Neutron Stars and Black Holes

Stars form in the collapse of interstellar dust and gas clouds under their mutual gravitational attraction. This process is going on even now, as witnessed, for example, by star-forming regions observed in the Orion nebula (Figure 15.1).

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This object is the remnant of an ancient supernova explosion and the stars being born from it contain elements like carbon, nitrogen, and oxygen formed during the lifetime of the progenitor star. Thus they are similar to our Sun, which also was formed, 5 billion years ago, from just such a "contaminated" cloud of dust and gas that collapsed and heated up until nuclear fusion reactions of hydrogen began to occur. The energy streaming from its center eventually stopped the collapse of the proto-Sun, which then reached "hydrostatic equilibrium" (Figure 15.2).

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The subsequent history of a star depends critically upon its mass. Very massive stars need to be very hot in order to reach hydrostatic equilibrium, and nuclear fusion occurs much faster in them because of this. Eventually the hydrogen gas in their cores is entirely fused into helium, within only about 50 million years for a 25 solar mass star. While the same thing will happen to our Sun, its expected lifetime is about 10 billion years. At this point, hydrogen-fusion energy disappears and the star again begins to collapse under gravitational attraction until the temperature at its core rises sufficiently for helium to fuse. (Helium fuses at a higher temperature than hydrogen because it has two electrical charges rather than one, so the electrostatic repulsion that must be overcome is greater.) When helium fusion begins, the star again reaches hydrostatic equilibrium but its outer atmosphere expands and cools thus forming a "red giant" star. In the case of the Sun, its surface will then be somewhere between the orbits of the Earth and Mars. The red-giant phase lasts for only about 100 million years until the helium fuel is used up. This time, however, the gravitational mass is too small to ignite fusion of heavier elements such as the carbon formed from helium fusion. The Sun will continue to contract and heat up until the atoms in its core are packed so tightly together that the repulsion of their electron clouds stops the collapse. In this process its outer layers are completely ejected, forming a " planetary nebula " with an extremely hot "white dwarf" star at its center (Figure 9.4). The radius of the Sun will go from the present 7 × 10⁵ km to only 1 × 10⁴ km, and its density will then be about 350,000 times that of water. Over many billions of years, the white dwarf will cool and eventually disappear from view. This is the fate awaiting stars with masses less than about 3.5 times the mass of the Sun.

The 25 solar mass star mentioned above will also go through the helium-fusing stage, forming a red supergiant star, but will run out of helium fuel in its core after only about 1 million years. Big stars live fast and die young! It also has a very different fate in store for it, since such stars have enough mass to ignite fusion of successively heavier elements. In a relatively short time, it will fuse first carbon, then magnesium, then silicon, and finally sulfur to form iron. According to calculations of this process, the resulting object has an iron core with layers of successively lighter elements arranged in an onion-like structure (Figure 9.2). However, the star now has a serious problem. According to well-understood laws of nuclear physics, fusion of iron does not generate energy. Instead, it acts like a nuclear refrigerator instead of a nuclear furnace. The star, which once was stabilized by the energy of fusion, now has no way to prevent its collapse under the influence of gravity and it first implodes and then bounces back and is destroyed in a supernova explosion. The elements "cooked up" in its various fusion stages are ejected into the surrounding interstellar medium, perhaps to be incorporated into later generations of stars and, in the case of the Sun, into the bodies of plants and animals on the Earth. Observations of supernova remnants have detected all these elements, including iron. Calculations have also shown that the elements heavier than iron are formed in the very energetic supernova explosion itself. The exact details of this process are the subject of ongoing research in nuclear astrophysics. However, at least a portion of the inner core of the supernova may escape this dispersion, resulting in the formation of more exotic objects as discussed below.

Compression of the matter in the supernova core can push through the repulsion of the electron clouds discussed in Section 15.1, forming a state of matter in which the atomic nuclei are packed tightly together and the electrons are pushed back into the protons to form a neutron star . This object has a density of about 5 × 10¹⁴ times that of water, corresponding to the entire mass of the Sun compressed into only a 10 km radius! Because of the physical law of conservation of angular momentum, the star's rate of spin increases dramatically during the collapse, up to as much as 40,000 revolutions per minute. Also, if like the Sun the progenitor star had a significant magnetic field, the compression will cause it to increase, typically to something like a magnetic field (at its surface) that's a trillion times that of the Earth. However, neutron stars having hundreds of times larger magnetic fields, called magnetars , have been detected. Their fields are so powerful that they could kill a person from 1000 km away by warping the atoms in living flesh! However, they pose a threat at much greater distances than that because of the powerful flares they eject. Due to these magnetic fields, neutron stars are relatively easy to detect because they form what are known as pulsars . The first of these was discovered in July of 1967 by Jocelyn Bell . Using a radiotelescope , she found that the signal from a peculiar object pulsed on and off in a very regular pattern, at a rate of approximately one pulse per second. It was initially called LGM-1 (for "little green men") because of the suggestion that the signal might be an attempt at communication from intelligent inhabitants of a distant star. However, it was soon realized that this was not the case and the explanation for the regular pulsation was more prosaic but almost as interesting. Flares emitted from the surface of a neutron star are channeled through its magnetic poles, just as the charged particles emitted by our Sun during a solar flare are directed by the Earth's magnetic field to form the Aurora Borealis and the Aurora Australis when they strike the Earth's atmosphere. In the case of neutron stars, these flares emit radio waves along the magnetic poles. If, as in the case of the Earth, the magnetic poles are not aligned with the geometric poles of rotation, then the corresponding radio emissions rotate in space at the frequency of rotation of the neutron star. If the Earth happens to be in a location such that these radio waves sweep past it, then the result is a pulsing emission much like the flashing light you see from a lighthouse (Figure 15.3).

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This discovery was awarded the 1974 Nobel Prize in physics but Jocelyn Bell, who was a postdoctoral scholar in the laboratory of Martin Ryle and Antony Hewish , controversially did not share in the prize. Pulsars are very accurate clocks. In fact, the most accurate clock in the world (at the time) was the " pulsar clock " installed in Gdansk, Poland in 2011. However, they do slow down over long periods of time due to the emission of energy, and this slowdown has also been detected (Section 16.2).

If the mass of a neutron star exceeds about 1.5–3 solar masses (the exact value is at present not well known), then its gravitational pressure will be so great that even neutrons can't withstand it. The surface of the neutrons is breached and the collapse continues catastrophically forming, a " black hole " which is a region of spacetime from which gravity prevents anything, even light, from escaping. The possibility that this might happen was discussed by John Michell in 1783 and at about the same time by Pierre Laplace , but the first solution of Einstein's general theory of relativity showing that these objects could actually exist was published by Karl Schwarzschild in 1916. The principle can best be understood in analogy with what happens to objects thrown upward from the surface of the Earth. As we all know, a baseball thrown directly upward will slow down under the influence of gravity, eventually stop, and then fall back to Earth. However, a rocket ship traveling fast enough (at the "escape velocity" of about 25,000 miles per hour) will break free from the Earth's gravity and leave without ever falling back to Earth. It turns out that the square of the escape velocity from the surface of an object is dependent only on the ratio of the mass of the object to its radius ($\mathit{\unicode[Book Antiqua]{x76}}$ ² = 2GM/R, where G is Newton's gravitational constant). If the mass M is large enough or the radius R small enough (or both), then the speed $\mathit{\unicode[Book Antiqua]{x76}}$ will equal the speed of light and not even light can escape from the surface of the object. Since the special theory of relativity states that nothing can travel through space faster than light, nothing can escape the gravitational pull at the "surface" of a black hole. Schwarzschild calculated the so-called " Schwarzschild radius " of a non-rotating black hole of mass M (R_s = 2GM/c², where c is the speed of light) which depends only on the mass of the object. For example, R_s for the Earth turns out to be only 9 mm. If some process were to compress the Earth into a ball of this radius, it would become a black hole. The more massive the black hole, the bigger it is, but the Schwarzschild radius is not like the radius of a ball bearing. All of the matter within R_s falls into the center of the black hole, forming a so-called singularity , and the Schwarzschild radius only represents the boundary from within which nothing can escape, called the " event horizon ."

Since no light, and therefore no information, can come from inside R_s, only quantities that are "conserved" according to the laws of physics can remain after the black hole is formed. In particular, these are the mass, charge (if any) and angular momentum (rotation) of the matter within its surface. This is known as the " no-hair theorem ," and it implies that a black hole cannot have a magnetic field. If a neutron star prior to its collapse possessed a magnetic field, that field will be radiated away as electromagnetic radiation during the collapse. That makes it quite difficult to detect a black hole resulting from a supernova explosion, unlike the case of a neutron star. Only a few have been detected, and they are in binary systems where matter from a neighboring star is pulled into a black hole generating a lot of energy. An example is the object known as Cygnus X-1 (Figure 15.4).

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However, there is another way in which black holes can form. The density of stars in the center of a galaxy is very high and they can collide and merge, eventually forming a "supermassive" black hole. In the case of the Milky Way, observations have shown that there exists a black hole at its center with a mass about 4 million times that of the Sun. Analysis of the motion of stars around this object, known as Sagittarius A* , have confirmed its mass and radius demonstrating that it must be a black hole. Supermassive black holes having masses up to billions of times that of the Sun have been detected in other galaxies, from the energetic jets that emerge from them as stars fall into the black hole. An example is the galaxy known as Cygnus A (Figure 15.5). It is thought that every galaxy may well have a supermassive black hole at its center.

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15.5.1. Imaging a Black Hole

On 2019 April 10, the members of the Event Horizon Telescope (EHT) collaboration announced the first ever image of such a black hole (Figure 15.6), at the center of the supergiant elliptical galaxy M87 in the constellation Virgo, one of the most massive (6.5 × 10⁹ solar masses), relatively nearby (16.5 Mpc) galaxies in our neighborhood. The EHT consisted of eight radio-telescopes situated around the world in Arizona, Hawaii, Mexico, Chile, Spain, and at the South Pole. This giant array of telescopes was necessary to produce a high-resolution image of the core of M87 by putting together data taken at the same time by the individual telescopes. This process generates an "effective aperture" that is basically equivalent to a single telescope approximating the size of the Earth. However, putting together the images takes time, and the data were actually collected in 2017 and 2018.

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The black hole itself is not visible in this image since nothing can escape from its interior. What can be seen is the hot glowing gas surrounding its Schwarzschild radius. However, recent calculations have shown that within this ring is a photon ring consisting of photons orbiting the black hole.

15.5.2. Gravitational Tidal Effects

15.5.3. Rotating Black Holes

Steven Hawking deduced some fundamental and important properties of the energetics of black holes from a consideration of a fundamental principle of physics, the second law of thermodynamics. In order to discuss this, we need to introduce the concept of " entropy ," which is a measure of the "disorder" of a system. As an example, consider the number of different sequences that can result from flipping n coins. For n = 3, there are eight possibilities, for n = 4 there are 16 possibilities, and for n = 100 there are 2¹⁰⁰ possibilities. The entropy for a random toss of 100 coins is log(2¹⁰⁰) = 30.1, where "log" indicates the logarithm to the base 10 (Section 1.3). In this example, 10^30.1 = 2¹⁰⁰. Logarithms are used because one rapidly gets very big numbers for most systems, and large entropy means a large degree of disorder. The second law of thermodynamics states that the entropy of an isolated system always increases. That doesn't mean that disorder is inevitable. For example, I could choose to arrange the sequences of coin tosses in some predetermined order, thereby lowering its entropy. However, it takes energy (mental and physical) for me to do this, and the second law says that the total entropy of the system (me plus the coins) increases in this ordering process. My use of internal and external energy increases my entropy more than enough to compensate for the decrease in the entropy of the coins. Viewed in this way, the order of living beings on the Earth (as only one example) is made possible by energy from the Sun, which in the process increases the Sun's entropy. The second law is an extremely powerful tool, and it's closely bound up with the nature of time. For example, consider watching a movie of a glass bottle shattering under the impact of a rock. If you run the movie backwards, it looks strange. Why? Because the disordered glass fragments rearrange themselves into an ordered bottle...and this never happens by itself. The direction of time-flow is related to the increase in entropy. Another thing to consider: no physical principle other than the second law prevents all the air molecules in a room from gathering by chance in one corner...which would be a disaster for us, of course. In fact, this can happen by chance but the probability it will happen is vanishingly small. The second law deals with the probabilities of such rare events. As such, it is different from physical laws such as Newton's laws of motion because these make absolute predictions. The second law is more akin to probabilistic theories such as quantum mechanics in which only the probability that a certain event will occur can be calculated.

Jacob Bekenstein was one of the first to notice that there is a deep connection between the entropy of a black hole and the area of its event horizon. In fact, he showed that the entropy of a black hole is equal to the ratio of this area to the square of the Planck distance. For a 10 solar mass black hole, R_s = 30 km, A = 2.8 × 10⁹ m², and the entropy is 10⁷⁹ (not 79...the log has already been taken)! This is an enormous amount of disorder for an object that is supposed to be simple. Remember: "a black hole has no hair." Initially this was viewed as a problem, which was one of the things that retarded the study of black hole thermodynamics. In retrospect, it is clear that this result is mandated by the second law. For example, we could take all the gas molecules in a room, which have a very big entropy because of their random motions, and dump them into a black hole. In the process, a lot of entropy disappears from the neighborhood because all we can know about the air molecules within the black hole is their total mass (and electric charge and angular momentum if they had any). Their random motions are not visible anymore. The second law says that the total entropy of the universe must increase, though, so the entropy of the black hole has to go up. In fact, it is now understood that the entropy of a black hole is a measure of the number of different ways we can form it, which is very large. This disorder must reside inside the black hole, and according to Bekenstein's theorem the area of the event horizon, which increases in size as more mass is added, is a measure of the disorder contained within it.

We now come to Steven Hawking's insight. In 1974, he put it all together when he showed that not only was the area of its event horizon related to the entropy of a black hole, but also the surface gravity at R_s was a measure of an "effective temperature" of the black hole. In the process, he showed that black holes can radiate not only photons but also particles of many kinds, and the spectrum of this radiation was given by the Planck formula. This is now known as " Hawking radiation ." How does it work? The process goes something like this: a quantum fluctuation produces, e.g., an electron–antielectron pair near the event horizon, which is then torn apart by the tidal gravity of the black hole. This makes the "virtual" particles real by taking energy from the gravitational field. One of the pair then drops into the hole and the other radiates into space. Of course, the same thing can happen with virtual photons, so the hole also radiates electromagnetic radiation. In Hawking's words: "black holes ain't black." What are the properties of this radiation? First of all, Hawking showed that the temperature of a black hole is inversely proportional to its mass. The temperature of a 4 solar mass black hole, for example, is 1.5 × 10⁻⁸ K. This is much less than the temperature of the cosmic microwave background, which means that in fact the black hole is a net absorber of radiation from the CMB. All "normal" black holes are heavier and therefore even colder than this. However, there may be an exception to the rule. Hawking speculated that " primordial black holes " might have been made in the Big Bang and survived to the present. Hawking radiation implies a loss of energy (and therefore mass) from the black hole. But a smaller black hole has a higher temperature and the emission of radiation goes as the fourth power of the temperature, so the hole heats up as it shrinks and therefore loses mass even faster. How long does it take for the black hole to completely evaporate? It can be shown that the lifetime of a black hole (neglecting absorption of the CMB) is proportional to the cube of its mass. For a 10 solar mass black hole, this works out to 2 × 10⁷⁸ s. The present age of the universe is about 5 × 10¹⁷ s so we don't have to worry about evaporation of "normal" black holes. However, one can ask for the mass of a black hole that lives for the current age of the universe. The answer turns out to be 1.3 × 10¹¹ kg, corresponding to a radius of about 20% of that of a proton. Now the Hawking process is a prescription for a " thermal runaway " in which most of the energy is emitted close to the end, leading to an explosion. Black holes of this mass should be exploding right now. Events such as this are being looked for, but so far none have been seen. There is also a problem called the " black hole information paradox " associated with evaporation of a black hole. As mentioned above, almost all the information associated with matter falling into a black hole disappears. This was not a problem at first when it was thought that the lifetime of a black hole is infinite. The "lost" information resides inside the black hole. However, the Hawking process doesn't take any information from the inside of the black hole, so where does this information go when the black hole evaporates? Loss of information turns out to be a serious problem for physics, and the community is currently divided on whether the information is actually lost. Recent progress on this issue announced in November of 2019 may have resolved the paradox, at least in certain simple models of gravity.

All the matter falling through the event horizon of a black hole should form a gravitational singularity at its center. Classically, this would be a one-dimensional point object where all the mass is concentrated into an infinitely small space and spacetime has infinite curvature. All physical laws break down in such an extreme environment. Quantum mechanics tells us that this could never happen since the size of the singularity will be the Planck length. Nevertheless, it takes a quantum theory of gravity (currently unavailable) in order to understand the nature of the singularity. Even so, various speculations about black holes and singularities have been put forward. As with many of the speculations surrounding string theory, these are on the boundary between physics and metaphysics. Among them is the " white hole ," a hypothetical region of spacetime from which matter and energy emerge from a singularity. In this sense, it's the reverse of a black hole, and possibly the place where the information lost in a black hole reappears. Such a solution of the general theory of relativity has been found, but there are no known physical processes by which a white hole could form. More intriguing is the Einstein–Rosen Bridge or "wormhole," which is like a tunnel with its two ends at separate points in spacetime (Figure 15.7). If such a thing existed, it could form a shortcut from one point in space to another very far distant point, allowing for travel at speeds much faster than light.

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As an alternative, it could also make time travel possible or even allow travel into another brane in the multiverse. However, it's been shown that wormholes that could actually be traversed from one end to another would only be possible if they could be stabilized by an exotic form of matter with negative energy density to open up the singularity at the center. Physicists such as Steven Hawking and Kip Thorne have argued that properties of quantum mechanics might allow for such things. These ideas play a big role in the movie " Interstellar " for which Kip Thorne was creative director and executive producer.

Another possibility under discussion is that the quantum foam (Figure 10.2) at the earliest stages of the Big Bang actually was formed from tiny wormholes that continually appear and disappear according to the laws of quantum mechanics. Finally, there's the possibility that our entire universe began from a singularity and is itself a black hole! This, as might be expected, is quite controversial, and Steven Hawking has put forward a no-boundary hypothesis in which the universe evolved in a smooth way from a single point which was not actually a singularity, in much the same way as the geometric poles of the Earth are unique points that are not actually singularities. Imagine tracing the intersection of the Earth with a plane moving downward from the North Pole toward the Equator. The result would be a single, one-dimensional point that then expanded smoothly into a circle. At the Equator, the circle would reach its maximum diameter and begin to shrink until it again became a point as the plane reached the South Pole. This is in fact part of the content of the no-boundary hypothesis, which predicts that the universe will eventually collapse into a point, after a very long time.