I know this odd, but I was asked the other day, around the Higgs Boson news how does a Black Hole work? I needed an answer that was beyond the populist one of the gravity is so strong nothing can escape, including light and the actual mathematical physics of General Relativity for the formation of Black Holes, the Large Scale Structure of Spacetime and Gravitation descriptions of objects in strong fields, I used to know how to use in graduate school.
But first, the notion of an object (a Star) where no light escapes is not new:
A luminous star, of the same density as the Earth, and whole diameter should be two hundred and fifty time larger than that of the Sun, would not, in consequence of its attraction, allow any of its rays to arrive at us; it is therefore possible that the largest luminous bodies in the universe may, through cause, be invisible - P. S. Laplace (1798)
The easiest way to derive the conditions for the formation of a Black Hole is to show the properties in the populist description, the ones we see in the movies like Star Trek - nothing, even light can escape. But we need some math to go with this it avoid the populist trap of not being able to calculate anything.
So let's pretend we live on a planet and have a small rock in our hands. We throw or shot the rock straight up and observe the motion through the simple equations of potential and kinetic energy.
This rock has kinetic energy of,
where m is the mass of the rock and v is the velocity of the rock once it leaves my hand.
The potential energy of the thrown rock comes from the gravitional pull on and is,
where M is the mass of the planet and r is the radius of the planet. In order for the rock to escape from our planet's gravitational field and not fall back to the place where I threw it, the total energy of the rock, E=K+U must be greater than zero (0).
Since nothing can travel faster that the speed of light c, remember Einstein's special relativity, the maximum kinetic energy, K is ½mc2. The rock can never escape from the surface of the planet if,
If this inequality is TRUE, then the planet has sufficient gravity to be a black hole. Since the little m, the mass of the rock on in the case of light, the photon (which is zero), cancels on the right side inequality, we can make this test for photons as well. So again, if the inequality if TRUE for the equation on the right, no photons are leaving this place and we live on a Black Hole.
So to figure out if you live on a Black Hole, just determine its mass, M, the gravitional constant G, which is assumed to be universal, which is G=6.67384(80)x10-11m3kg-1s-2
