Chapter 13
In This Chapter
Creating a spacetime geometry
Developing and expanding the theory of black holes
Trying to find a black hole
Figuring out how black holes are made
Understanding that black holes emit radiation
Wondering if time travel is possible
E instein’s general theory of relativity shows how light can be trapped in a black hole — a star with such strong gravity that nothing can escape. Although his own theory of general relativity led to the prediction of the existence of black holes, Einstein never liked the idea. He thought that black holes “smelled wrong” and died before one was actually discovered.
In this chapter, I explain how the idea of black holes emerged, starting two centuries before Einstein’s theory. I discuss how our thinking about black holes has evolved since Einstein’s time. And what about the possibility that black holes may allow time travel? Stick with me — I get to that topic as well.
Finding the Geometry of Spacetime from Einstein’s Field Equation
Einstein completed his general theory of relativity in 1915, after struggling for eight years in search of the right equations. When he was finally finished, he called it a “theory of incomparable beauty” and “the most valuable discovery of my life.” Physicists since that time have agreed with him. In a seminar in Trieste, Italy, in 1968, the famous English scientist Paul Dirac said that the general theory of relativity was “probably the greatest scientific discovery ever made.”
The general theory of relativity is summarized in an equation, called Einstein’s field equation, which says that the curvature of spacetime is determined by matter and energy. Einstein’s field equation breaks down into a set of ten separate equations that are extremely difficult to solve. Even today, few exact solutions to these equations have been obtained.
Measuring how spacetime warps
According to general relativity and Einstein’s field equation, the sun stretches the spacetime around it, warping it in such a way that it changes the motion of anything traveling through that spacetime (see Chapter 12 for details). The sun isn’t the only thing that warps spacetime. Everything, even you or me or an apple, warps spacetime. But you can’t measure the amount of warping that occurs from ordinary objects. It’d be extremely hard to measure the warping even for the Earth. Only really large objects, like the sun, the stars, and galaxies, can warp spacetime to a measurable extent. The sun, for example, warps spacetime by just two parts in a million.
The warping of spacetime can be measured directly. (As I explain in Chapter 12, Arthur Eddington’s expedition to measure the bending of starlight by the sun in 1919 did just that.) But the warping can also be calculated from the theory, with Einstein’s field equation. That’s how Einstein discovered it and how he predicted what the exact bending of light that Eddington measured was going to be.
Einstein’s first book
After the publication of his final paper on general relativity, Einstein thought he needed to write a complete and clear presentation of the theory so that other physicists could study and understand it. The elements of the theory were scattered in many of the papers that he wrote during the eight years he spent developing it. Many of those papers contained mistakes that he corrected later on, or were dead ends that he backed out from. He wanted to put the theory all together in a single and coherent article, but he felt that perhaps someone else ought to spare the time to do it.
Einstein knew that Hendrik Antoon Lorentz was excellent at presenting this kind of work and dropped hints to try to convince him to write it. He told Lorentz that he had “unfortunately been denied by nature the gift of communication, with the result that what I write may be correct but is highly indigestible.”
Lorentz didn’t bite, and Einstein had to write the article. He finished it in March 1916, and it ended up being 50 pages long. Einstein submitted it to the Annalen der Physik with the title “The Foundation of the General Theory of Relativity.” The publisher of the journal decided to print it as a separate brochure, which became Einstein’s first book.
Einstein’s field equation gave physicists a powerful tool to calculate the exact way that stars and galaxies move and interact, and how the entire universe evolves. The field equation of general relativity provides answers to all these questions. Getting the answers, however, isn’t easy.
Developing Schwarzschild’s geometry
The first attempt at using Einstein’s field equation was made by the great astrophysicist Karl Schwarzschild in December of 1915. Schwarzschild read Einstein’s final paper on the general theory of relativity in the November 1915 issue of the Proceedings of the Prussian Academy of Sciences. Right away, he set out to calculate what the theory would predict about stars.
Schwarzschild divided the problem into two parts:
First, he restricted the problem to the exterior of the star. In a few days, he had the exact solution to the field equations for the curvature of spacetime outside any star. (To be able to perform the calculation, Schwarzschild confined the solution to a star that doesn’t spin.)
The first solution was very elegant. Schwarzschild immediately sent it to Einstein, who was pleasantly surprised. Einstein replied right away, telling Schwarzschild that he “had not expected that one could formulate the exact solution to the problem in such a simple way.” Einstein presented the paper on Schwarzschild’s behalf at the meeting of the Prussian Academy of Sciences on January 13, 1916.
A few days later, Einstein received a second paper from Schwarzschild, this time with the calculations for the geometry of spacetime inside any star. Einstein presented the second Schwarzschild paper at the Academy a few weeks later.
In these two papers, Schwarzschild established the geometry of spacetime around stars. Within a few years, the Schwarzschild geometry became the standard working tool for physicists.
What’s remarkable about this work is that Schwarzschild wasn’t in his comfortable office at the Potsdam Astrophysical Observatory near Berlin (where he was the director) when he did it. Instead, he was in the Russian front of World War I, having volunteered to serve in the German army. He performed his calculations under the rough conditions of war, in severe weather, and with serious health problems.
Sadly, Schwarzschild wouldn’t live to see the remarkable implications of what he had done. He contracted a disease in the Russian front and died at the age of 41, only four months after Einstein presented his second and last paper on general relativity.
Formulating the Black Hole Idea
The idea of a black hole didn’t originate with Einstein or with Schwarzschild. It actually dates back to the late 1700s when the British scientist John Michell wrote a paper about what he thought would happen to the light from a star that shrunk below a certain size but kept all its mass.
Trapping light
Michell was thinking about Isaac Newton’s idea that light was made up of tiny particles — or corpuscles, as Newton called them. These corpuscles should have very small masses, and they should feel the gravitational attraction of the star.
Think of a corpuscle of light as being a tiny ball. If you throw a ball upward from the surface of the Earth, it slows down as it goes up, eventually stops, and falls back down (as shown in Figure 13-1). If you throw it harder, with a greater speed, it goes higher before stopping and returning. If you throw it at 11 kilometers (7 miles) per second, the ball will leave Earth and never return. This speed of 11 kps is called the escape velocity.
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Figure 13-1: Gravity slows the ball down until it stops and falls back, unless you achieve the escape velocity (right). |
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If you could stand on the surface of the sun, you’d need to throw the ball at about 617 kps (385 mps) to have it leave the sun and not fall back. Because the speed of light is much higher than that, Michell thought that the corpuscles of light leaving the sun wouldn’t be affected much by the sun’s gravity. (The speed of light was known with some accuracy at Michell’s time; see Chapter 7 for details.)
But what if a star existed that was much smaller than the sun but had the same mass? In this case, the escape velocity would be larger than the sun’s. Michell used Newton’s universal law of gravity to calculate the escape velocities of small, massive stars. The smaller the star, the closer its surface is to its center, the stronger gravity is at the surface, and the larger the escape velocity is.
Predicting dark stars
Michell realized that it was possible for a star to exist that was small enough and massive enough that its escape velocity would be the speed of light. A star smaller than that and with the same mass would have a gravitational field so strong that light would not be able to leave it. Light would be trapped inside. The corpuscles of light would rise up at first, then slow down, stop, and fall back to the surface of this star (see Figure 13-2).
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Figure 13-2: Light will be trapped in a star that is small and massive enough. |
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Michell calculated this critical size of the star, the size at which the escape velocity is the speed of light. A star with the mass of the sun would have to be just under 6 km (3.7 mi) across in order to be a dark star — a star from which light cannot escape. A star with a mass twice as large as the sun would have a critical diameter twice as large, or about 12 km. For a star with a mass equal to three suns, the critical diameter triples (see Figure 13-3).
Michell thought that the universe should contain a large number of stars smaller than the critical size. These dark stars would be invisible to us — they would be black holes.
Michell presented his calculations and ideas at the Royal Society of London in November of 1783. A few years later, the French mathematician and philosopher Pierre Simon de Laplace mentioned the possible existence of these dark stars in his popular book The System of the World, without mentioning Michell. (Laplace, a brilliant and accomplished man, was sometimes reluctant to give credit to others.)
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Figure 13-3: Critical sizes for stars of different masses. |
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In the early 1800s, the corpuscle view of light was questioned by the experiments of Thomas Young, who showed that light is actually a wave. His wave theory of light, which explained all the observations about the behavior of light, became universally accepted. (In Chapter 7, I describe Young’s experiment. As I explain in Chapter 16, Einstein later showed that light is more complicated than what Young proposed.) By the end of the 19th century, scientists believed that because light is a wave and not a particle, it couldn’t be affected by gravity. And that was the end of the dark star idea. For a while.
Reintroducing the theory of black holes
The idea of dark stars, or black holes (as we call them today), was still on ice when Schwarzschild solved Einstein’s field equations and proposed his geometry. The Schwarzschild geometry revived the idea of black holes, because it predicted that there is a critical size for each star that depends on the star’s mass. The critical size that Schwarzschild calculated was the same as what Michell had come up with: 6 km (3.7 mi) in diameter, or 3 km in radius, for a star the same mass as the sun. This critical radius is now called the Schwarzschild radius. If a star was smaller than this critical value, light would be trapped. The black hole idea came back.
But here’s the problem with the way Michell had been thinking of dark stars: You can’t treat light like you do balls that you throw up in the air. The balls slow down because of gravity. But Einstein’s special theory of relativity says that light always travels at the same speed; gravity won’t slow it down.
How did Schwarzschild’s calculation revive Michell’s idea, then?
Einstein’s relativity opened the door. In Newton’s mechanics, space and time are absolute, but the speed of light can change. In Einstein’s relativity, space and time are relative, but the speed of light is absolute. (Check out Chapters 9 and 10 to refresh your memory.)
As I explain in Chapter 12, according to the theory of general relativity, the light emitted from a strong gravitational field is Doppler-shifted to stretched waves. The stronger the gravitational field, the greater the shift. Light isn’t slowed down as the field increases; it continues to travel at the same speed that special relativity mandates (c). But its wavelength changes.
Look at Figure 13-4, where each star has the same mass but a different radius. The middle star has a smaller radius than the top star and, therefore, a stronger gravitational field. Its light wave is stretched in comparison with the light wave of the top star.
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Figure 13-4: Light emitted from a gravitational field is Doppler-shifted to stretched waves. Time also runs more slowly. |
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If a star has a strong enough gravitational field, the light becomes stretched out so much that it’s flat — its wavelength is infinitely long (it can’t be measured). The bottom star in Figure 13-4 shows this situation. Because there is no wave, there is no light. The gravitational field that stretches the wavelength to infinity corresponds to that of a star at the Schwarzschild radius.
Another way to consider this situation is to realize that in a gravitational field, time is slowed down (see Chapter 12). The stronger the gravitational field, the slower time flows. (For example, a clock runs ever-so-slightly faster in the attic than in the basement, because gravity is stronger in the basement.) At the Schwarzschild radius, as seen from the outside, time is dilated an infinite amount; time doesn’t flow. From our point of view outside the star, light doesn’t exist. Because the star is still emitting light, light must be trapped. The star is a black hole.
Facing Einstein’s skepticism
Einstein was very pleased with the first exact solution to his field equation that Schwarzschild had obtained. Scientists could now use this solution to calculate the gravitational properties of planets and stars. For example, the orbit of Mercury was quickly calculated as the result of Schwarzschild’s work. Einstein had used approximation methods to calculate Mercury’s orbit as the first proof of the power of the general theory (see Chapter 12). But he hadn’t obtained an exact solution. Thanks to Schwarzschild, the exact solution became possible. When the calculation was done, the result matched what Einstein had previously obtained.
However, Einstein wasn’t happy with the idea of black holes. He didn’t think that the universe was made that way. He didn’t like the fact that his equation broke down at the center of the black hole or at the surface of the Schwarzschild radius. (Before the term black holes came into being, these entities were called Schwarzschild singularities.)
In 1935, Einstein published a calculation that he interpreted as showing that black holes couldn’t exist. Einstein chose a cluster of particles held together by gravity and moving in circular orbits around a center, forming a sphere. He then made the cluster smaller and smaller, with the particles moving faster and faster to maintain equilibrium. When the radius of his sphere reached about 1.5 times the Schwarzschild radius, his calculations showed that the particles would need to move at speeds greater than the speed of light to avoid being pulled in, with the whole thing collapsing into a single point. And nothing could move faster than the speed of light. “The essential result of this investigation,” wrote Einstein in his paper, “is a clear understanding as to why the ‘Schwarzschild singularities’ do not exist.”
Einstein’s calculation was correct, which obviously created another problem for the black hole theory. But even Einstein made a mistake every once in a while.
Einstein assumed that he needed to keep the whole cluster of orbiting particles from collapsing into itself. At the time, that was the only logical thing to do, based on observations. No one had yet discovered the possibility that a star could collapse. To prevent the cluster from collapsing, the particle’s speeds had to increase beyond the speed of light, which would contradict the theory of special relativity. But, by not allowing the cluster to collapse, Einstein missed the whole point. (This wasn’t the first time that he didn’t let his equations take him to new places. I describe the other time in Chapter 18.)
Studying Collapsing Stars
Despite Einstein’s skepticism about the existence of black holes, other scientists continued to use the foundation laid by the general theory and Schwarzschild’s calculations to study the behavior of the stars.
For example, a few years after Einstein tried to prove that black holes couldn’t exist, J. Robert Oppenheimer and his graduate students at the University of California, Berkeley were trying to find out what would happen to a star after its nuclear fuel was spent. Oppenheimer, who would later lead the team of scientists that invented the nuclear bomb, was at the time one of the top U.S. scientists.
Oppenheimer knew that the Indian physicist Subrahmanyan Chandrasekhar had recently calculated that a star with a mass that was just under 50 percent larger than the sun’s would contract into a white dwarf, a star with the mass of the sun and the size of the Earth. Because you can fit more than 1 million Earths in the volume of the sun, that’s an astonishing contraction. The contraction isn’t sudden, however; it takes place over millions of years.
Identifying other extreme stars
Oppenheimer wanted to know what would happen to stars that were even larger. His calculations showed that a larger star would collapse into a neutron star, a star composed mostly of neutrons (one of the particles that make up the nucleus of an atom). Neutrons were discovered in 1932, and within a year, two physicists at Caltech — Fritz Zwicky and Walter Baade — proposed that stars made up of only neutrons could be the end result of the collapse of stars larger than the ones Chandrasekhar had studied. They suggested that neutron stars form in supernova explosions. Their proposal was generally ignored until Oppenheimer and his graduate student performed their calculations.
Real neutron stars were discovered in the 1960s and have been studied since. They are so dense that if you could bring to Earth a piece of a neutron star the size of sugar cube, it would weigh 100 million tons.
J. Robert Oppenheimer
Oppenheimer graduated from Harvard in 1922 with a B.S. in chemistry but switched to physics in graduate school. He decided to attend grad school in Europe, which was the mecca of theoretical physics. He attended the University of Göttingen and obtained his PhD working with Paul Dirac, Max Born, and other luminaries.
After graduating, he found himself in high demand, with offers from Caltech, Berkeley, Harvard, and two European universities. Berkeley intrigued him because, at the time, it didn’t have a theoretical physics program, and he saw the opportunity of building it himself. But he feared being isolated. He decided to accept the offers from both Berkeley and Caltech, spending half a year at each. Caltech was the place to go and “be checked if I got too far off base,” he said.
Theorizing the ultimate collapse
Oppenheimer’s calculations showed that there was an upper limit to the size of the stars that would collapse to neutron stars. The range of masses was from 11/2 to 3 solar masses. (Stars smaller than 11/2 solar masses collapse to white dwarfs.) What would happen to stars larger than 3 solar masses?
Oppenheimer gave that problem to Hartland Snyder, another of his graduate students. He calculated that if the original star had a mass larger than 3 solar masses, when its fuel was all spent, the star would start collapsing and continue collapsing without stopping. Snyder and Oppenheimer wrote in a paper that “when all . . . sources of energy are exhausted, a sufficiently heavy star will collapse; [its contraction] will follow indefinitely.”
Oppenheimer was puzzled; “the results have been very odd,” he said to a friend. And physicists found the results fascinating. The paper written by Snyder and Oppenheimer was published in 1939. However, at the time, political events in Europe were forcing physicists to think about the possibility of nuclear fusion. Neutrons were at the heart of these nuclear processes. No one could take time to think about nuclear processes in stars anymore. Oppenheimer himself was soon called to lead the Manhattan Project.
Reviving Interest in Black Holes
After World War II, very few scientists remembered Oppenheimer’s papers about collapsing stars. Oppenheimer himself was involved in other matters.
During the 1950s, interest in general relativity began to grow. In 1955, for the first time, there was an international conference on relativity theory, which was followed by a series of international conferences on relativity and gravitation that continues today.
Calculating spacetime for rotating black holes
In 1963, Roy Kerr, a New Zealander who was then working at the University of Texas, found a new solution to Einstein’s field equation, this time describing the spacetime curvature outside a rotating star. Soon, Brandon Carter, Roger Penrose, and other relativists found that this solution described the spacetime geometry not just of a rotating star but of a rotating black hole.
Kerr’s black hole was an extension to Schwarzschild’s black holes, which didn’t rotate. It was a more general solution and had very important new properties. Kerr’s work rekindled interest in black hole physics.
Discovering quasars and pulsars
In 1960, U.S. astronomer Allan Sandage used a 200-inch telescope at Mt. Palomar, north of San Diego, California (which was the largest optical telescope in the world at the time), to discover a “star” at the location of a very strong radio source that had been detected a year earlier. A couple of things were odd with this discovery. First, ordinary stars usually are not strong sources of radio emissions. Second, the star’s spectrum had several features that astronomers couldn’t identify, which is not normally the case.
Soon, other such “stars” were discovered. The most famous one was named 3C 273. Astronomers soon discovered the reason for the strange features in the spectrum of 3C 273 and the other “stars”: The spectrum was Doppler-shifted due to the star’s extremely large speed moving away from Earth. Calculations showed that 3C 273 was moving at 45,000 kps, which is 15 percent the speed of light. The objects in question weren’t stars. They were new strong radio sources. Astronomers named them quasi-stellar radio sources, or quasars. Soon, astronomers found other quasars that weren’t radio emitters. (It turns out that only about 10 percent of quasars are radio emitters.)
Quasars are astonishing objects. Most are very far away, more than 3 billion light-years from Earth. They’re extraordinarily bright, which allows them to be seen from Earth at those large distances. They’re actually brighter than many galaxies. Our galaxy, the Milky Way, shines with the light of 25 billion suns. 3C 273 is as bright as 35 trillion suns.
In 1968, the British astronomer Donald Lynden-Bell, who was working at Caltech at the time, proposed that this incredible energy output was powered by an extremely massive black hole that pulls in surrounding gases. As these gases fall, gravitational energy is released in the form of radiation. That’s the consensus today.
At about this time in England, astronomers discovered a new pulsating radio source that they named pulsar. (See the sidebar “Little green men.”) Eventually, many other such pulsars were found. They turned out to be neutron stars that are spinning very fast.
These discoveries fueled the interest of physicists and astrophysicists in black holes. In the past three decades, black hole physics has been an extremely active area of research.
Little green men
In 1967, Jocelyn Bell was working toward her PhD in astronomy at Cambridge University under Anthony Hewitt. Her thesis was related to the study of radio emissions from several astronomical sources. She had just finished helping to build a large array of radio antennas in a four-acre field in the English countryside and was collecting data when she noticed that one of the antennas was picking up a very regular signal from one particular location in the sky. The signals were all one and one-third seconds apart. She immediately telephoned Hewitt, who rushed over and observed the phenomenon. For some time, half-jokingly, they called the signals LGM, for Little Green Men.
Before Christmas, Bell went to Hewitt’s home to discuss ways to announce the discovery without setting off a media circus. They couldn’t think of any. She went back home that evening feeling upset about trying to finish her PhD dissertation while “some silly lot of little green men had to choose my aerial and my frequency to communicate with us.”
After dinner, Bell went back to the lab and found that her telescope had picked up a similar signal from another part of the sky. That ruled out the LGM idea in her mind. But not for the British tabloid press. After Hewitt announced the discovery, reporters swarmed the place, trying to interview the young, attractive female astronomer who’d been talking to extraterrestrials.
Bell’s radio sources were named pulsars, and many more have been detected since. Astronomers later discovered that pulsars are rapidly rotating neutron stars that emit strong radio sources from their polar regions.
Starting the Hunt
After Oppenheimer’s discovery in 1939 that stars with masses larger than 3 solar masses will keep collapsing, a few scientists began to consider the possibility of finding real black holes. But it was tough to figure out how to go about it since you really can’t see one.
In 1964, Yakov Zel’dovich in the Soviet Union came up with a way to see a black hole. He knew that a wind of gas blows off the surfaces of many stars. The sun, for example, has its solar wind, made mostly of protons and electrons, which streams off of the sun in all directions at speeds of about 400 kps (about 1 million mph). The solar wind isn’t particularly strong when compared with the wind of many other stars.
If a star with a strong solar wind was located near a black hole, the strong gravity of the hole would pull in this gas, heating it up. Zel’dovich calculated that the gas would heat up to several million degrees. Hot objects radiate energy. Normally, this energy is in the infrared part of the spectrum (see Chapter 7) and sometimes in the visible spectrum. When the temperature is in the millions of degrees, the radiated energy is in the form of x-rays. Zel’dovich thought that he’d be able to “see” black holes by detecting these x-ray emissions near a star.
There was a problem, though (there’s always a problem). You can’t detect x-rays from space down here on the ground. And in 1964, there weren’t any space telescopes that could detect them in space.
Several resourceful scientists tried sending x-ray detectors in rockets and were able to detect some x-ray sources. However, the real search for black holes started in 1970 with NASA’s launch of the satellite Uhuru, dedicated to x-ray astronomy. This satellite discovered 339 x-ray stars. Uhuru was followed eight years later by NASA’s Einstein x-ray telescope.
With Uhuru and Einstein, scientists discovered several black hole candidates. The most promising was named Cygnus X-1, which scientists call Cyg X-1 (see the sidebar “Betting on black holes”).
There are several other candidates today. Scientists are now trying to detect the energy flowing into a black hole. It they succeed, that will be a more direct method of detecting black holes. One day, we’ll be as certain that black holes exist as we are that the sun exists.
Understanding How the Universe Makes Black Holes
The constellation of Orion the hunter has three bright stars in the middle that are supposed to represent the belt of the hunter. Three dimmer stars below the belt represent the sword of Orion. A dim, middle star in the belt is not actually a star but a large cloud of gas called the Orion Nebula. This nebula, which is located at 1,500 light-years from us and is about 15 light-years across, is a birthplace of stars.
In this nebula, some regions of higher concentration of dust separate themselves into large spheres. These spheres, or protostars, begin to shrink because of gravity, and the temperature rises. At some point, the temperature is high enough for nuclear fusion to start. The star is born.
Giants
That’s how our sun started its life some 5 billion years ago. The sun is a middle-aged star and will shine for another 5 billion years before using up all its nuclear fuel. After 10 billion years, with no more fuel to spend, the sun (and any other star with a mass up to about 4 solar masses) will begin to contract. Without fuel, the sun’s core won’t be able to withstand gravity. This contraction will heat up the core again, which will begin to radiate energy out. The sun’s outer layers will then start expanding, eventually forming a red giant.
When this second phase is over, gravity will start to compress the core of the sun once more until it becomes a white dwarf. In this stage, the sun will get a second lease on life and will radiate for a few million years more. After that, it will be all over. The sun will become a black dwarf, a burned-out mass — a dead star.
There are no dead stars of this kind anywhere yet. The universe hasn’t lived long enough for a star like the sun to live out its life and die.
Supernovas
Not all stars live out their lives like the sun. Stars that are more massive explode into supernovas. One was discovered in our back yard in 1987. It was seen in the Large Magellanic Cloud, a satellite galaxy to our Milky Way. The core of supernovas like this will collapse into a neutron star.
Much more massive stars, with cores of more than 3 solar masses, will also collapse. As Oppenheimer and Snyder discovered theoretically, in this situation, nothing stops the collapse. The star becomes a black hole.
Betting on black holes
Scientists calculated that Cyg X-1 should have a mass of about 6 solar masses, in the right range for black holes. In 1974, everything pointed toward Cyg X-1 being a black hole. In spite of this, renowned physicist Stephen Hawking of Cambridge University made a bet with the U.S. physicist Kip Thorne of Caltech that Cyg X-1 wasn’t a black hole. If Hawking won, Thorne would buy him a one-year subscription to Private Eye magazine. If Thorne won, Hawking would buy him a one-year subscription to Penthouse magazine.
In his best-seller A Brief History of Time, Hawking explains that his bet was an insurance policy for him. He’d spent a lot of time working on black holes, and if black holes didn’t exist, he’d at least win his subscription to Private Eye. In his well-known book Black Holes and Time Warps, Thorne explains that although his wife, mother, and siblings were mortified by the stakes, he didn’t think he’d win. He was convinced that Cyg X-1 was a black hole but didn’t think he could prove it.
When Hawking and Thorne made their bet, scientists estimated that they were about 80 percent certain that Cyg X-1 was a black hole. In 1988, when A Brief History of Time was published, the level of certainty had gone up to 95 percent, but Hawking hadn’t accepted defeat.
One night in 1990, while Thorne was in Moscow, Hawking and his friends broke into Thorne’s office at Caltech, found the bet, took it out of its frame, and wrote the note of concession on it with Hawking’s thumbprint as signature. The certainty that Cyg X-1 was a black hole was still 95 percent.
Most scientists agree that Hawking was right in conceding. Cyg X-1 is very likely a black hole. Today, the evidence is still about the same. The certainty isn’t 100 percent because Cyg X-1 is being detected indirectly, and you can always find another less likely but possible explanation for it.
So, What Is a Black Hole Anyway?
Do scientists know what the properties of a black hole are?
During the 1960s and 1970s, the “golden age” of black hole research, many scientists spent a great deal of time performing complicated calculations with the general relativity equations, trying to come up with what they thought black holes should be like. They also did computer simulations and modeling to help guide some of the complicated calculations.
Out of all this activity came several interesting results. The most important one is that a black hole is always perfectly spherical, regardless of the shape of the star that originated it. If the star has a mountain, the black hole ends up round. If the star is deformed in any way — even if it is a square star — the black hole that it forms is still round (see Figure 13-5). What happens to the mountains or the deformations? They are radiated away, in the form of gravitational waves.
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Figure 13-5: Black holes are round, no matter what shape the stars were. |
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These answers seem simple, but to get to them required years of extremely complicated calculations and discussions among the scientists involved. These scientists also discovered that if the original star had a magnetic field, the resulting black hole sucks this field down behind its event horizon, the surface of the sphere at the Schwarzschild radius or the boundary of a black hole. No magnetic field lines remain sticking out of the black hole.
Black holes have no hair
In fact, nothing sticks out of a black hole: no mountains, protrusions, or magnetic fields. This finding prompted the renowned U.S. physicist John Wheeler, who earlier coined the term black hole, to declare that a black hole has ho hair. With this apparently crude statement, Wheeler wanted to drive home the concept of how a black hole modified many physical properties to end up in a very simple form.
The phrase caught on fairly quickly at international conferences and meetings. When it came time for scientists to publish research papers on this topic in respected journals, things became a bit more difficult. The editor of the prestigious U.S. journal Physical Review told the scientists that his journal would not permit such obscenities. Some European journals and Russian journals, published in languages where the phrase had similar connotations, also refused to use it.
Physicists continued to use the phrase in their conference presentations. Eventually, the phrase lost its lewd interpretation and became synonymous with what Wheeler intended. The journal editors gave in, and now you can find it in countless scholarly papers and books. You’ll be hard-pressed to find a popular book dealing with black holes that doesn’t use it.
If black holes have no hair, do they have anything?
It turns out that they can’t really get rid of everything. (They still have some hair!) Black holes must keep all the things that physics says cannot be destroyed — things that obey what are called conservation laws. Physics says that black holes must keep their masses, electric charges, and spin. If they were to get rid of those things, physicists would have to revise all their laws and start over again. (They’d rather stay with the laws that they’ve discovered and see how far they can go.)
In 1965, the well-known British physicists Stephen Hawking and Roger Penrose showed that spacetime becomes infinite inside a black hole. This means that gravity inside a black hole is infinite. This infinite stretching of spacetime is called a singularity (see Figure 13-6).
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Figure 13-6: Spacetime is stretched to infinity inside a black hole. |
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“Black holes ain’t so black”
Black holes seem to swallow everything within sight and don’t let anything come out. Even light is trapped inside black holes.
In 1971, Yakov Zel’dovich, one of the founders of the Soviet Union’s nuclear weapons program, came up with the strange idea that a black hole sends off radiation as well as particles. Didn’t everyone know that nothing comes out of a black hole? Why was Zel’dovich claiming that something does come out?
Zel’dovich was attempting to apply some ideas of quantum physics to the study of black holes, which are studied with general relativity. However, Zel’dovich didn’t know enough general relativity to perform the actual calculations for black holes, so he made calculations for a rotating metal sphere instead. His analysis showed that the sphere must radiate a tiny amount of energy because of its interaction with the surrounding spacetime.
Zel’dovich published his idea in a prestigious physics journal, but no one paid much attention to it. Most thought that it was crazy.
In 1973, Stephen Hawking visited the Soviet Union to discuss physics with the Russian physicists. When Hawking and Zel’dovich met, they discussed Zel’dovich’s idea of radiating black holes. Hawking was interested but didn’t like the way Zel’dovich had used quantum physics and general relativity. Hawking decided to do it right. When he was done, he confirmed what Zel’dovich had said, that a spinning black hole emits energy and particles. But he went further. Hawking showed that even after the hole stops spinning, it continues to radiate and emit particles. His discovery was coined Hawking radiation.
As Hawking likes to say, “Black holes ain’t so black.” Initially, Hawking maintained that the radiation that comes out of the black hole is random and carries no information about what went in. Other scientists disagreed. In 1997, Hawking and Kip Thorne bet John Preskill of Caltech that if an encyclopedia is sucked in by a black hole, the information would be lost forever. The Hawking radiation emitting from the black hole would be meaningless; it wouldn’t contain any of the information in the encyclopedia.
Recently, Hawking came around and accepted that he’d been wrong. His calculations are showing that Hawking radiation is related to the information that goes into the black hole. The contents of the encyclopedia should slowly come out in the radiation. Hawking conceded his bet at the conference where he presented his paper. He gave Preskill a baseball encyclopedia.
Journey into a black hole
If an astronaut were brave enough to travel into a black hole, what would he see?
Imagine an interstellar spaceship arriving to the region near a black hole of 4 solar masses. If the ship commander does it right, he can park his spaceship in an orbit around the black hole. The astronauts on this ship won’t feel anything different from what they feel and see when in orbit about the sun, for example.
When an intrepid astronaut leaves the ship on a small runabout and heads toward the black hole, things begin to move away from the ordinary. Suppose the astronaut is reporting back to the station at ten-second intervals. At first, the signals received at the ship come in every ten seconds, as planned. When the astronaut in the runabout gets close to the black hole’s event horizon, the intervals received by the spaceship become longer and longer. Instead of one every 10 seconds, they are now coming every 20, 45, 90 seconds. Soon, the ship’s astronauts are waiting minutes to get the next signal. Then, the minutes become hours, days, months. Eventually, they find themselves waiting a couple years before the next signal arrives. Finally, years pass without receiving a signal. The astronaut passed the event horizon and went into the black hole.
To the astronaut, however, time is flowing as usual. Every ten seconds, she sends a signal. She crosses the event horizon, and her watch keeps ticking at the normal rate. She dutifully continues sending her signals.
But the signals have been Doppler-shifted because of the enormous gravitational field of the black hole. From the spaceship, the astronaut’s time is slowed down. When the astronaut crosses the event horizon, her signal is Doppler-shifted an infinite amount, which is another way of saying that the signal disappears.
Although the astronaut in the runabout sees her watch running as always, things are far from normal for her. The worst problem she has is the enormous tidal forces that are tearing her craft and herself apart. This imaginary trip doesn’t have a happy ending. Our astronaut won’t survive for long.
Contemplating Time Travel
Could black holes offer the possibility of time travel? As I discuss in Chapter 12, time travel to the future is possible. In fact, we all do it (to a very small degree) all the time. Every time you get on an airplane, a train, or a car, you are moving relative to those who stay behind. According to general relativity, time flows more slowly in the accelerated frame. Since you need to accelerate to reach cruising speed, your time will pass at a slower rate compared to those who stay behind. And while you’re traveling at cruising speed relative to us, special relativity says that your time flows more slowly.
At the end of your trip, you’ll be younger than those who didn’t travel with you. However, the age difference is measured in extremely small fractions of a second. On a flight from Los Angeles to Tokyo at 1,000 kph (625 mph), you travel ten nanoseconds into the future, which is the same as saying that you’ll be ten nanoseconds younger. If you get on a spaceship that travels at 0.99995c and go for a nice 10-year tour of our neighborhood in the galaxy, you’ll travel 1,000 years into the future. When you return to Earth, you won’t recognize the place!
Visiting the past
If you travel to the future, can you get back? Or better yet, can you travel to the past from the present?
Special relativity allows you to see the past. In fact, this phenomenon happens all the time. When you look at the sun, you’re seeing it as it existed 8 minutes ago. That’s how long it takes light to travel from the sun to the Earth. The images that we get from a NASA rover on Mars also travel at the speed of light. The scientists at Jet Propulsion Laboratory (JPL) see the rover as it existed 20 minutes earlier. For all they know, the rover malfunctioned 15 minutes ago.
When you see your friend across the table, you’re seeing her as she was 3 nanoseconds ago. She doesn’t change that much in 3 nanoseconds, so there’s no difference in her appearance. But if she travels to another planetary system 20 light-years away, the image that you receive of her will be the one she sent 20 years ago.
What you really want to know is if you can visit the past and come back to the present. We’re talking real time travel, like in Back to the Future or Peggy Sue Got Married.
The short answer is: Perhaps, but not with present-day technology.
Exploring wormholes
In 1988, Kip Thorne and Michael Morris at Caltech and Ulvi Yurtsever at the University of Michigan published the first serious paper on the possibility of building a time machine. It was entitled “Wormholes, Time Machines, and the Weak Energy Condition,” and it was published in the prestigious Physical Review Letters, a journal with a very strict acceptance policy. In the paper, the three physicists presented their conclusions about their study on the possibilities of time travel.
The idea for the study began with Carl Sagan, who at the time was writing his science fiction novel and movie script Contact. Sagan’s story deals with time travel, and he wanted to know from Thorne if time travel was scientifically feasible. Sagan was an astronomer and was well acquainted with the theory of relativity and black holes and what the theories say today about time travel. However, he wanted to know from the top scientists working on relativity and black holes whether time travel is possible for a sufficiently advanced civilization.
Einstein had looked at a related problem. In 1935, Einstein and his collaborator, the physicist Nathan Rosen, used general relativity to examine the shape of spacetime near a very massive star. They found that spacetime became warped into a tunnel, a hole in the universe. Einstein examined the equations more carefully and realized that this tunnel would lead to another region of the universe (see Figure 13-7). Einstein was disturbed with this strange solution.
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Figure 13-7: The Einstein– Rosen bridge, connecting two parts of the universe. |
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Einstein knew that getting through the tunnel required speeds greater than the speed of light. Because that wasn’t allowed by the special theory, he took his conclusions to be a mathematical quirk, with no physical reality. Later, the Einstein–Rosen bridge, as it is called, appeared in other solutions to Einstein’s field equation. But Einstein wasn’t concerned with the reappearance of the bridge, because relativity didn’t allow travel through the bridge.
In 1963, when Kerr obtained his solution to Einstein’s field equation describing a rotating black hole, the Einstein–Rosen bridge idea was revived. Kerr found that his spinning black hole doesn’t collapse to a point, like Schwarzschild’s. Instead, it collapses to a ring. If that’s the case, then you could in principle travel through the tunnel. You’d have to do it very carefully, moving through the middle of the tunnel, not deviating much from the axis (see Figure 13-8). The Einstein–Rosen bridge is a wormhole connecting two regions of the universe.
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Figure 13-8: An astronaut traveling through an Einstein– Rosen bridge. |
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Using a time machine
Thorne and his collaborators realized that the Einstein–Rosen bridge could be used as a time machine. If you move one of the openings of the bridge to the neighborhood of a neutron star while the other opening remains here, the much greater gravity of the neutron star would slow down the flow of time.
If the time difference between the two openings of the wormhole is 100 years, an astronaut entering our opening to the wormhole will come out at the other end, 100 years into the future. When the astronaut returns, he will be back in our time.
However, Thorne warns us that for the wormhole to be passable, it must contain some undiscovered exotic matter. Thorne was looking to see if any laws of physics would prohibit the existence of a time machine. The exotic matter that he required for the wormhole doesn’t exist today, but the laws of physics don’t prevent it from existing. This exotic matter must generate antigravity or gravitational repulsion to allow the astronaut to pass through the wormhole without having it collapse on himself. Gravitational repulsion can be generated with negative energy or negative pressure.
Physicists have discovered negative energy in some quantum systems. Back in 1948, the Dutch physicist Hendrik Casimir had predicted the existence of these systems. The Casimir effect was observed for the first time in 1958 (see Chapter 19). Today, scientists are trying to apply this effect to operate micro-machines. Thorne’s exotic matter with negative energy that can produce antigravity may someday be discovered.
Prohibiting time loops
Suppose that scientists one day are able to solve all of the engineering problems and construct a time machine using the Einstein–Rosen bridge. Is time travel really possible with all the paradoxes that it creates?
Imagine that you travel back in time and find yourself witnessing a man about to commit a crime. You decide to alert a policeman and the man is killed in a shootout. Later on you discover that the man was your own grandfather and that he hadn’t even met your grandmother at the time. Since he is dead, your parents aren’t going to be born, and you won’t ever exist to travel back in time and alert the policeman.
You could also travel back to January 1905 and meet Albert Einstein at his home in Bern, perhaps the evening after he returned from his long discussion with Michele Besso about the problems with electromagnetism and the principle of relativity (see Chapter 9). You can then tell him that the solution is to keep the principle of relativity and to make the speed of light constant for all nonaccelerated observers. Then you leave his apartment, climb on your time machine, and return to the present. The next day, Einstein tells Besso that he has figured out the problem and proceeds to develop relativity.
Who discovered relativity? You told Einstein how to do it. Einstein listened to you and developed it. Then he published his theory and became famous. You learned about his discovery in this book and then traveled back in time to tell him how to do it.
Equally disconcerting is the time traveler from the future who studies how the time machine he is about to use is made. Then he travels back in time and meets the inventor of the time machine and tells him how to do it.
All these paradoxes prompted Stephen Hawking to propose that there seems to exist a Chronology Protection Agency that protects history and prevents time travelers from changing the past. His chronology protection hypothesis, which comes out of general relativity calculations that he performed, outlaws these time loops.
However, the theory of relativity permits these time loops. Hawking’s chronology protection hypothesis would have to be supported by some new physics. Physicists have suggested that perhaps quantum mechanics may support it. Some calculations indicate that if particles travel to their own past, the interaction with the previous form of the particle creates a runaway surge of energy that destroys the wormhole.
Is time travel possible? The present laws of physics don’t prohibit it, but they don’t support it either. New physics has to be discovered to answer the question.
If time travel is possible, where are the time travelers? As Stephen Hawking says, “the best evidence we have that time travel is not possible, and never will be, is that we have never been invaded by hordes of tourists from the future.”
