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Week #10 Notes Black Holes: The End of Space and Time The Formation of a Stellar-Mass Black Hole Astronomers had long assumed that the most massive stars would somehow lose enough mass to wind up as white dwarfs. When the discovery of pulsars (neutron stars) ended this prejudice, it seemed more reasonable that black holes could exist. If more than 2 or 3 times the mass of the Sun remains after the supernova explosion, the star collapses through the neutron-star stage. We know of no force that can stop the collapse. (In some cases, the supernova explosion itself may fail, so the remaining mass can be much larger than 2 or 3 solar masses.) The Formation of a Stellar-Mass Black Hole We may then ask what happens to an evolved 5-, 10-, or 50-solarmass star as it collapses, if it retains more than 2 or 3 solar masses. As far as we know, it must keep collapsing, getting denser and denser. Einstein’s general theory of relativity suggests that the presence of mass or energy warps (bends) space and even time in its vicinity (see figure); recall our discussion of the Sun’s gravity in Section 10.3. The greater the density of the material, the more severe is the warp. The Formation of a Stellar-Mass Black Hole Eventually, when the mass has been compressed to a certain size, light from the star can no longer escape into space. We say that the star has become a black hole, specifically a stellar-mass black hole. Why do we call it a black hole? The star has withdrawn from our observable Universe, in that we can no longer receive radiation from it. We think of a black surface as a surface that reflects none of the light that hits it. Similarly, any radiation that hits a black hole continues into its interior and is not reflected or transmitted out. In this sense, the object is perfectly black. The Photon Sphere As the star collapses, two effects begin to occur. Although we on the surface of the star cannot notice the effects ourselves, a friend on a planet revolving around the star could detect them and radio information back to us about them. For one thing, our friend could see that our flashlight beam is redshifted. Second, our flashlight beam would be bent by the gravitational field of the star (see figure, left). If we shine the beam straight up, it would continue to go straight up. But the further we shine it away from the vertical, the more it would be bent from the vertical. When the star reaches a certain size, a horizontal beam of light would not escape (see figure, right). The Photon Sphere From this time on, only if the flashlight is pointed within a certain angle of the vertical does the light continue outward. This angle forms a cone, with its apex at the flashlight, and is called the exit cone (see figure). As the star grows smaller yet, we find that the flashlight has to be pointed more directly upward in order for its light to escape. The exit cone grows smaller as the star shrinks. When we shine our flashlight in a direction outside the exit cone, the light is bent sufficiently that it falls back to the surface of the star. When we shine our flashlight exactly along the side of the exit cone, the light goes into orbit around the star, neither escaping nor falling onto the surface. The Photon Sphere The sphere around the star in which light can orbit is called the photon sphere. Its radius is calculated theoretically to be 4.5 km for each solar mass present. As the star continues to contract, theory shows that the exit cone gets narrower and narrower. It is thus 45 km in radius for a star of 10 solar masses, for example. Light emitted within the exit cone still escapes. The photon sphere remains at the same height even though the matter inside it has contracted further, since the total amount of matter within has not changed. The Event Horizon We might think that the exit cone would simply continue to get narrower as the star shrinks. But Einstein’s general theory of relativity predicts that the cone vanishes when the star contracts beyond a certain size. Even light traveling straight up can no longer escape into space, as was worked out in 1916 by Karl Schwarzschild while solving Einstein’s equations. The radius of the star at this time is called the Schwarzschild radius. The imaginary surface at that radius is called the event horizon (see figure). (A horizon on Earth, similarly, is the limit to which we can see.) Its radius is exactly ⅔ times that of the photon sphere, 3 km for each solar mass. Formally, the equation for the Schwarzschild radius is R=2GM/c 2, where M is the star’s mass, G is Newton’s constant of gravitation, and c is the speed of light. A Newtonian Argument We can visualize the event horizon in another way, by considering a classical picture based on the Newtonian theory of gravitation. A projectile launched from rest must be given a certain minimum speed, called the escape velocity, to escape from the other body, to which it is gravitationally attracted. For example, we would have to launch rockets at 11 km /sec (40,000 km /hr) or faster in order for them to escape from the Earth, if they got all their velocity at launch. (From a more massive body of the same size as Earth, the escape velocity would be higher.) A Newtonian Argument Note that the escaping object still feels the gravitational pull of the other body; one cannot “cut off ” or block gravity. Now imagine that this body contracts; we are drawn closer to the center of the mass. As this happens, the escape velocity rises. If the body contracts to half of its former radius, for instance, the gravitational force on its surface increases by a factor of four, since gravity follows an inverse-square law; the corresponding escape velocity therefore also increases (although not by a factor of four). A Newtonian Argument When all the mass of the body is within its Schwarzschild radius, the escape velocity becomes equal to the speed of light. Thus, even light cannot escape (see figure). If we begin to apply the special theory of relativity, which deals with motion at very high speeds, we might then reason that since nothing can go faster than the speed of light, nothing can escape. Black Holes in General Relativity Now let us return to the picture according to the general theory of relativity, which explains gravity and the effects caused by large masses. A black hole curves space (actually, space-time) to such a large degree that light cannot escape; it remains trapped within the event horizon. The size of the Schwarzschild radius of the event horizon is directly proportional to the amount of mass that is collapsing: R=2GM/c 2. A star of 3 solar masses, for example, would have a Schwarzschild radius of 9 km. A star of 6 solar masses would have a Schwarzschild radius of 18 km. Black Holes in General Relativity One can calculate the Schwarzschild radii for less massive stars as well, although the less massive stars would be held up in the white dwarf or neutron-star stages and not collapse to their Schwarzschild radii. The Sun’s Schwarzschild radius is 3 km. The Schwarzschild radius for the Earth is only 9 mm; that is, the Earth would have to be compressed to a sphere only 9 mm in radius in order to form an event horizon and be a black hole. Black Holes in General Relativity Anyone or anything on the surface of a star as it passes its event horizon would not be able to survive. An observer would be stretched out and torn apart by the tremendous difference in gravity between his head and feet (see figure). This resulting force is called a tidal force, since this kind of difference in gravity also causes tides on Earth (recall our discussion in Chapter 6). The tidal force on an observer would be smaller near a more massive black hole than near a less massive black hole: Just outside the event horizon of a massive black hole, the observer’s feet would be only a little closer to the center (in comparison with the Schwarzschild radius) than the head. 14.3b Black Holes in General Relativity If the tidal force could be ignored, the observer on the surface of the star would not notice anything particularly wrong locally as the star passed its event horizon, except that the observer’s flashlight signal would never get out. (On the other hand, the view of space outside the event horizon would be highly distorted.) Once the star passes inside its event horizon, it continues to contract, according to the general theory of relativity. Nothing can ever stop its contraction. In fact, the classical mathematical theory predicts that it will contract to zero radius—it will reach a singularity. Quantum effects, however, probably prevent it from reaching exactly zero radius. Black Holes in General Relativity Even though the mass that causes the black hole has contracted further, the event horizon doesn’t change. It remains at the same radius forever, as long as the amount of mass inside is constant. Note that if the Sun were turned into a black hole (by unknown forces), Earth’s orbit would not be altered: The masses of the Sun and Earth would remain constant, as would the distance between them, so the gravitational force would be unchanged. Indeed, the gravitational field would remain the same everywhere outside the current radius of the Sun. Only at smaller distances would the force be stronger. Time Dilation According to the general theory of relativity, if you were far from a black hole and watching a space shuttle carrying some of your friends fall into it, your friends’ clocks would appear to run progressively slower as they approached the event horizon. (Eventually they would get torn apart by tidal forces, as discussed above, but here we will ignore this complication.) From your perspective, time would be slowing down (or “dilated”) for them; indeed, it would take an infinite amount of time (as measured by your clock) for them to reach the horizon. From your friends’ perspective, in contrast, no such time dilation occurs; it takes a finite amount of time for them to reach and cross the event horizon, and shortly thereafter they hit the singularity. Time Dilation On the other hand, if your friends were to approach the event horizon and subsequently escape from the vicinity of the black hole, they would have aged less than you did (for example, only 3 months instead of 30 years). This is a method for jumping into the future while aging very little! However, this strategy doesn’t increase longevity—your friends’ lives would not be extended. Locally, they would not read more books or see more movies than you would in the same short time interval (3 months in our case). In contrast, while viewing you from the vicinity of the black hole, they would see you read many books and watch many movies, signs that you are aging much more than they are (30 years in our case). Rotating Black Holes Once matter is inside a black hole and reaches the singularity, it loses its identity in the sense that from outside a black hole, all we can tell is the mass of the black hole, the rate at which it is spinning (more precisely, its angular momentum), and what total electric charge it has. These three quantities are sufficient to completely describe the black hole. Rotating Black Holes The theoretical calculations about black holes we have discussed in previous sections are based on the assumption that black holes do not rotate. But this assumption is only a convenience; we think, in fact, that the rotation of a black hole is one of its important properties. It took decades before Einstein’s equations were solved for a black hole that is rotating. (The realization that the solution applied to a rotating black hole came after the solution itself was found.) In this more general case, an additional special boundary—the stationary limit —appears, with somewhat different properties from the original event horizon. Within the stationary limit, no particles can remain at rest even though they are outside the event horizon. Rotating Black Holes The equator of the stationary limit of a rotating black hole has the same diameter as the event horizon of a nonrotating black hole of the same mass. But a rotating black hole’s stationary limit is squashed. The event horizon touches the stationary limit at the poles. Since the event horizon remains a sphere, it is smaller than the event horizon of a non-rotating black hole (see figure). Rotating Black Holes The space between the stationary limit and the event horizon is the ergosphere. This is the region in which particles cannot be at rest. In principle, we can get energy and matter out of the ergosphere. For example, if one sends an object into the ergosphere along an appropriate trajectory, and part of the object falls into the black hole at the right moment, the rest of the object can fly out of the ergosphere with more energy than the object initially had. The extra energy was gained from the rotation of the black hole, causing it to spin more slowly. Rotating Black Holes A black hole can rotate up to the speed at which a point on the event horizon’s equator is traveling at the speed of light. The event horizon’s radius is then half the Schwarzschild radius. If a black hole rotated faster than this, its event horizon would vanish. Unlike the case of a non-rotating black hole, for which the singularity is always unreachably hidden within the event horizon, in this case distant observers could receive signals from the singularity. Such a point is called a naked singularity. Since so much energy might erupt from a naked singularity, we can conclude from the fact that we do not find signs of them that there are probably none in our Universe. Thus, each black hole cannot exceed its maximum rate of rotation (or a naked singularity could be seen). Passageways to Distant Lands? In science fiction movies (such as Contact), it is often claimed that one can travel through a black hole to a distant part of our Universe in a very short amount of time, or perhaps even travel to other universes. This misconception arises in part from diagrams such as the figure for two nonrotating black holes. One black hole is connected to another black hole by a tunnel, or wormhole (officially called an “Einstein-Rosen bridge”), and it appears possible to traverse this shortcut. Passageways to Distant Lands? However, this map is misleading; it does not adequately describe the structure of space–time inside a black hole. In particular, one would need to travel through space faster than the speed of light (which nobody can do) to avoid the singularity and end up in a different region. Thus, non-rotating black holes definitely seem to be excluded as passageways to distant lands. In the case of a rotating black hole, on the other hand, travel through the wormhole at speeds slower than that of light, avoiding the singularity, initially seems feasible. In fact, it appears as though one could travel back to one’s starting point in space, possibly arriving at a time prior to departure! This is quite disturbing, since “causality” could then be violated. For example, the traveler could affect history in such a way that he or she would not have been born and could not have made the journey! Passageways to Distant Lands? More detailed analysis, however, shows that this favorable geometry of a rotating black hole is only valid for an idealized black hole into which no material is falling (or has previously fallen). As soon as an object actually tries to traverse the wormhole, the passageway closes! One would need to have a very exotic form of matter with anti-gravitating properties to keep the wormhole open. There is no evidence for the existence of such matter, at least not in the form where it can be gathered. (In Chapter 18, we will introduce the concept of “dark energy,” which causes the expansion of the Universe to accelerate. But this energy is uniformly spread throughout space, and cannot be concentrated into a small volume.) Detecting a Black Hole A star collapsing to become a black hole would blink out in a fraction of a second, so the odds are unfavorable that we would actually see the crucial stage of star collapse as it approached the event horizon. And a black hole is too small to see directly. But all hope is not lost for detecting a black hole. Though the black hole disappears, it leaves its gravity behind. It is a bit like the Cheshire Cat from Alice in Wonderland, which fades away leaving only its grin behind (see figure). Cygnus X-1: The First Plausible Black Hole We can determine masses only for certain binary stars. When we search the position of the x-ray sources, we look for a single-lined spectroscopic binary Then, if we can show that the companion is too faint to be a normal, main-sequence star, it must be a collapsed star. If, further, the mass of the unobservable companion is greater than 3 solar masses, it is likely to be a black hole, assuming that the general theory of relativity is the correct theory of gravity and that our present understanding of matter is correct. To definitively show that the black-hole candidate is indeed a black hole, however, additional evidence is needed, as we will discuss below. Cygnus X-1: The First Plausible Black Hole The first and most discussed, though no longer the most persuasive, case is named Cygnus X-1 (see figure), where X-1 means that it was the first x-ray source to be discovered in the constellation Cygnus. A 9th-magnitude star called HDE 226868 has been found at its location. This star has the spectrum of a blue supergiant; its mass is uncertain but is thought to be about 20 times that of the Sun. Its radial velocity varies with a period of 5.6 days, indicating that the supergiant and the invisible companion are orbiting each other with that period. Cygnus X-1: The First Plausible Black Hole From the orbit, it is deduced that the invisible companion must probably have a mass greater than 7 solar masses and less than about 13 solar masses. This range makes it very likely that it is a black hole. But if the visible star is abnormal, and doesn’t really have a mass of 20 times that of the Sun, the invisible companion could be less massive, so the case for it being a black hole is not absolutely conclusive. Other Black-Hole Candidates In 1992, an even more convincing case was found: The dark object in the x-ray binary star V404 Cygni has a mass of at least 6 Suns, but probably closer to 12 solar masses (see figure). One of the authors (A.F.) has found four additional black-hole candidates with more than 5 times the Sun’s mass, as well as two somewhat weaker cases for black holes. By mid-2005, about two dozen good or excellent stellar-mass black-hole candidates in binary systems had been identified and measured in our Galaxy. Supermassive Black Holes Although the stellar-mass black holes discussed above provided the best initial evidence for the existence of black holes, we now have firm observational support for black holes containing millions or billions of solar masses. Let us discuss these super-massive black holes briefly in this black-hole chapter, and say more about them in Chapter 17, when we consider quasars and active galaxies. The more mass involved, the lower the density needed for a black hole to form. For a very massive black hole, one containing hundreds of millions or even billions of solar masses, the density would be so low when the event horizon formed that it would be close to the density of water. Supermassive Black Holes Thus if we were traveling through the Universe in a spaceship, we couldn’t count on detecting a black hole by noticing a volume of high density, or by measuring large tidal effects. We could pass through the event horizon of a high-mass black hole without being stretched tidally to oblivion, though visual effects would certainly be bizarre. We would never be able to get out, but it would be over an hour before we would notice that we were being drawn into the center at a rapidly accelerating rate. Where could such a supermassive black hole be located? The center of our Milky Way Galaxy almost certainly contains a black hole of about 3.7 million solar masses. Though we do not observe radiation from the black hole itself, the gamma rays, xrays, and infrared radiation we detect come from the gas surrounding the black hole. (See the image opening this chapter.) Supermassive Black Holes Other galaxies and quasars (see Chapter 17) are also probable locations for massive black holes. The Hubble Space Telescope is being used to take images of galaxies with the highest possible resolution, and is finding extremely compact, bright cores at the centers of some of them. Many of these cores probably contain black holes that have a large concentration of stars around them, making their central regions unusually luminous. Supermassive Black Holes The Hubble Space Telescope has also been able to take spectra very close to the centers of certain galaxies, though that particular instrument on the Hubble is currently broken. The Doppler shifts on opposite sides of the galaxies’ cores, especially in disks of gas surrounding these cores, show how fast these points are revolving around the centers of the galaxies (see figure). The speed, in turn, shows how much mass is present in such a small volume. Only a giant black hole can have so much mass in such a small volume. Supermassive Black Holes It now seems that most galaxies have a supermassive black hole at their centers. A majority of these galaxies look relatively normal, but some are “active” (Chapter 17), giving off exceptionally large amounts of electromagnetic radiation from very small volumes. The Chandra X-ray Observatory, the XMMNewton Mission, and the Hubble Space Telescope, working together, are greatly increasing our knowledge of such supermassive black holes. Chandra has found evidence of a jet in one galaxy (see figure, top) and confirmed the presence of a supermassive black hole at the center of the Andromeda Galaxy (see figure, bottom). X-ray spectra it took showed features that very strongly suggested that the objects are black holes. Supermassive Black Holes Radio techniques that provide very high resolution can be used to study the jets of gas emitted from the vicinity of supermassive black holes. The jet from the galaxy M87, the central galaxy in the Virgo Cluster of galaxies, has been imaged in various ways (see figure). The image has been traced so close to the central object that it shows a widening of the jet. This widening seems to indicate that the jet comes from the accretion disk rather than from the central black hole itself. Moderation in All Things Until recently, many scientists doubted that there were black holes with masses intermediate between those of stars—say, 10 times the Sun’s mass—and those in the centers of all or most galaxies—perhaps a million to a billion times the Sun’s mass. But one of the latest advances in black-hole astrophysics is the discovery of so-called intermediate-mass black holes, with masses of “only” 100 –10,000 times the Sun’s, bridging this gap. Chandra X-ray images reveal luminous, flickering sources in some galaxies, suggesting that intermediate-mass black holes are accreting material sporadically from their surroundings. Moderation in All Things One possible place for such black holes to form is in the centers of dense clusters of stars, perhaps even young globular clusters, which retain the black holes after they form. Indeed, in 2002, measurements of the motions of stars in the central regions of two old globular clusters seemed to imply the presence of intermediate-mass black holes, although the interpretation of the data is still somewhat controversial. Moderation in All Things Chandra observations of one x-ray source in the galaxy M74 (see figure) discovered strong variations in its x-ray brightness that repeated almost periodically about every two hours. Because they are almost, but not quite, periodic, the phenomenon is known as “quasi-periodic oscillations.” The object is one of many known as “ultraluminous x-ray sources,” which are 10 to 1000 times stronger in x-rays than neutron stars or black holes of stellar mass. These sources are being observed from both NASA’s Chandra and the European Space Agency’s XMM-Newton. Gamma-Ray Bursts: Birth Cries of Black Holes? One of the most exciting astronomical fields in the middle of this first decade of the new millennium is the study of gamma-ray bursts. These events are powerful bursts of extremely short-wavelength radiation, lasting only about 20 seconds on average. Each gamma-ray photon, to speak in terms of energies instead of wavelength, carries a relatively large amount of energy. Gamma-ray bursts flicker substantially in gamma-ray brightness and each of them has a unique light curve (see figure). The joke so far has been “If you see one gamma-ray burst, you’ve seen one gamma-ray burst,” a far cry from “you’ve seen them all.” How Far Away Are Gamma-Ray Bursts? Several other telescopes have also been automated to make follow-up observations of gamma-ray bursts. For example, A robotic telescope at Lick Observatory that is able to slew over to the position of a gamma-ray burst and take relatively deep images within a few tens of seconds after the outburst (see figure). Mini Black Holes We have discussed how black holes can form by the collapse of massive stars. But theoretically a black hole should result if a mass of any amount is sufficiently compressed. No object containing less than 2 or 3 solar masses will contract sufficiently under the force of its own gravity in the course of stellar evolution. The density of matter was so high at the time of the origin of the Universe (see Chapter 19) that smaller masses may have been sufficiently compressed to form mini black holes. Mini Black Holes Stephen Hawking (see figure), an English astrophysicist, has suggested their existence. There is no observational evidence for a mini black hole, but they are theoretically possible. Mini black holes the size of pinheads would have masses equivalent to those of asteroids. Hawking has deduced that small black holes can emit energy in the form of elementary particles (electrons, neutrinos, and so forth). The mini black holes would thus evaporate and eventually disappear, with the final stage being an explosion of gamma rays. Mini Black Holes Hawking’s idea that black holes radiate may seem to be a contradiction to the concept that mass can’t escape from a black hole. But when we consider effects of quantum physics, the simple picture of a black hole that we have discussed up to this point is not sufficient. Physicists already know that “virtual pairs” of particles and antiparticles can form simultaneously in empty space, though they disappear by destroying each other a short time later. Hawking suggests that a black hole so affects space near it that the particle or antiparticle disappears into the black hole, allowing its partner to escape (see figure). Photons, which are their own antiparticles, appear as well. Mini Black Holes As the mass of the mini black hole decreases, the evaporation rate increases, and the typical energy of emitted particles and photons increases. The final result is an explosion of gamma rays. Although “gamma-ray bursts” have indeed been detected in the sky (see the preceding section), their observed properties are not consistent with the explosions of mini black holes. For example, gamma-ray bursts flicker substantially in gamma-ray brightness, with each one being different (see figure). Such behavior is not expected of exploding black holes. 14.11 Mini Black Holes Instead, as we have seen, gamma-ray bursts may result from neutron stars merging to form black holes, or from supernovae in which the collapsing core forms a black hole. Gamma-ray bursts are extremely powerful, and probably have something to do with the formation of black holes, but unfortunately for Hawking they are not evidence for exploding mini black holes.