Neutron stars are peculiar objects. Nevertheless, theory predicts that they are in equilibrium, as are most other stars. For neutron stars, however, equilibrium does not mean a balance between the inward pull of gravity and the outward pressure of hot gas. Instead, as we have seen, the outward force is provided by the pressure of tightly packed neutrons. Squeezed together, the neutrons form a hard ball of matter that not even gravity can compress further. Or do they? Is it possible that given enough matter packed into a small enough volume, the collective pull of gravity can eventually crush any opposing pressure? Can gravity continue to compress a massive star into an object the size of a planet, a city, a pinhead—even smaller? The answer, apparently, is yes.
We have seen that the eventual evolution of a star depends critically on its mass. Low-mass stars leave behind a compact remnant known as a white dwarf. 
(Sec. 20.3) High-mass stars can also produce a compact remnant, in the form of a neutron star. The laws of physics make specific predictions about the masses of these core remnants. A white dwarf must be less than about 1.4 solar masses—the so-called Chandrasekhar mass, beyond which the electrons cannot support the core against its own gravity. (Sec. 21.3) Similarly, neutron stars resulting from supernovae have masses between about 1.4 and 3 solar masses.* The lower limit of 1.4 solar masses comes from stellar evolution: the iron core of an evolved star must exceed the Chandrasekhar mass for core collapse to begin and a supernova to occur. The (quite uncertain) upper limit of 3 solar masses is the neutron-star equivalent of the Chandrasekhar mass—beyond 3 solar masses not even tightly packed neutrons can withstand the star's gravitational pull.
*(The dividing lines at 1.4 and 3 solar masses are somewhat uncertain because they ignore the effects of magnetism and rotation, both of which are surely present in the cores of evolved stars. Because these effects can compete with gravity, they influence the evolution of stars. (Sec. 19.1) In addition, we do not know for certain how the basic laws of physics might change in regions of very dense matter that is both rapidly spinning and strongly magnetized. However, we expect that these dividing lines will shift generally upward when magnetism and rotation are included, because even larger amounts of mass will then be needed for gravity to compress stellar cores into neutron stars or black holes.)
In fact, we know of no force that can counteract gravity beyond the point at which neutron degeneracy pressure is overwhelmed. If enough material is left behind after a supernova, as may happen in the case of an extremely massive progenitor star, gravity finally wins, and the central core collapses forever. As the core shrinks, the gravitational pull in its vicinity eventually becomes so great that even light itself is unable to escape. The resultant object therefore emits no light, no radiation, no information whatsoever. Astronomers call this bizarre end point of stellar evolution, in which a massive core remnant collapses in on itself and vanishes forever, a black hole.
Newtonian mechanics—up to now our reliable and indispensable tool in understanding the universe—cannot adequately describe conditions in or near black holes (More Precisely 2-2). To comprehend these collapsed objects we must turn instead to the modern theory of gravity, Einstein's general theory of relativity (see More Precisely 22-1). The Einsteinian description of the universe is equivalent to Newton's for situations encountered in everyday life, but the two theories diverge radically in circumstances where speeds approach the speed of light and in regions of intense gravitational fields.
Despite the fact that general relativity is necessary for a proper description of black-hole properties, we can still usefully discuss some aspects of these strange bodies in more or less Newtonian terms. Let's reconsider the familiar Newtonian concept of escape speed—the speed needed for one object to escape from the gravitational pull of another—supplemented by two key facts from relativity: (1) nothing can travel faster than the speed of light, and (2) all things, including light, are attracted by gravity.
To explore how gravity attracts even light, let's reconsider the concept of escape speed—the speed needed for one object—a molecule, a baseball, a rocket, whatever—to escape the gravitational pull of another. In Chapter 2 we noted that escape speed is proportional to the square root of a body's mass divided by the square root of its radius. (Sec. 2.7) Earth's radius is 6400 km, and the escape speed from Earth's surface is just over 11 km/s.
Now consider a hypothetical experiment in which Earth is squeezed on all sides by a gigantic vise. As our planet shrinks under the pressure its mass remains the same. The escape speed increases because the radius is decreasing. Suppose Earth were compressed to one-fourth its present size. The proportionality just mentioned for escape speed then predicts that the escape speed would double (because 
). Any object escaping from this hypothetically compressed Earth would need a speed of at least 22 km/s.
Imagine compressing Earth some more. Squeeze it, for example, by an additional factor of 1000, making its radius hardly more than a kilometer. Now a speed of about 630 km/s would be needed to escape. Compress Earth still further, and the escape speed continues to rise. If our hypothetical vise were to squeeze Earth hard enough to crush its radius to about a centimeter, then the speed needed to escape its surface would reach 300,000 km/s. But this is no ordinary speed—it is the speed of light, the fastest speed allowed by the laws of physics as we currently know them.
Thus, if by some fantastic means the entire planet Earth could be compressed to less than the size of a grape, the escape speed would exceed the speed of light. And because nothing can exceed that speed, the compelling conclusion is that nothing—absolutely nothing—could escape from the surface of such a compressed body. Even radiation—radio waves, visible light, X-rays, photons of all wavelengths—would be unable to escape the intense gravity of our reshaped Earth. With no photons leaving, our planet would be invisible and uncommunicative—no signal of any sort could be sent to the universe beyond. The origin of the term black hole becomes clear. For all practical purposes, such a supercompact Earth could be said to have disappeared from the universe! Only its gravitational field would remain behind, betraying the presence of its mass, now shrunk to a point.*
*(In fact, we now know that regardless of the composition or condition of the object that formed the hole, only three physical properties can be measured from the outside—the hole's mass, charge, and angular momentum. All other information is lost once the infalling matter crosses the event horizon. Thus, only three numbers are required to describe completely a black hole's outward appearance. In this chapter we will consider only holes that formed from nonrotating, electrically neutral matter. Such objects are completely specified once their masses are known.)
Astronomers have a special name for the critical radius at which the escape speed from an object would equal the speed of light and within which the object could no longer be seen. It is the Schwarzschild radius, after Karl Schwarzschild, the German scientist who first studied its properties. The Schwarzschild radius of any object is simply proportional to its mass. For Earth, it is 1 cm; for Jupiter, at about 300 Earth masses, it is about 3 m; for the Sun, at 300,000 Earth masses, it is 3 km. For a 3—solar mass stellar core remnant, the Schwarzschild radius is about 9 km. As a convenient rule of thumb, the Schwarzschild radius of an object is simply 3 km multiplied by the object's mass, measured in solar masses. Every object has a Schwarzschild radius. It is the radius to which the object would have to be compressed for it to become a black hole. Put another way, a black hole is an object that happens to lie within its own Schwarzschild radius.
The surface of an imaginary sphere with radius equal to the Schwarzschild radius and centered on a collapsing star is called the event horizon. It defines the region within which no event can ever be seen, heard, or known by anyone outside. Even though there is no matter of any sort associated with it, we can think of the event horizon as the "surface" of a black hole.
A 1.4—solar mass neutron star has a radius of about 10 km and a Schwarzschild radius of 4.2 km. If we were to keep increasing the star's mass, the star's Schwarzschild radius would grow, although its actual physical radius would not. In fact the radius of a neutron star decreases slightly with increasing mass. By the time our neutron star's mass exceeded about 3 solar masses, it would lie just within its own event horizon, and it would collapse of its own accord. It would not stop shrinking at the Schwarzschild radius—the event horizon is not a physical boundary of any kind, just a communications barrier. The remnant would shrink right past it to ever-diminishing size on its way to being crushed to a point.
Thus, provided that at least 3 solar masses of material remain behind after a supernova explosion, the remnant core will collapse catastrophically, diving below the event horizon in less than a second. The core simply "winks out," disappearing and becoming a small dark region from which nothing can escape—a literal black hole in space. According to theory, this is the fate of any star whose main-sequence mass exceeds about 20-30 times the mass of the Sun.
An alternative way of seeing the significance of the Schwarzschild radius is to consider what happens to rays of light emitted at different distances from the event horizon of a black hole. Imagine moving a light source closer and closer to the hole, as shown in Figure 22.11. Let's suppose that the source emits radiation uniformly in all directions.
Figure 22.11 This light source, moving closer and closer to a black hole, emits radiation in all directions. (a) At large distances from the hole, most of the light (marked in blue) escapes into space. (b) As the distance decreases, more and more of the radiation is deflected by the black hole's gravity onto paths that intersect the event horizon, and this radiation is trapped (shown in red). (c) At a distance of 1.5 Schwarzschild radii, photons can orbit the hole on circular trajectories, which mark the photon sphere. (d) Eventually, when the source reaches the horizon, all the light it emits is destined to enter the hole.
At large distances from the black hole (Figures 22.11a,b) essentially all the radiation eventually escapes into space. Only that portion of the beam that is aimed directly at the hole is captured. As the light source moves closer, however, the effect of the hole's strong gravity becomes evident. Some photons that would have missed the hole if light traveled in straight lines are instead deflected onto paths that cross the event horizon, and these photons are trapped by the black hole.
At 1.5 Schwarzschild radii (Figure 22.11c), exactly half the radiation emitted by our light source escapes into space. Photons emitted perpendicular to the line joining the source to the center of the hole move in circular orbits at this radius, never escaping the hole's gravity but never crossing the event horizon. The surface on which these photons move as they travel on their circular paths is called the photon sphere.
Closer still (Figure 22.11d), the amount of deflection continues to increase, and the fraction of the radiation that escapes into space steadily decreases until, close to the event horizon, only a thin sliver of our beam is able to escape. At the event horizon itself, the sliver vanishes completely. All light rays from that surface, whatever their initial direction, are destined to enter the hole. The black hole's immense gravitational field dominates the trajectories of all particles, even photons, in its vicinity.
