 Given their bizarre properties, we might question whether a black hole could even exist. This effectively comes down to the question, is it possible for an object to undergo gravitational collapse? In the early days of relativity theory, most physicists, including Einstein, intuitively felt this was impossible. That some physical process would prevent the density of matter from growing large enough to create an event horizon. Even a star is a spherically symmetric ball of gas. At any spherical boundary within the star, there will be a gravitational force pressing inward, representing the weight of all the stellar material outside the boundary. We'll quantify the effect of gravity by the amount of energy required to pull the star apart. Viewing the star in cross-section, imagine we somehow take hold of the top layer of material. And working against gravity, pull it away to infinity. We continue to pull the star apart, layer by layer, until nothing is left. Calculating the total work required to disassemble the star, we find that it varies as the star mass m squared over the star radius r, multiplied by a constant, which we don't bother with here. We define the gravitational energy of the star to be the negative of this work of disassembly. General systems tend toward the state of lowest energy. If gravity was the only force acting, this would be an energy of minus infinity at r equals zero. Gravity wants the star to collapse. For the star to resist collapse and remain in equilibrium, the gravitational force must be offset by an outward pressure of equal magnitude. The primary sources of stellar pressure are first, gas pressure, analogous to earth's atmospheric pressure, although under much more extreme conditions. Gas pressure varies with temperature. At higher temperatures, gas particles are, on average, moving faster, and so transfer more momentum when they collide. Second is radiation pressure, due to transfer of momentum by the thermal emission, absorption, and scattering of electromagnetic waves inside the star. Now, both of these sources of pressure vary with temperature. At non-zero temperature, a star will radiate power into space. This must be made up by an internal power source, otherwise the star will cool. Nuclear fusion is the primary source of this power. As light nuclei fuse to form heavier nuclei, energy is released, but only up to a point. For nuclei heavier than iron and nickel, fusion requires a net input of energy, such as can occur in a supernova explosion. So even though it may take billions of years, a star is ultimately destined to exhaust all internal sources of energy and grow cold, bringing gas and radiation pressure to an end. What remains is the phenomenon sometimes referred to as the stability of matter, the fundamental explanation of why solid matter is solid, the reason why we don't fall through the floor. To be completely rigorous, we would need to solve for the quantum mechanical state of all nuclei and electrons in the frozen star and the resulting forces opposing gravity. Instead, what we'll consider is a very rough, back-of-the-envelope type calculation of what is expected to be the dominant effect at extreme densities, so-called electron degeneracy pressure. We first consider a non-relativistic analysis. Let's denote the star's radius and mass by big R and big M. Assume that each electron, with mass little m, is constrained to a region of radius little r, and that there are n sub e electrons per unit mass. In most elements, there is approximately one electron per proton neutron pair, so this is more or less a fixed value. The total number of electrons in the star is that n sub e times the star mass. This is also the volume of the star divided by the volume per electron, which varies as the cube of big R over little r. Solving this expression for little r, we find that it varies as big R over the cube root of big M. Here and in what follows, we drop physical constants and focus on the effects of the star's radius and mass. The uncertainty principle of quantum mechanics implies that if an electron is constrained to a region of dimension little r, it must have a momentum p on the order of Planck's constant h over r. Using the above expression, this varies as the cube root of the star's mass over its radius. The non-relativistic expression for kinetic energy of an electron is its momentum squared over twice its mass. This varies as the two-thirds power of star mass over the square of star radius. This kinetic energy is not dependent on the star's temperature, it's an intrinsic quantum mechanical property, even at absolute zero. Multiplying by the number of electrons, we get the total kinetic energy. Since the number of electrons varies as the star's mass, we obtain a total kinetic energy of big M to the five-thirds power over big R squared. We've already seen that the gravitational energy varies as the star's mass squared over its radius. Combining these two terms and removing a common factor, we see that for any mass, the total energy grows arbitrarily large as the radius grows arbitrarily small. From this, we conclude that gravitational collapse is impossible. Now we look at a relativistic analysis. The steps are the same until we get to the electron kinetic energy. The relativistic expression for kinetic energy is square root of p squared plus M squared minus M. For small p, this agrees with the classical expression, but for very large p, which corresponds to very small star radius, it behaves like p and not like p squared. The result is that the total kinetic energy varies as one over the star radius, not one over the radius squared. Combining this with the gravitational energy, we get a total energy that varies as a constant over the star radius. We see that if the star's mass is small, the constant will be positive, so the energy would again approach positive infinity as the radius approaches zero. However, if the mass is large enough, the constant will be negative, and the energy will approach negative infinity as the radius approaches zero. That is, the star will undergo gravitational collapse. This phenomenon is called the Chandrasekhar limit. After Subramanian Chandrasekhar, who derived it rigorously in a series of papers starting in 1931, he was able to solve for the ultimate radius of a star as a function of its mass, and showed that this radius is zero if the mass exceeds a certain limit. Current calculations set this limit at about 1.4 times the mass of our sun. Now, the process of stellar evolution is typically dynamic and often violent, resulting in some of the star's mass being ejected as a planetary nebula, supernova remnant, or in some other phenomenon. However, if the mass of the remaining core is above the Chandrasekhar limit, then it will not be able to resist gravitational collapse. But what does it collapse to? The answer is, not necessarily a black hole. By the 1930s, a theoretical picture of nuclear physics was emerging. Nuclear were known to contain protons and neutrons. It was thought possible that under certain conditions an electron could be captured by a nucleus, converting one of the protons to a neutron with the emission of a neutrino. This process of electron capture was observed and reported in 1937 by Luis Alvarez. The implication was that in a collapsing star all electrons and protons might combine in this manner, leaving an object composed of nothing but neutrons, what we now call a neutron star. This led to a consideration of the stability of neutron stars through neutron degeneracy pressure. In 1939, Oppenheimer and Volkov analyzed the problem and showed that there is an upper limit to the mass of a neutron star, beyond which it will undergo gravitational collapse. They concluded, actual stellar matter after the exhaustion of thermonuclear sources of energy will, if massive enough, contract indefinitely. This limit is not as well defined as the Chandra Shekhar limit, because the physics is more extreme and not as well understood. It's thought to be less than about three solar masses. Currently, the largest known neutron star has an estimated mass about twice that of the sun. A neutron star over this limit might collapse to a black hole. Another hypothetical possibility is the transformation of neutrons into a collection of free quarks, forming a quark star. Currently, there is no substantial evidence of quark stars. Even if such things did exist, they would have their own upper mass limit. And since quarks are elementary particles, there should be no further transformations possible, and we would finally have a collapse into a black hole. So, it seems that there is no theoretical process that can stop a large enough star from ultimately forming a black hole through gravitational collapse. Someone who pushed back against the idea of black holes was Albert Einstein. Someone ironically, since it was his theory that predicted them. In 1939, he published a paper claiming, The Schwarzschild Singularity does not appear for the reason that matter cannot be concentrated arbitrarily, and this is due to the fact that otherwise, the constituting particles would reach the velocity of light. His argument was based on assuming the particles of a collapsing star would follow essentially circular orbits. He remarked, Although the theory given here treats only clusters whose particles move along circular paths, it does not seem to be subject to reasonable doubt that more general cases will have analogous results. But, Einstein was wrong, and we've already seen why in our previous discussion of orbits around a black hole. It's true that there is a radius outside the event horizon, where the velocity of a circular orbit is the speed of light. And it's true that no circular orbits smaller than this are possible. However, it does not follow that an in-falling particle passing through this radius would have to travel faster than light. In fact, we've already calculated such orbits. They're not circular, they're spirals. And, as we've shown, a particle can fall in radially. Both of these processes are well described by relativity theory. Regardless of how bizarre they may have seemed to Einstein and others, no known physical process has been found that could keep a black hole from forming. Given that they are theoretically possible, the obvious question then is, can we detect black holes? By definition, we cannot see a black hole. Classically, they do not emit any radiation for us to detect. So any evidence for their existence will have to be circumstantial. What they do produce is a very strong gravitational field. If a visible object moves through that field, it will exhibit an acceleration that indicates both the mass and distance of the black hole. Due to breakthroughs in observational technology, astronomers are now able to view and track stars near the center of our Milky Way galaxy. What they've seen are several stars orbiting an invisible central object. From these orbits, the object's mass has been determined to be some 4 million solar masses. Moreover, the orbit's place and upper limit on its size, roughly the size of our solar system. A supermassive black hole is the only known object consistent with these observations. Other evidence for black holes can come from an accretion disk. If material falls into a black hole, it can accelerate to near the speed of light. These high speeds and violent collisions produce tremendous heat, which in turn generates radiation. Temperatures of millions of degrees are possible, which can produce intense x-ray emissions. One source of an accretion disk can be a companion star in mutual orbit with the black hole. The radiation burst associated with material falling into an accretion disk is called an x-ray nova. However, this by itself is not evidence of a black hole. The intense gravity of a neutron star can also produce an accretion disk in x-ray nova. But there's a key difference illustrated in this figure. As material spirals in through the accretion disk, the radiation it produces is ever more redshifted. If the central object is a black hole, this redshifting continues without end, and the material disappears into the event horizon. However, a neutron star has a solid surface, and the in-spirits of a black hole star has a solid surface, and the in-spiriling material ultimately strikes this, producing an additional characteristic x-ray burst. It's the lack of this final burst that provides evidence that the central object has no solid surface. And the only massive object without a solid surface is a black hole. Finally, gravitational waves were recently observed by the laser interferometer Gravitational Wave Observatory. These bear the characteristic signature of merging black holes, and cannot be explained by any other known process.