At what point would the Earth become a black hole?

Pretty much anything could, in theory, become a black hole if it were squeezed down small enough. At what point would the Earth become a (very unimpressive) black hole in space? We're going to figure it out, and tell the tale of the Schwarzschild Radius.

Karl Schwarzschild is one of those people who make us resentfully conscious of just how little we get done during our days. He was a well-respected professor of astrophysics in Germany, one of the major centers of physics research. When World War I broke out, he enlisted in the army, and was sent to the Russian front. There wasn't really a pleasant front in World War I, but Russia was extra-tough, and everyone would have understood if he had just concentrated on staying alive. Instead, he got some reading material, which happened to be Einstein's Theory of General Relativity. This theory has Einstein stating that massive objects distort spacetime, and that distortion is what we perceive as gravity. Schwarzschild considered this, and thought that a massive enough object packed into a small enough space could distort spacetime enough that not even light could escape it. As there was nothing currently on his plate (probably literally), he decided to go ahead and work out that relation of mass and the space it was packed into.

He came up with the idea that the radius within which an object became a black hole could be calculated by doubling the object's mass, multiplying it by the universal gravitational constant, and dividing the entire thing by the speed of light squared. In other words:

R=(2GM)/(c^2)

This technically means that anything can become a black hole, as long as it's compressed down enough. "Technically" isn't the same as "practically." Recently scientists were surprised that a star forty times as massive as the sun failed to produce a black hole after its collapse. But, if we had some preternatural mass-hugger out there, almost anything could be compressed into black hole if it got pushed within its own Schwarzschild Radius. It would have to be a hell of a hug — to get the sun to turn into a black hole, it would need to be pushed down to about three kilometers in radius. The Earth would need to be pressed into having a radius of 8.7 millimeters. One could think of that as very unimpressive, but I like to think of it as being like a tiny space assassin.

Sadly, Schwarzschild died of infection within a year of coming up with his eponymous calculation. He never got to see black holes, or wide acceptance of his idea. But he certainly got a lot done.

Image: NASA/JPL-Caltech

Via PBS, ESO, and Cosmos.