by David Ando

Purpose:

The purpose of this page is to give an easy to understand explanation of what black holes are.

What are Black Holes?

A black hole is a super dense object that has an intense gravitational pull. There are two parts to a black hole, a singularity and a event horizon. If you were to take a slice of a black hole right through its center it would look like this:

The event horizon is where the force of gravity becomes so strong that even light is pulled into the black hole. Although the event horizon is part of a black hole, it is not a tangible object. If you were to fall into a black hole, it would be impossible for you to know when you hit the event horizon. For a mathematical derivation of the radius of a event horizon see below.

The singularity is not really a tangible object either. According to the General Theory of Relativity the Singularity is a point of infinite space time curvature. This means that the force of gravity has become infinitely strong at the center of a black hole. Everything that falls into a black hole by passing the event horizon, including light, will eventually reach the singularity of a black hole. Before something reaches the singularity it is torn apart by intense gravitational forces. Even the atoms themselves are torn apart by the gravitational forces.

Formation of a Black Hole:

Imagine a star which is much more massive than our sun, and which has a mass, called the critical mass, which is large enough to cause a black hole to form. What keeps this star from collapsing onto itself and becoming a black hole? The answer is that there is an intense pressure caused by nuclear reactions within the sun. When the fuel that feeds the nuclear reactions gets used up the massive star cannot support itself anymore. It then collapses to form a black hole.

It is interesting to note that when a black hole is formed by a collapsing star it is actually impossible to watch the final steps of the formation of the black hole from a stationary external reference frame. An external reference frame is a place where one watches the formation of the black hole from far away, like an astronomer on Earth. In addition, it is impossible to see any object fall into a black hole. This is not to say that everything appears to freeze just before entering a black hole. As an object falls into a black hole it gets increasingly dimmer and dimmer from the point of view of an outside observer. By the time an object gets to the edge of a black hole, it will be completely black. This effect, called a gravitational redshift, is caused by the immense gravity near the outside of a black hole.

Cool things about Black Holes:

First of all, if you get close enough to a black hole you will see the back of your own head! This effect, called an Einstein ring, is caused by the intense gravity around a black hole. When you are near a black hole at certain distances the light that leaves from the back of your head will travel though space that is bent so much by gravity that it will enter your eyes.

Another cool thing about black holes is that they might be able to destroy information. The destruction of information is not allowed by quantum mechanics, so Hawking concludes that the usual rules of quantum mechanics cannot apply for black holes!! John Preskill of Caltech has written a paper about this that should be accessible to almost any reader. The paper is located at http://xxx.lanl.gov/abs/hep-th/9209058 and you will need the freely distributed Adobe Acrobat Reader 3.0 to read the PDF document located under other formats.

Evidence for the existence of Black Holes:

This is an interesting problem. How do you prove the existence of something that cannot be observed by definition? There are actually many methods used to see if black holes really exist in our universe. The first method is to look for objects in our universe that have a lot of mass, but are very small. For example we can prove that there exists a black hole in an astronomical object called M87. This object weighs three billion times more than our sun, but takes up a volume no larger than our solar system.

Another method of finding black holes is to look for an acceleration of matter. Since black holes have such strong gravitational fields, they accelerate anything that gets near them to great speeds. Rapid acceleration of an object can be observed by looking for doppler shifts in the light given off by an accelerating object. You can see a picture of the doppler signature of a black hole at a NASA web page here.

The Derivation of the Radius of the Event Horizon of a Black Hole:

Now for some real fun. First we need to know equations for the kinetic energy and potential energy of an object. Kinetic energy, or KE, is energy of movement, and it is given by

, where m is the mass of the object and v is the velocity of the object. Potential energy, or PE, is energy of location in a gravitational field and it is given by the equation

, where m₁ is the mass of the first mass and m₂ is the mass of the second mass. G is the gravitational constant, which is equal to 6.67 * 10^-11, and r is the distance between the centers of mass of m₁ and m₂.

The total energy of the object is given by the sum of the kinetic and potential energies. By the law of conservation of energy the total energy will never change. This gives us the following relationship between kinetic and potential energies at two different points: . This means that the total energy for a object is going to be equal at points 1 and 2.

Lets say you are at point one which is a distance r₁ from the center of mass of a planet. The center of mass of a planet is the center of the planet and we assume that you are located on the surface of the planet. This would make r₁ the radius of the planet. In calculating your potential energy we make your mass equal to m₁ and the mass of the planet equal to m₂. To fully escape this planet you give yourself an initial velocity of v₁. To keep this problem simple you are not allowed to accelerate after you get your initial velocity of v₁.

    Since you want to fully escape the gravitational pull of the planet you will have to go an infinite distance from the planet. This means that point two will have an infinite distance for the value of r. Now it is important to note that the potential energy is zero as the distance between the two masses goes to infinity. Now that we know that the potential energy at point two is zero, what is the kinetic energy at point two? If we want to find the minimum velocity to escape the planet the final kinetic energy at point two should be zero. Any kinetic energy left over would be a waste.
    Here is a summary of the situation so far:

    The next step is to do a little algebra. We know that . Now substitute in the equations for the KE and PE. The equation then becomes: . To find the velocity that you need to reach in order to fully escape a planet solve the previous equation for v_1. This gives the escape velocity for an object equal to .

Now for the part that you have all been waiting for! What is the radius of the event horizon of a black hole? To do this you need to modify the escape velocity equation. First, substitute the speed of light, which is the maximum escape velocity allowed by the laws of physics, for the velocity. Next, solve the escape velocity equation for the radius. The radius that you solve for will be the closest distance you can get to the center of mass of a black hole before you will not able to escape the black hole, even if you travel at the maximum speed allowed by physics. The radius of the event horizon of a black hole is therefore: . The letter c stands for the speed of light, G for the gravitational constant, m is the mass of the black hole, and r is the radius of the event horizon.

Movies of black holes:

This site has some very interesting movies of black holes. It is located at http://www.ncsa.uiuc.edu/Cyberia/NumRel/MoviesEdge.html. The web site has movies of gravitational waves and an awesome movie of two black holes colliding into each other, something that is still not fully understood in detail.