$$\require{cancel}$$

# 13.31 Properties of Black Holes

black hole is a very unusual state of matter. To see how strange it is, let’s begin with the idea of escape velocity. A full treatment requires general relativity, but we can get an approximate answer using Newtonian gravity. The escape velocity of an object of mass M and radius R is:

v = √(2GM/R).

In this equation G is the universal gravitational constant or 6.67 × 10-11 Newton m2kg-2. If M is in kilograms and R is in meters, then v will be in meters per second. The escape velocity is higher for more massive objects of the same size or for a smaller object of the same mass — both correspond to a higher density. If you insert numbers for the Earth, you will see that the escape velocity of the Earth is about 11 kilometers per second. For the Sun, the escape velocity is about 600 kilometers per second.

To calculate the size of a black hole, set the escape velocity to be equal to the speed of light. So, v = c = √(2GM/R). Squaring both sides and rearranging the equation gives:

RS = 2GM/c2

This is the Schwarzschild radius of a black hole. Let's see what we get for an object with the mass of the Sun. In this case, M = 2 × 1030 kilograms and c = 3 × 108 meters per second, so RS = 2 × 6.67 × 10-11 × 2 × 1030 / (3 × 108)2 = 2960 meters, or about 3 kilometers. Crush the Sun down to the size of a small town and it would be a black hole. The Schwarzschild radius is proportional to mass so in the realistic example of a 3 solar mass stellar core, the black hole radius is 9 kilometers.

Any object can become a black hole if it is sufficiently compressed. For example, Earth would become a black hole if it were compressed in an enormous vise down to a radius of 1 centimeter! Jupiter would be a black hole if its mass were contained in a region 6 meters across. A human would be a black hole if concentrated into a region 10-25 meters across — far smaller than a proton. None of these interesting possibilities has been found in the universe. As far as we know, nature can only make black holes out of objects that are star-sized or larger.

Mass is a fundamental property of a black hole and it is a property we can measure. What other properties does a black hole have? All stars rotate, so when a star collapses by a large factor the conservation of angular momentum dictates that the spin rate will speed up. Angular momentum is proportional to orbital velocity times radius, so if the angular momentum is constant, then orbital velocity must be inversely proportional to radius. Since a faster orbital velocity means a shorter orbital time, the rotation period reduces proportionally to the radius as an object shrinks. The Sun rotates every 25 days and has a radius of 700,000 kilometers. So if it were to shrink to black-hole size of 3 kilometers, it would rotate every (3/700,000) × (25 × 24 × 3600) = 9 seconds! We can see that very rapid rotation must be a property of any star that becomes a black hole.

black hole has no other properties that we can measure. Notice, for example, that we cannot find out what a black hole is made of, because all information on the chemical abundance is lost when the material enters the event horizon. The fact that information is lost in a black hole has an intriguing consequence. Jakob Beckenstein and Stephen Hawking showed in the 1970s that black holes carry enormous amounts of entropy. High entropy means a state of high disorder. Black holes consuming matter is just another example of the universal tendency toward disorder.

You may have heard speculation about black holes. Could someone survive a trip into a black hole? Do objects that disappear into black holes appear elsewhere in the universe? What can we say about wormholes in space? These ideas are the subject of books, movies, and TV shows, and they are fun to speculate about. But the truth is that we have no theory to describe what happens beyond the event horizon of a black hole. Since the event horizon is an information barrier, and since we have no black holes we can experiment on, we cannot apply the principles of the scientific method. For now, we must be content with speculation.