## Friday, March 29, 2013

### Black Holes 101

Most of what I discuss on this blog has to do directly with gravitational waves.  This time I'd like to talk about one of their most talked about exotic sources: black holes.  Black holes are an exemplary source because they are highly concentrated mass.  Just add a touch of accelerated motion and gravitational waves are emitted in abundance (well, it's not quite that simple and "abundance" is a relative term, but you get the idea).  But what are the fundamental concepts that add up to the existence of black holes?  That's what we are focusing on now.

1.  Escape velocity

You've probably noticed that the harder you throw an ball straight up in the air, the higher it goes.  We also know that the farther away the ball gets from the Earth, the lower the gravitational attraction is between the ball and the Earth.  When you connect these two concepts, you can imagine that there is a speed at which you can throw the ball up and it will never come back down.  This is called the escape velocity:

We have discussed escape velocity before on this blog (specifically when discussing the conditions an object must have to be 'eaten' by a black hole).  In this equation, G is the gravitational constant, M is the mass of the 'thing' you are trying to escape, and r is the distance you are from the center of the 'thing'.  The bigger the mass of the 'thing', M, is, the faster the object must be thrown to escape it, ve.

Going back to throwing a ball up in the air from the Earth's surface, you would need to throw that ball about 25,000 mph so that the ball would not come back to the Earth (good luck with that)!

2.  The speed of light is the universal 'speed limit'

Nothing can travel faster than the speed of light (in a vacuum), represented by c.  No matter, energy, or information about the Universe can travel faster than that.  That is pretty much the long and the short of this concept.

 [Source: Knight Science Journalism at MIT blog]

This 'speed limit' comes from Einstein's special relativity and the effect of simultaneity.  Perhaps I will write a longer post about this in the future, but all that is important now is to recognize that if something needed to travel faster than the speed of light to communicate information to you, then you are never going to know about it.

3.  Light is affected by gravity

Light is both a particle and a wave.  As a particle, it has no mass.  Since gravity acts between two masses, it may be surprising that light can be affected by gravity at all!  But it does.  This effect is called gravitational lensing and I've written about this previously here

CONCLUSION:  How these concepts form a black hole

The simplest black hole is called a Schwarzschild black hole.  This is a black hole that has no electrical charge and is not rotating - it's just "there" meaning that there won't be anything to complicate our black hole situation.  For now, let us think of our black hole as having mass but no volume.  You can think of this as being the ultimate implosion.  Since this mass has no volume, there isn't any surface to it.  Eventually, as we get closer and closer to where the mass is centered, the escape velocity will become so large that the escape velocity will be greater than the speed of light.  And since light is indeed affected by gravity, that means that nothing will be able to escape the black hole.  We can even figure out what this distance is from the equation for escape velocity by setting ve to the speed of light, c, and solving for r (the distance away from the mass where the escape velocity equals the speed of light):

This distance from a black hole where light will not be able to emerge from a black hole is also called the event horizon and this sphere around the black hole is what is being referred to when we talk about the size of the back hole.  For this simple Schwarzschild black hole, it is also known as the Schwarzschild radius.  You can also think of this radius as how small a mass would have to be to become a black hole.

So, how big would the Schwarzschild radius be for some things we are familiar with?  Well, for a black hole with the mass of our Sun it would be just about 3 km or about 1.9 miles.  For a black hole with the mass of the Earth it would be 8.87 mm or a little under 11/32" (0.349 in).