Solution

The change in g, Ag, is given by Ag = (dg/dr) Ar where Ar is the distance over which the change is to be found. In this case, Ar is the height of the astronaut, which we will take to be 2m. We find dg/dr from equation (8.13) to be

8.4.2 Approaching a black hole

What is it like to fall into a black hole? We consider two astronauts. One approaches the black hole, and the other stays a safe distance away. The various steps are indicated in Fig. 8.12.

We assume that the astronaut approaching the black hole can send out signals in various directions, including back to the other astronaut. As the first astronaut approaches the black hole, the first thing the distant astronaut would notice is the redshift in the signals received. The magnitude of the redshift increases as the first astronaut becomes closer to the Schwarzschild radius.

Before the Schwarzschild radius is reached, another effect becomes noticeable. The paths of photons sent out by the first astronaut are not straight lines. They bend. The only direction in which the astronaut can aim a beam and not have

(Note that the minus sign means that gravity is stronger at the feet than at the head.) For Ar = 2 X 102 cm, we have

This is two billion times the acceleration of gravity at the surface of the Earth. The astronaut would be pulled apart with a force of over a billion times the astronaut's weight!

The tidal force, dg/dr, is proportional to M/r3, just as is the density. Therefore, the tidal force will be less for more massive black holes, falling to more tolerable values for very massive black holes (see Problem 8.9.)

Exit Cone

Exit Cone

Approaching a black hole.Astronaut 1 approaches the black hole while astronaut 2 stays behind. In each frame, the emitted and received wave correspond to a beam sent from 1 to 2.

it bend is straight up. If the beam is not aimed sufficiently close to the vertical, the bending will be so great that the light will not escape. Only light aimed into a cone about the vertical, called the exit cone, will escape. As the first astronaut moves closer to the Schwarzschild radius, the exit cone becomes smaller. At a distance equal to (3/2)RS, photons aimed horizontally go into orbit around the black hole. The sphere of orbiting photons is called the photon sphere. If you were to look straight out, along the horizon, you would see the back of your head.

The second astronaut never actually sees the first astronaut reach the Schwarzschild radius. The gravitational time dilation is so great that, as RS is approached, the second astronaut thinks that it takes the first astronaut an infinite amount of time to reach RS. The time dilation makes the first astronaut appear to slow down as RS is approached.

From the point of view of the first astronaut, there is no such respite. The Schwarzschild radius is reached very quickly. If the black hole is of sufficiently small mass, the tidal forces would tear the first astronaut apart. However, if the black hole is massive enough, the tidal forces might be survived and the astronaut crosses RS. When this happens, we say that the astronaut has crossed the event horizon. If the black hole is massive enough, the astronaut might not notice anything unusual, except that escape is impossible!

Once inside the black hole, the inevitable journey to the center continues. The gravitational time dilation is so great that time passes slowly. However, the headlong rush through space continues. Outside the black hole, it is time that rushes on while distance is covered slowly. It is as if crossing the event horizon has interchanged to roles of space and time.

The second astronaut can tell nothing about what is going on inside the black hole. In fact, the only properties of a black hole that can be deduced are its mass, radius, electric charge and angular momentum. (So far, we have assumed zero angular momentum. We will discuss rotating black holes below.) The external simplicity of black holes is summarized in a theorem that states that black holes have no hair.

So far we have been discussing non-rotating black holes. The structure of a rotating black hole

Ergosphere

Plane perpendicular to axis of rotation through rotating mass

Ergosphere

Plane perpendicular to axis of rotation through rotating mass

Side View

Rotating black hole.The structure of the surface is complicated, so we show two cuts. In the upper figure, we see the intersection of various surfaces with the plane perpendicular to the axis of rotation. At the center is a disk singularity. (This is just a disk and doesn't extend above or below the plane.) There are two infinite redshift surfaces and two event horizons. Between the event horizons, the roles of space and time are reversed.The region between the outer infinite redshift surface and the outer event horizon is called the ergosphere.The lower figure shows a side view.

Side View

Rotating black hole.The structure of the surface is complicated, so we show two cuts. In the upper figure, we see the intersection of various surfaces with the plane perpendicular to the axis of rotation. At the center is a disk singularity. (This is just a disk and doesn't extend above or below the plane.) There are two infinite redshift surfaces and two event horizons. Between the event horizons, the roles of space and time are reversed.The region between the outer infinite redshift surface and the outer event horizon is called the ergosphere.The lower figure shows a side view.

is somewhat more complicated than that of a non-rotating black hole, and is depicted schematically in Fig. 8.13. The situation shown is for the case in which the angular momentum per unit mass, J/M, is less than GM/c. For the case shown, there are two infinite redshift surfaces instead of a single event horizon. Between the two surfaces, the roles of space and time are reversed, just as inside the event horizon in the non-rotating case. The region between the outer infinite redshift surface and the event horizon is called the ergos-phere. The name results from the fact that there is a way to extract energy from the black hole by moving particles through the ergosphere in the correct trajectory.

8.4.3 Stellar black holes

In Chapter 11 we will see that some types of stars evolve to a point were nothing can support them. Such a star will collapse right through the

Schwarzschild radius for its mass, and will become a black hole. Black holes would be a normal state for the evolution of some stars. How would we detect a stellar black hole? We obviously could not see it directly. We could not even see it in silhouette against a bright source, since the area blocked would be only a few kilometers across. We have to detect stellar black holes indirectly. We hope to see their gravitational effects on their surrounding environment. This is not a hopeless task, since we might expect to find a reasonable number in binary systems. We will discuss the probable detection of black holes in binary systems in Chapter 12.

8.4.4 Non-stellar black holes

Black holes that have masses much less than a solar mass are called mini black holes. We think that mini black holes might have formed when the density of the universe was much higher than it is now. (The conditions in the early universe will be discussed in Chapter 21.) These may still exist. The British physicist Stephen Hawking has found that there is a mechanism by which mini black holes could actually evaporate. Hawking is studying the relationship between gravity and quantum mechanics, and the process he has proposed is a quantum mechanical one.

This mechanism involves a different concept of a vacuum than we are accustomed to seeing. In classical physics, a vacuum is simply nothing. In quantum mechanics it is possible to make something out of nothing, if you don't do it for long. It amounts to borrowing energy for a brief time interval. The more energy you borrow, the less time you can borrow it for. It is related to the uncertainty principle (which we discussed in Chapter 3). We have talked about the uncertainty principle as it relates to momentum and position. However, it also relates to energy and the lifetime of a state. It says that if the state has a lifetime At, then the energy of the state is uncertain by an amount AE, given by

Annihilated

Electron-Positron Pair Created

Annihilated

Electron-Positron Pair Created

Free

Free

Event Horizon

Captured

Event Horizon

Captured

Pair production. (a) The process in free space.An electron and positron are created out of nothing, but quickly come back together to annihilate. (b) Near a black hole, one of the particles can be captured before they can annihilate, and the other escapes, carrying energy away from the black hole.

The longer lived a state, the more accurately its energy can be determined. Since the energy of a state is uncertain by AE, it is possible for us to have this extra amount of energy and not detect it.

As a result of the uncertainty principle, a quantum mechanical vacuum is a very busy place. At any place it is possible to create a particle-antiparticle pair (Fig. 8.14). (We will discuss antiparitcles in Chapter 21.) It requires an energy equal to 2mc2, where m is the mass of the particle (and the antiparticle). The pair can exist for at most a time h/[(2-n-)mc2]. Before the time is up, they must find each other and annihilate. Since electrons have masses that are much less than protons, an electron-positron (antielectron) pair will live longer than a proton-antiproton pair. We can therefore think of a vacuum as being made up of continuously appearing and disappearing electron-positron pairs (with a small contribution from heavier particle-antiparticle pairs). The phenomenon is called vacuum polarization.

When an electron-positron pair is created just outside a black hole, it is possible for one of the particles to be pulled into the black hole before the two recombine. The other particle will continue moving away from the black hole. The two cannot recombine. The particles then exist for much longer than the time limit for violating conservation of energy. We must therefore make up the energy from somewhere. This process actually reduces the mass of the black hole. The black hole shrinks slightly. For mini black holes this energy loss can be a significant fraction of the mass of the black hole. Eventually, the black hole shrinks to the point where it disappears in a small burst of gamma radiation. The more massive a black hole is when it starts out, the longer it will live. An estimate for the lifetime of a black hole of mass M (in grams) is (10~26 s)(M3). So, a black hole of about 1014 g would have a liftime of about 1010 yr, a little less than the age of the uni verse. In the lifetime of the universe, black holes smaller than some given mass should have disappeared. Those at that mass should just be dying now. Some physicists have suggested that when this happens we should be able to see the burst of gamma radiation.

At the other end of the mass scale, much larger than stellar black holes, are maxi black holes. They probably result from large amounts of material gathering together in a small region. In Chapter 19, we will see evidence for 108 to 109 M0 black holes being present in the centers of many galaxies.

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