According to Stephen Hawking, if we start with a volume of space, say, a public library, and add books and more books, eventually the total number of books will become so massive they will collapse into a black hole. This is a little difficult to verify experimentally, but we can derive a black hole from Einstein's field equations. Below are the variables we will need:
In the diagram below, the blue circle represents the compressed mass (library books); the yellow and blue circle represent the mass's initial volume (library shelf space); The largest circle has a Scharzschild radius. As more and more mass is added, the blue circle shrinks and the singularity radius (r) shrinks as well.
Let's begin the derivation with equation 1:
Equation 1 has second-order tensors. We want to convert these to easy-to-work-with scalars (aka: invariant zero-order tensors). We can do this by contracting the indices. At equation 3 we pull the metric tensor (g) out of the Ricci tensor (R). At equation 4, the contravariant and covariant indices (i) cancel and vanish.
Now let's do the subtraction at equation 4 to get equation 5:
At equation 6 we set g equal to 8pi, so we can divide both sides of equation 5 by 8pi to get equation 7:
At 8 we set R equal to 1/r^2 and make a substitution to get equation 9:
The energy-stress tensor (T) has units of energy density. We do what we must to convert energy density to mass density (equations 10 and 11). We do a little algebra at 12 and 13 to get the Scharzschild radius (equation 13).
Taking equation 12 and applying limits gives the black-hole equation 14:
Just as Stephen Hawking said: If we keep adding library books (mass), the library's radius (r) shrinks and collapses into a black-hole singularity.
Below are bonus equations that show the maximum potential velocity is light speed and the maximum rest-mass energy is still mc^2.