Jillian's Guide to Black Holes: Forming - Types - Outside - Inside - Finding - References - Websites

Only the brave and the curious continue! I shall put forth such ideas as Schwarzschild's metric for spacetime and the Schwarzschild radius equation!

I'm kidding. They're not really that bad, but I feel the need to warn you that I'm going to actually show two equations, and I know seeing equations frightened me when I was surfing for black hole pages. No real math here; I just think it's important to show 'em.

Schwarzschild Radius
Rs = 2MG/c2

M stands for mass
G is Newton's constant coefficient of gravity, 6.67 x 10-11
c is the speed of light, 3 x 108

What the heck does this equation mean? It's for calculating the Schwarzschild radius of an object. What does that have to do with black holes? Well, when an object is compressed below its Schwarzschild Radius, an event horizon forms around it. Sound familiar? The event horizon of a black hole is just Rs = r. 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 Metric
ds2= - (1 -Rs/r)dt2 + dr2/(1 - Rs/r) + r2d2 + r2sin2d2

t stands for time
r stands for the radius
is like latitude
is like longitude

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.

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 d and d factors are part of the whole spherical geometry transformation thing and aren't special to the metric. They make the math work out right, yes they do.

 

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