Black Holes

A black hole is a region of space that has so much mass concentrated in it that there is no way for a nearby object to escape its gravitational pull. Suppose that you are standing on the surface of a planet. You throw a rock straight up into the air. Assuming you don't throw it too hard, it will rise for a while, but eventually the acceleration due to the planet's gravity will make it start to fall down again. If you threw the rock hard enough, though, you could make it escape the planet's gravity entirely. It would keep on rising forever. The speed with which you need to throw the rock in order that it just barely escapes the planet's gravity is called the escape velocity. The escape velocity depends on the mass of the planet. If the planet is extremely massive, then its gravity is very strong, and the escape velocity is high. A lighter planet would have a smaller escape velocity. The escape velocity also depends on how far you are from the planet's center. The closer you are, the higher the escape velocity. The Earth's escape velocity is 11.2 kilometers per second (about 25,000 m.p.h.), while the Moon's is only 2.4 kilometers per second (about 5300 m.p.h.)

Now imagine an object with such an enormous concentration of mass in such a small radius that its escape velocity was greater than the velocity of light. Then, since nothing can go faster than light, nothing can escape the object's gravitational field. Even a beam of light would be pulled back by gravity and would be unable to escape.

In 1784, the English geologist John Michell realized that it would be theoretically possible for gravity to be so overwhelmingly strong that nothing, not even light traveling at 186,000 miles a second, could escape. To generate such gravity, an object would have to be very massive and unimaginably dense. At the time, the necessary conditions for "dark stars", as Michell called them, seemed physically impossible. His ideas were published by the French mathematician and philosopher Pierre Simon Laplace in two successive editions of an astronomy guide, but were dropped from the third edition. In Laplace's 1795 edition, he put forward the following equation saying what the mass and radius would have to be to form a black hole.

Vesc = (2GM/r)1/2 = c

Where Newton had used the motion of the moon around the earth as a guiding example in his work, Einstein used this deviation of Mercury. The central ideas of general relativity were already in place. At issue was the exact form of the curvature term G in the Einstein equation G = kT. A precise relativistic model of the sun's gravitational field was not needed. Einstein used a simple polynomial approximation. Late in 1915 he succeeded, and the 43 second lag was eliminated. A few weeks later, in 1916, Einstein, working in Berlin, received a paper from Karl Schwarzschild, an astronomer who, though no longer young, was serving in the German army in Russia. Hospitalized by an illness that soon proved mortal, Schwarzschild had time to discover the desired precise relativistic model, and Schwarzschild spacetime replaced the Newtonian model as the best description of the gravitational field of an isolated spherically symmetrical star. But only a few theorists were familiar with relativity, and significant experimental tests were not possible in laboratories.

In 1916, Karl Schwarzschild decided to compute the gravitational fields of stars using Einstein's new field equation. Schwarzschild limited the complexity of the problem by assuming the star was perfectly spherical, gravitationally collapsed, and did not rotate. His calculations yielded a solution aptly called a Schwarzschild singularity. The name "black hole" was invented in 1968 by John Archibald Wheeler. Before Wheeler, these objects were often referred to as "black stars" or "frozen stars". The first episodes of Star Trek had been made before this, and contain the phrase "black star".

Almost immediately after Einstein developed general relativity, Karl Schwarzschild discovered a mathematical solution to the equations of the theory that described such an object. It was only much later, with the work of such people as Robert Oppenheimer, Volkoff, and H. Snyder in the 1930's, that people thought seriously about the possibility that such objects might actually exist in the Universe. These researchers showed that when a sufficiently massive star runs out of fuel, it is unable to support itself against its own gravitational pull, and it should collapse into a black hole.

In 1939, Robert Oppenheimer and H. Snyder said in their paper, "When all thermonuclear sources of energy are exhausted, a sufficiently heavy star will collapse. Unless [something can somehow] reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely"...past white dwarfs, past neutron stars, to an object cut off from communication with the rest of the universe. Such discoveries redirected attention to the Schwarzschild model of the exterior of a star. It had generally been assumed that the model becomes singular at the Schwarzschild radius r* = 2m. This seemed of no great significance since in the case of our sun, for example, whose radius is about 700,000 km, the Schwarzschild model's presumed singularity is buried in at r* = 3 km. But for black holes it was gradually realized that radius r* = 2m is not a singularity but rather the "horizon" from which nothing, not even light, can emerge.

In 1954, the Reports of the State University of Kazan, a city 300 miles east of Moscow in the then Soviet Union, contained this year a classification of spacetimes given by the young physicist A. Z. Petrov. The families of radially ingoing and outgoing light rays in the Schwarzschild model show that it has what is now called Petrov type D. Petrov's classification was slow in making its way into the mainstream, but was crucial to the next major development.

All stars rotate. For a very slowly turning star like our sun this is not very important, but when a star collapses, conservation of angular momentum implies that its rate of rotation increases. Thus a neutron star, for example, can be expected to rotate at a fantastic rate. It is believed that pulsars are rapidly rotating neutron stars.

But the star producing Schwarzschild spacetime does not rotate. In 1915, this static model had been found in only a few weeks, so it must have been considered fairly easy to set it spinning. However, years passed without success. Then in 1963, the British-educated New Zealand physicist Roy Kerr, working at the University of Texas, figured out an ingenious way of doing it. Bearing in mind that Schwarzschild spacetime has Petrov type D, he did not aim directly at the elusive rotating model, but instead examined an algebraically simple class of type D metric tensors. The long-sought metric appeared. Kerr's minimal one-and-a-half page announcement of his discovery was followed two years later by elaborate detailed calculations.

In 1967, R. H. Boyer and R. W. Lindquist made Kerr spacetime more accessible by introducing the elegant coordinate system that now bears their names. In the same paper they found the maximally extended Kerr spacetimes and investigated their geodesics. However, a full analysis of Kerr geodesics become possible only with the discovery of a fourth geodesic first-integral by Brandon Carter, a student of Stephen Hawking at the University of Cambridge, in 1968. Carter's paper remains the best brief exposition of the global properties of Kerr spacetime.

In general relativity, gravity is a manifestation of the curvature of spacetime. Massive objects distort space and time, so that the usual rules of geometry don't apply anymore. Near a black hole, this distortion of space is extremely severe and causes black holes to have some very strange properties. In particular, a black hole has something called an event horizon. This is a spherical surface that marks the boundary of the black hole. You can pass in through the horizon, but you can't get back out. In fact, once you've crossed the horizon, you're doomed to move inexorably closer and closer to the singularity at the center of the black hole.

You can think of the horizon as the place where the escape velocity equals the velocity of light. Outside of the horizon, the escape velocity is less than the speed of light, so if you fire your rockets hard enough, you can give yourself enough energy to get away. But if you find yourself inside the horizon, then no matter how powerful your rockets are, you can't escape.

The horizon has some very strange geometrical properties. To an observer who is sitting still somewhere far away from the black hole, the horizon seems to be a nice, static, unmoving spherical surface. But once you get close to the horizon, you realize that it has a very large velocity. In fact, it is moving outward at the speed of light. That explains why it is easy to cross the horizon in the inward direction, but impossible to get back out. Since the horizon is moving out at the speed of light, in order to escape back across it, you would have to travel faster than light. You can't go faster than light, and so you can't escape from the black hole. The horizon is in a certain sense sitting still, but in another sense it is flying out at the speed of light.

Once you're inside of the horizon, spacetime is distorted so much that the coordinates describing radial distance and time switch roles. That is, "r", the coordinate that describes how far away you are from the center, is a timelike coordinate, and "t" is a spacelike one. One consequence of this is that you can't stop yourself from moving to smaller and smaller values of r, just as under ordinary circumstances you can't avoid moving towards the future, that is, towards larger and larger values of t. Eventually, you will hit the singularity at r = 0. Trying to avoid it is like trying to avoid traveling forward in time.

There is good observational evidence from X-ray observations and from the Hubble Space Telescope that there are massive black holes, with masses more than a million times that of the Sun, exist in the centers of some galaxies In addition, there is also evidence for black holes which result from the final stage of a star's life. These galactic black holes, so called because they are in our Galaxy, are usually 3 - 10 times the mass of the Sun. They also often orbit a companion star, and by observing the X-rays emitted from the region near the black hole and/or the visible light from the companion, a mass can be determined for the black hole. If it is larger than the accepted mass for a neutron star, about 1.5 times the mass of the Sun, then astronomers generally agree the object is a black hole. There are quite a number of these objects known in our galaxy. Perhaps the first object to be generally recognized as a black hole is the X-ray binary star Cygnus X-1. Its effect on its companion star suggested as early as 1971 that it must be a compact object with a mass too high for it to be a neutron star. That was two years after JohnWheeler coined the term "black hole".

Stephen Hawking determined that black holes evaporate. All around us are virtual pairs, pairs of particles and antiparticles that are continually coming in and out of the vacuum. The virtual pair is a fluctuation, one particle has some positive energy, the other has some negative one. It is only the particle with positive energy that can escape far way, and be seen as black hole radiation. Negative energy particle falls into the hole, and, through E = mc2, brings some negative mass to the hole. The black hole mass decreases, and the horizon shrinks. Soon the black hole evaporates.

The first hint that there might be a connection between black holes and thermodynamics came with the mathematical discovery in 1970 that the surface area of the event horizon, the boundary of a black hole, has the property that it always increases when additional matter or radiation falls into the black hole. Moreover, if two black holes collide and merge to form a single black hole, the area of the event horizon around the resulting black hole is greater than the sum of the areas of the event horizons around the original black holes. These properties suggest that there is a resemblance between the area of the event horizon of a black hole and the concept of entropy in thermodynamics. Entropy can be regarded as a measure of the disorder of a system or, equivalently, as a lack of knowledge of its precise state. The famous second law of thermodynamics says that entropy always increases with time.