the black hole, with no chance of coming back.

• r = 2M and r = 0 are points where the metric blows up - that's where we have encountered singularities. But fear not, for r = 2M can be removed with just a simple coordinate transformation - that's a coordinate singularity! r = 0 on the other hand, cannot be resolved, and that is called a physical singularity, which implies infinite curvature of spacetime.

The equations of motion define all the different types of paths you can have within the Schwarzschild spacetime - and they're all hidden behind the Lagrangian for the Schwarzschild black hole:

When solving the associated Euler-Lagrange equations, we encounter two constants of motion: angular momentum per unit mass and energy per unit mass:

Thus, our Lagrangian can be rewritten in terms of E and L:

• Veff is the effective potential. 

• ϵ is a constant that can attain one of two values, depending on the kind of particle that's orbitting around a black hole:

  • ϵ = 0 for light or other massless particles

  • ϵ = -1 for massive particles

Orbits of massive particles around the Schwarzschild Black Hole

Generally, our object moves through the effective potential with an 'effective energy' Eeff that is given by the right hand side of the equation above. This continues until it encounters a 'turning point', at which  Veff=Eeff.
Setting ϵ = -1, we can minimize Veff to solve for the possible radii of orbit at these turning points to find:

Equipped with this understanding of the mathematics behind the Schwarzschild black hole, play with the simulation and see what trajectories for light and massive objects you are able to find. Search for answers to these questions:

• Can light escape to infinity?  

• Can light move in a circular orbit around the black hole? 

• What's the smallest radius for which an object orbits the black hole? 

• Does increasing the angular momentum or decreasing it help an object escape to infinity? 

• Are you ever able to escape to infinity when the radial distance is less than 2M? 

Search for answers to these questions using the simulation, and see if you can make up, and then answer, your own questions!

You can also try and verify the facts presented here. Note that M=1 in the simulation.

References:

• "A First Course in General Relativity", Bernard F. Schutz.

• "Spacetime and Geometry: An Introduction to General Relativity", Sean Carroll.

• "Lecture Notes on General Relativity", Matthias Blau.