Jillian's Guide to Black Holes: Forming - Types - Outside - Inside - Finding - References - Websites
Schwarzschild radius and metric for spacetime
An event horizon will form around anything that is smaller than its Schwarzschild Radius, making it a black hole. What is this mysterious radius, and is it easy or difficult to get that small? If you know the mass of an object, you can calculate its Rs. For a human it is 1.5 x 10-27 meters per kilogram (for comparison a proton is 10-15 meters). Very tiny! It's tough to make black holes out of small things!
| Schwarzschild Radius | Rs = 2MG/c2 M stands for massG is Newton's constant coefficient of gravity, 6.67 x 10-11 m3 kg-1 s-2 c is the speed of light, 3 x 108 m/s |
What is the Schwarzschild metric?
It is Schwarzschild's solution to Eintstein's general relativity equation set. The metric describes the shape of spacetime outside of matter. Y'know, those cool curvey spacetime pictures in Scientific American. Once you hit matter, be it some gas, a star, a planet, or a rock, this metric no longer applies. The metric's kinda ... spherical. It looks a lot like an equation made for rectangular coordinates transformed into spherical coordinates, a standard calculus problem.
| Schwarzschild Metric | ds2= - (1 -Rs/r)dt2 + dr2/(1 - Rs/r) + r2dθ2 + r2sin2θdφ2 M stands for massG is Newton's constant coefficient of gravity, 6.67 x 10-11 m3 kg-1 s-2 c is the speed of light, 3 x 108 m/s |
What does all that messy stuff mean?
Well, the ds factor tells you how space changes, what it all looks like. The dt factor tells how time changes as spacetime changes. You can see that, if r = Rs, dt would be zero. That is to say that at the even horizon there would be no change in time. Makes sense; you can look at the event horizon as being the place where time "stops." The dr factor deals with how close to something you are. You'll notice that it "blows up" when r = Rs; the Schwarzschild metric does not apply beyond the event horizon. The dθ and dφ factors are part of the whole spherical geometry transformation thing and aren't special to the metric. They make the math work out right.
