In 1916 Albert Einstein presented his general theory of relativity, which included a new concept of the nature of gravitation. In his theory, a massive object actually changes the geometry of the space around it. Other objects sense this altered geometry and respond by being attracted to the first object. The general theory of relativity is beyond our scope in this chapter, but we can look at one of its most startling predictions: the existence of black holes, objects whose gravitational influence is so great that nothing—not even light—can escape them. We can understand the basic idea of a black hole by using Newtonian principles.
Black holes, the schwarzschild radius, and the event horizon
The first expression for escape speed in Eq. (13.29) suggests that a body of mass M will act as a black hole if its radius R is less than or equal to a certain critical radius. How can we determine this critical radius? You might think that you can find the answer by simply setting v = c in Eq. (13.29). As a matter of fact, this does give the correct result, but only because of two compensating errors. The kinetic energy of light is not mc2/2, and the gravitational potential energy near a black hole is not given by Eq. (13.9). In 1916, Karl Schwarzschild used Einstein’s general theory of relativity to derive an expression for the critical radius RS, now called the Schwarzschild radius. The result turns out to be the same as though we had set v = c in Eq. (13.29), so
Solving for the Schwarzschild radius RS, we find
If a spherical, nonrotating body with mass M has a radius less than RS, then nothing (not even light) can escape from the surface of the body, and the body is a black hole (Fig. 13.27). In this case, any other body within a distance RS of the center of the black hole is trapped by the gravitational attraction of the black hole and cannot escape from it.
The surface of the sphere with radius RS surrounding a black hole is called the event horizon: Since light can’t escape from within that sphere, we can’t see events occurring inside. All that an observer outside the event horizon can know about a black hole is its mass (from its gravitational effects on other bodies), its electric charge (from the electric forces it exerts on other charged bodies), and its angular momentum (because a rotating black hole tends to drag space—and everything in that space—around with it). All other information about the body is irretrievably lost when it collapses inside its event horizon.
A visit to a Black hole
At points far from a black hole, its gravitational effects are the same as those of any normal body with the same mass. If the sun collapsed to form a black hole, the orbits of the planets would be unaffected. But things get dramatically different close to the black hole. If you decided to become a martyr for science and jump into a black hole, the friends you left behind would notice several odd effects as you moved toward the event horizon, most of them associated with effects of general relativity.
If you carried a radio transmitter to send back your comments on what was happening, your friends would have to retune their receiver continuously to lower and lower frequencies, an effect called the gravitational red shift. Consistent with this shift, they would observe that your clocks (electronic or biological) would appear to run more and more slowly, an effect called time dilation. In fact, during their lifetimes they would never see you make it to the event horizon.
In your frame of reference, you would make it to the event horizon in a rather short time but in a rather disquieting way. As you fell feet first into the black hole, the gravitational pull on your feet would be greater than that on your head, which would be slightly farther away from the black hole. The differences in gravitational force on different parts of your body would be great enough to stretch you along the direction toward the black hole and compress you perpendicular to it. These effects (called tidal forces) would rip you to atoms, and then rip your atoms apart, before you reached the event horizon.
Detecting Black holes
If light cannot escape from a black hole and if black holes are as small as Example 13.11 suggests, how can we know that such things exist? The answer is that any gas or dust near the black hole tends to be pulled into an accretion disk that swirls around and into the black hole, rather like a whirlpool (Fig. 13.28, next page). Friction within the accretion disk’s gas causes it to lose mechanical energy and spiral into the black hole; as it moves inward, it is compressed together. This causes heating of the gas, just as air compressed in a bicycle pump gets hotter. Temperatures in excess of 106 K can occur in the accretion disk, so hot that the disk emits not just visible light (as do bodies that are “red-hot” or “white-hot”) but x rays. Astronomers look for these x rays (emitted by the gas material before it crosses the event horizon) to signal the presence of a black hole. Several promising candidates have been found, and astronomers now express considerable confidence in the existence of black holes.
Black holes in binary star systems like the one depicted in Fig. 13.28 have masses a few times greater than the sun’s mass. There is also mounting evidence for the existence of much larger supermassive black holes. One example lies at the center of our Milky Way galaxy, some 26,000 light-years from earth in the direction of the constellation Sagittarius. High-resolution images of the galactic center reveal stars moving at speeds greater than 1500 km>s about an unseen object that lies at the position of a source of radio waves called Sgr A* (Fig. 13.29).
By analyzing these motions, astronomers can infer the period T and semi-major axis a of each star’s orbit. The mass mX of the unseen object can be calculated from Kepler’s third law in the form given in Eq. (13.17), with the mass of the sun mS replaced by mX:
The conclusion is that the mysterious dark object at the galactic center has a mass of 8.2 * 1036 kg, or 4.1 million times the mass of the sun. Yet observations with radio telescopes show that it has a radius no more than 4.4 * 1010 m, about one-third of the distance from the earth to the sun. These observations suggest that this massive, compact object is a black hole with a Schwarzschild radius of 1.1 * 1010 m. Astronomers hope to improve the resolution of their observations so that they can actually see the event horizon of this black hole.
Other lines of research suggest that even larger black holes, in excess of 109 times the mass of the sun, lie at the centers of other galaxies. Observational and theoretical studies of black holes of all sizes continue to be an exciting area of research in both physics and astronomy.
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