Black Holes Discovered and Rejected
in which Einstein’s laws
of warped spacetime
predict black holes,
and Einstein rejects the prediction
“The essential result of this investigation,” Albert Einstein wrote in a technical paper in 1939, “is a clear understanding as to why the ‘Schwarzschild singularities’ do not exist in physical reality.” With these words, Einstein made clear and unequivocal his rejection of his own intellectual legacy: the black holes that his general relativistic laws of gravity seemed to be predicting.
Only a few features of black holes had as yet been deduced from Einstein’s laws, and the name “black holes” had not yet been coined; they were being called “Schwarzschild singularities.” However, it was clear that anything that falls into a black hole can never get back out and cannot send light or anything else out, and this was enough to convince Einstein and most other physicists of his day that black holes are outrageously bizarre objects which surely should not exist in the real Universe. Somehow, the laws of physics must protect the Universe from such beasts.
What was known about black holes, when Einstein so strongly rejected them? How firm was general relativity’s prediction that they do exist? How could Einstein reject that prediction and still maintain confidence in his general relativistic laws? The answers to these questions have their roots in the eighteenth century.
Throughout the 1700s, scientists (then called natural philosophers) believed that gravity was governed by Newton’s laws, and that light was made of corpuscles (particles) that are emitted by their sources at a very high, universal speed. That speed was known to be about 300,000 kilometers per second, thanks to telescopic measurements of light coming from Jupiter’s moons as they orbit around their parent planet.
In 1783 John Michell, a British natural philosopher, dared to combine the corpuscular description of light with Newton’s gravitation laws and thereby predict what very compact stars should look like. He did this by a thought experiment which I repeat here in modified form:
Launch a particle from the surface of a star with some initial speed, and let it move freely upward. If the initial speed is too low, the star’s gravity will slow the particle to a halt and then pull it back to the star’s surface. If the initial speed is high enough, gravity will slow the particle but not stop it; the particle will manage to escape. The dividing line, the minimum initial speed for escape, is called the “escape velocity.” For a particle ejected from the Earth’s surface, the escape velocity is 11 kilometers per second; for a particle ejected from the Sun’s surface, it is 617 kilometers per second? or 0.2 percent of the speed of light.
Michell could compute the escape velocity using Newton’s laws of gravity, and could show that it is proportional to the square root of the star’s mass divided by its circumference. Thus, for a star of fixed mass, the smaller the circumference, the larger the escape velocity. The reason is simple: The smaller the circumference, the closer the star’s surface is to its center, and thus the stronger is gravity at its surface, and the harder the particle has to work to escape the star’s gravitational pull.
There is a critical circumference, Michell reasoned, for which the escape velocity is the speed of light. If corpuscles of light are affected by gravity in the same manner as other kinds of particles, then light can barely escape from a star that has this critical circumference. For a star a bit smaller, light cannot escape at all. When a corpuscle of light is launched from such a star with the standard light velocity of 299,792 kilometers per second, it will fly upward at first, then slow to a halt and fall back to the star’s surface; see Figure 3.1.
Michell could easily compute the critical circumference; it was 18.5 kilometers, if the star had the same mass as the Sun, and proportionately larger if the mass were larger.
3.1 The behavior of light emitted from a star that is smaller than the critical circumference, as computed in 1783 by John Michell using Newton’s laws of gravity and corpuscular description of light.
Nothing in the eighteenth-century laws of physics prevented so compact a star from existing. Thus, Michell was led to speculate that the Universe might contain a huge number of such dark stars, each living happily inside its own critical circumference, and each invisible from Earth because the corpuscles of light emitted from its surface are inexorably pulled back down. Such dark stars were the eighteenth-century versions of black holes.
Michell, who was Rector of Thornhill in Yorkshire, England, reported his prediction that dark stars might exist to the Royal Society of London on 27 November 1783. His report made a bit of a splash among British natural philosophers. Thirteen years later, the French natural philosopher Pierre Simon Laplace popularized the same prediction in the first edition of his famous work Le Systeme du Monde, without reference to Michell’s earlier work. Laplace kept his dark-star prediction in the second (1799) edition, but by the time of the third (1808) edition, Thomas Young’s discovery of the interference of light with itself1 was forcing natural philosophers to abandon the corpuscular description of light in favor of a wave description devised by Christiaan Huygens—and it was not at all clear how this wave description should be meshed with Newton’s laws of gravity so as to compute the effect of a star’s gravity on the light it emits. For this reason, presumably, Laplace deleted the concept of a dark star from the third and subsequent editions of his book.
Only in November 1915, after Einstein had formulated his general relativistic laws of gravity, did physicists once again believe they understood gravitation and light well enough to compute the effect of a star’s gravity on the light it emits. Only then could they return with confidence to the dark stars (black holes) of Michell and Laplace.
The first step was made by Karl Schwarzschild, one of the most distinguished astrophysicists of the early twentieth century. Schwarz-schild, then serving in the German army on the Russian front of World War I, read Einstein’s formulation of general relativity in the 25 November 1915 issue of the Proceedings of the Prussian Academy of Sciences. Almost immediately he set out to discover what predictions Einstein’s new gravitation laws might make about stars.
Since it would be very complicated, mathematically, to analyze a star that spins or is nonspherical, Schwarzschild confined himself to stars that do not spin at all and that are precisely spherical, and to ease his calculations, he sought first a mathematical description of the star’s exterior and delayed its interior until later. Within a few days he had the answer. He had calculated, in exact detail, from Einstein’s new field equation, the curvature of spacetime outside any spherical, non-spinning star. His calculation was elegant and beautiful, and the curved spacetime geometry that it predicted, the Schwarzschild geometry as it soon came to be known, was destined to have enormous impact on our understanding of gravity and the Universe.
Schwarzschild mailed to Einstein a paper describing his calculations, and Einstein presented it in his behalf at a meeting of the Prussian Academy of Sciences in Berlin on 13 January 1916. Several weeks later, Einstein presented the Academy a second paper by Schwarzschild: an exact computation of the spacetime curvature inside the star. Only four months later, Schwarzschild’s remarkable productivity was halted: On 19 June, Einstein had the sad task of reporting to the Academy that Karl Schwarzschild had died of an illness contracted on the Russian front.
The Schwarzschild geometry is the first concrete example of spacetime curvature that we have met in this book. For this reason, and because it is so central to the properties of black holes, we shall examine it in detail.
If we had been thinking all our lives about space and time as an absolute, unified, four-dimensional spacetime “fabric,” then it would be appropriate to describe the Schwarzschild geometry immediately in the language of curved (warped), four-dimensional spacetime. However, our everyday experience is with three-dimensional space and one-dimensional time, un-unified; therefore, I shall give a description in which warped spacetime is split up into warped space plus warped time.
Karl Schwarzschild in his academic robe in Göttingen, Germany. [Courtesy AIP Emilio Segré Visual Archives.]
Since space and time are “relative” (my space differs from your space and my time from yours, if we are moving relative to each other2), such a split requires first choosing a reference frame—that is, choosing a state of motion. For a star, there is a natural choice, one in which the star is at rest; that is, the star’s own. reference frame. In other words, it is natural to examine the star’s own space and the star’s own time rather than the space and time of someone moving at high speed through the star.
As an aid in visualizing the curvature (warpage) of the star’s space, I shall use a drawing called an embedding diagram Because embedding diagrams will play a major role in future chapters, I shall introduce the concept carefully, with the help of an analogy.
Imagine a family of human-like creatures who live in a universe with only two spatial dimensions. Their universe is the curved, bowl-like surface depicted in Figure 3.2. They, like their universe, are two-dimensional; they are infinitesimally thin perpendicular to the surface. Moreover, they cannot see out of the surface; they see by means of light rays that move in the surface and never leave it. Thus, these “2D beings,” as I shall call them, have no method whatsoever to get any information about anything outside their two-dimensional universe.
These 2D beings can explore the geometry of their two-dimensional universe by making measurements on straight lines, triangles, and circles. Their straight lines are the “geodesics” discussed in Chapter 2 (Figure 2.4 and associated text): the straightest lines that exist in their two-dimensional universe. In the bottom of their universe’s “bowl,” which we see in Figure 3.2 as a segment of a sphere, their straight lines are segments of great circles like the equator of the Earth or its lines of constant longitude. Outside the lip of the bowl their universe is flat, so their straight lines are what we would recognize as ordinary straight lines.
If the 2D beings examine any pair of parallel straight lines in the outer, flat part of their universe (for example, Lt and L2 of Figure 3.2), then no matter how far the beings follow those lines, they will never see them cross. In this way, the beings discover the flatness of the outer region. On the other hand, if they construct the parallel straight lines L3 and L4 outside the bowl’s lip, and then follow those lines into the bowl region, keeping them always as straight as possible (keeping them geodesics), they will see the lines cross at the bottom of the bowl. In this way, they discover that the inner, bowl region of their universe is curved.
The 2D beings can also discover the flatness of the outer region and the curvature of the inner region by measuring circles and triangles (Figure 3.2). In the outer region, the circumferences of all circles are equal to π (3.14159265 . . .) times their diameters. In the inner region, circumferences of circles are less than π times their diameters; for example, the large circle drawn near the bowl’s bottom in Figure 3.2 has a circumference equal to 2.5 times its diameter. When the 2D beings construct a triangle whose sides are straight lines (geodesics) and then add up the triangle’s interior angles, they obtain 180 degrees in the outer, flat region, and more than 180 degrees in the inner, curved region.
Having discovered, by such measurements, that their universe is curved, the 2D beings might begin to speculate about the existence of a three-dimensional space in which their universe resides—in which it is embedded. They might give that three-dimensional space the name hyperspace, and speculate about its properties; for example, they might presume it to be “flat” in the Euclidean sense that straight, parallel lines in it never cross. You and I have no difficulty visualizing such a hyperspace; it is the three-dimensional space of Figure 3.2, the space of our everyday experience. However, the 2D beings, with their limited two-dimensional experience, would have great difficulty visualizing it. Moreover, there is no way that they could ever learn whether such a hyperspace really exists. They can never get out of their two-dimensional universe and into hyperspace’s third dimension, and because they see only by means of light rays that stay always in their universe, they can never see into hyperspace. For them, hyperspace would be entirely hypothetical.
3.2 A two-dimensional universe peopled by 2D beings.
The third dimension of hyperspace has nothing to do with the 2D beings’ “time” dimension, which they might also think of as a third dimension. When thinking about hyperspace, the beings would actually have to think in terms of four dimensions: two for the space of their universe, one for its time, and one for the third dimension of hyperspace.
We are three-dimensional beings, and we live in a curved three-dimensional space. If we were to make measurements of the geometry of our space inside and near a star—the Schwarzschild geometry-—we would discover it to be curved in a manner closely analogous to that of the 2D beings’ universe.
We can speculate about a higher-dimensional, flat hyperspace in which our curved, three-dimensional space is embedded. It turns out that such a hyperspace must have six dimensions in order to accommodate curved three-dimensional spaces like ours inside itself. (And when we remember that our Universe also has a time dimension, we must think in terms of seven dimensions in all.)
Now, it is even harder for me to visualize our three-dimensional space embedded in a six-dimensional hyperspace than it would be for 2D beings to visualize their two-dimensional space embedded in a three-dimensional hyperspace. However, there is a trick that helps enormously, a trick depicted in Figure 3.3.
Figure 3.3 shows a thought experiment: A thin sheet of material is inserted through a star in its equatorial plane (upper left), so the sheet bisects the star leaving precisely identical halves above and below it. Even though this equatorial sheet looks flat in the picture, it is not really flat. The star’s mass warps three-dimensional space inside and around the star in a manner that the upper left picture cannot convey, and that warpage curves the equatorial sheet in a manner the picture does not show. We can discover the sheet’s curvature by making geometric measurements on it in our real, physical space, in precisely the same way as the 2D beings make measurements in the two-dimensional space of their universe. Such measurements will reveal that straight lines which are initially parallel cross near the star’s center, the circumference of any circle inside or near the star is less than π times its diameter, and the sums of the internal angles of triangles are greater than 180 degrees. The details of these curved-space distortions are predicted by Schwarzschild’s solution of Einstein’s equation.
To aid in visualizing this Schwarzschild curvature, we, like the 2D beings, can imagine extracting the equatorial sheet from the curved, three-dimensional space of our real Universe, and embedding it in a fictitious, flat, three-dimensional hyperspace (lower right in Figure 3.3). In the uncurved hyperspace, the sheet can maintain its curved geometry only by bending downward like a bowl. Such diagrams of two-dimensional sheets from our curved Universe, embedded in a hypothetical, flat, three-dimensional hyperspace, are called embedding diagrams.
3.3 The curvature of the three-dimensional space inside and around a star (upper left), as depicted by means of an embedding diagram (lower right). This is the curvature predicted by Schwarzschild’s solution to Einstein’s field equation.
It is tempting to think of hyperspace’s third dimension as being the same as the third spatial dimension of our own Universe. We must avoid this temptation. Hyperspace’s third dimension has nothing whatsoever to do with any of the dimensions of our own Universe. It is a dimension into which we can never go and never see, and from which we can never get any information; it is purely hypothetical. Nonetheless, it is useful. It helps us visualize the Schwarzschild geometry, and later in this book it will help us visualize other curved-space geometries: those of black holes, gravitational waves, singularities, and worm-holes (Chapters 6, 7, 10, 13, and 14).
As the embedding diagram in Figure 3.3 shows, the Schwarzschild geometry of the star’s equatorial sheet is qualitatively the same as the geometry of the 2D beings’ universe: Inside the star, the geometry is bowl-like and curved; far from the star it becomes flat. As with the large circle in the 2D beings’ bowl (Figure 3.2), so also here (Figure 3.3), the star’s circumference divided by its diameter is less than π. For our Sun, the ratio of circumference to diameter is predicted to be less than π by several parts in a million; in other words, inside the Sun, space is flat to within several parts in a million. However, if the Sun kept its same mass and were made smaller and smaller in circumference, then the curvature inside it would become stronger and stronger, the downward dip of the bowl in the embedding diagram of Figure 3.3 would become more and more pronounced, and the ratio of circumference to diameter would become substantially less than π.
Because space is different in different reference frames (“your space is a mixture of my space and my time, if we move relative to each other”), the details of the star’s spatial curvature will be different as measured in a reference frame that moves at high speed relative to the star than as measured in a frame where the star is at rest. In the space of the high-speed reference frame, the star is somewhat squashed perpendicular to its direction of motion, so the embedding diagram looks much like that of Figure 3.3, but with the bowl compressed transversely into an oblong shape. This squashing is the curved-space variant of the contraction of space that Fitzgerald discovered in a universe without gravity (Chapter 1).
Schwarzschild’s solution to the Einstein field equation describes not only this curvature (or warpage) of space, but also a warpage of time near the star—a warpage produced by the star’s strong gravity. In a reference frame that is at rest with respect to the star, and not flying past it at high speed, this time warpage is precisely the gravitational time dilation discussed in Chapter 2 (Box 2.4 and associated discussion): Near the star’s surface, time flows more slowly than far away, and at the star’s center, it flows slower still.
In the case of the Sun, the time warpage is small: At the Sun’s surface, time should flow more slowly by just 2 parts in a million (64 seconds in one year) than far from the Sun, and at the Sun’s center it should flow more slowly than far away by about 1 part in 100,000 (5 minutes in one year). However, if the Sun kept its same mass and were made smaller in circumference so its surface was closer to its center, then its gravity would be stronger, and correspondingly its gravitational time dilation—its warpage of time—would become larger.
One consequence of this time warpage is the gravitational redshift of light emitted from a star’s surface. Since the light’s frequency of oscillation is governed by the flow of time at the place where the light is emitted, light emerging from atoms on the star’s surface will have a lower frequency when it reaches Earth than light emitted by the same kinds of atoms in interstellar space. The frequency will be lowered by precisely the same amount as the flow of time is slowed. A lower frequency means a longer wavelength, so light from the star must be shifted toward the red end of the spectrum by the same amount as time is dilated on the star’s surface.
At the Sun’s surface the time dilation is 2 parts in a million, so the gravitational redshift of light arriving at the Earth from the Sun should also be 2 parts in a million. This was too small a redshift to be measured definitively in Einstein’s day, but in the early 1960s, technology began to catch up with Einstein’s laws of gravity: Jim Brault of Princeton University, in a very delicate experiment, measured the red-shift of the Sun’s light, and obtained a result in nice agreement with Einstein’s prediction.
Within a few years after Schwarzschild’s untimely death, his spacetime geometry became a standard working tool for physicists and astrophysicists. Many people, including Einstein, studied it and computed its implications. All agreed and took seriously the conclusion that, if the star were rather large in circumference, like the Sun, then spacetime inside and around it should be very slightly curved, and light emitted from its surface and received at Earth should be shifted in color, ever so slightly, toward the red. All also agreed that the more compact the star, the greater must be the warpage of its spacetime and the larger the gravitational redshift of light from its surface. However, few were willing to take seriously the extreme predictions that the Schwarzschild geometry gave for highly compact stars (Figure 3.4):
The Schwarzschild geometry predicted that for each star there is a critical circumference, which depends on the star’s mass—the same critical circumference as had been discovered by John Michell and Pierre Simon Laplace more than a century earlier: 18.5 kilometers times the mass of the star in units of the mass of the Sun. If the star’s actual circumference is larger than this critical one by a factor of 4 (upper part of Figure 3.4), then the star’s space will be moderately curved as shown, time at its surface will flow 15 percent more slowly than far away, and light emitted from its surface will be shifted toward the red end of the spectrum by 15 percent. lf the star’s circumference is smaller, just twice the critical one (middle part of Figure 3.4), its space will be more strongly curved, time at its surface will flow 41 percent more slowly than far away, and light from its surface will be redshifted by 41 percent. These predictions seemed acceptable and reasonable. What did not seem at all reasonable to physicists and astrophysicists of the 1920s, or even as late as the 1960s, was the prediction for a star whose actual circumference was the same as its critical one (bottom part of Figure 3.4). For such a star, with its more strongly curved space, the flow of time at the star’s surface is infinitely dilated; time does not flow at all—it is frozen. And correspondingly, no matter what may be the color of light when it begins its journey upward from the star’s surface, it must get shifted beyond the red, beyond the infrared, beyond. radio wavelengths, all the way to infinite wavelengths; that is, all the way out of existence. In modern language, the star’s surface, with its critical circumference, is precisely at the horizon of a black hole; the star, by its strong gravity, is creating a black-hole horizon around itself.
3.4 General relativity’s predictions for the curvature of space and the redshift of light from three highly compact stars with the same mass but different circumferences. The first is four times larger than the critical circumference, the second is twice as large as critical, and the third has its circumference precisely critical. In modern language, the surface of the third star is a black-hole horizon.
The bottom line of this Schwarzschild-geometry discussion is the same as that found by Michell and Laplace: A star as small as the critical circumference must appear completely dark, when viewed from far away; it must be what we now call a black hole. The bottom line is the same, but the mechanism is completely different:
Michell and Laplace, with their Newtonian view of space and time as absolute and the speed of light as relative, believed that for a star just a bit smaller than the critical circumference, corpuscles of light would very nearly escape. They would fly up to great heights above the star, higher than any orbiting planet; but as they climbed, they would be slowed by the star’s gravity, then halted somewhere short of interstellar space, then turned around and pulled back down to the star. Though creatures on an orbiting planet could see the star by its slow-moving light (to them it would not be dark), we, living far away on Earth, could not see it at all. The star’s light could not reach us. For us the star would be totally black.
By contrast, Schwarzschild’s spacetime curvature required that light always propagate with the same universal speed; it can never be slowed. (The speed of light is absolute, but space and time are relative.) However, if emitted from the critical circumference, the light must get shifted in wavelength an infinite amount, while traveling upward an infinitesimal distance. (The wavelength shift must be infinite because the flow of time is infinitely dilated at the horizon, and the wavelength always shifts by the same amount as time is dilated.) This infinite shift of wavelength, in effect, removes all the light’s energy; and the light, thereupon, ceases to exist! Thus, no matter how close a planet might be to the critical circumference, creatures on it cannot see any light at all emerging from the star.
In Chapter 7, we shall study how the light behaves as seen from inside a black hole’s critical circumference, and shall discover that it does not cease to exist after all. Rather, it simply is unable to escape the critical circumference (the hole’s horizon) even though it is moving outward at the standard, universal speed of 299,792 kilometers per second. But this early in the book, we are not yet ready to comprehend such seemingly contradictory behavior. We must first build up our understanding of other things, as did physicists during the decades between 1916 and 1960.
During the 1920s and into the 1930s, the world’s most renowned experts on general relativity were Albert Einstein and the British astrophysicist Arthur Eddington. Others understood relativity, but Einstein and Eddington set the intellectual tone of the subject. And, while a few others were willing to take black holes seriously, Einstein and Eddington were not. Black holes just didn’t “smell right”; they were outrageously bizarre; they violated Einstein’s and Eddington’s intuitions about how our Universe ought to behave.
In the 1920s Einstein seems to have dealt with the issue by ignoring it. Nobody was pushing black holes as a serious prediction, so there was not much need on that score to straighten things out. And since other mysteries of nature were more interesting and puzzling to Einstein, he put his energies elsewhere.
Eddington in the 1920s took a more whimsical approach. He was a bit of a ham, he enjoyed popularizing science, and so long as nobody was taking black holes too seriously, they were a playful thing to dangle in front of others. Thus, we find him writing in 1926 in his book The Internal Constitution of the Stars that no observable star can possibly be more compact than the critical circumference: “Firstly,” he wrote, “the force of gravitation would be so great that light would be unable to escape from it, the rays falling back to the star like a stone to the Earth. Secondly, the redshift of the spectral lines would be so great that the spectrum would be shifted out of existence. Thirdly, the mass would produce so much curvature of the space-time metric that space would close up round the star, leaving us outside (i.e. nowhere).” The first conclusion was the Newtonian version of light not escaping; the second was a semi-accurate, relativistic description; and the third was typical Eddingtonian hyperbole. As one sees clearly from the embedding diagrams of Figure 3.4, when a star is as small as the critical circumference, the curvature of space is strong but not infinite, and space is definitely not wrapped up around the star. Eddington may have known this, but his description made a good story, and it captured in a whimsical way the spirit of Schwarzschild’s spacetime curvature.
In the 1930s, as we shall see in Chapter 4, the pressure to take black holes seriously began to mount. As the pressure mounted, Eddington, Einstein, and others among the “opinion setters” began to express unequivocal opposition to these outrageous objects.
In 1939, Einstein published a general relativistic calculation that he interpreted as an example of why black holes cannot exist. His calculation analyzed the behavior of an idealized kind of object which one might have thought could be used to make a black hole. The object was a cluster of particles that pull on each other gravitationally and thereby hold the cluster together, in much the same way as the Sun holds the solar system together by pulling gravitationally on its planets. The particles in Einstein’s cluster all moved in circular orbits around a common center; their orbits formed a sphere, with particles on one side of the sphere pulling gravitationally on those on the other side (left half of Figure 3.5).
3.5 Einstein’s evidence that no object can ever be as small as its critical circumference. Left: If Einstein’s spherical cluster of particles is smaller than 1.5 critical circumferences, then the particles’ speeds must exceed the speed of light, which is impossible. Right: If a star with constant density is smaller than 9/8 = 1.125 critical circumferences, then the pressure at the star’s center must be infinite, which is impossible.
Einstein imagined making this cluster smaller and smaller, trying to drive its actual circumference down toward the critical circumference. As one might expect, his calculation showed that the more compact the cluster, the stronger the gravity at its spherical surface and the faster the particles must move on its surface to prevent themselves from being pulled in. If the cluster were smaller than 1.5 times the critical circumference, Einstein’s calculations showed, then its gravity would be so strong that the particles would have to move faster than the speed of light to avoid being pulled in. Since nothing can move faster than light, there was no way the cluster could ever be smaller than 1.5 times critical. “The essential result of this investigation,” Einstein wrote, “is a clear understanding as to why the ‘Schwarzschild singularities’ do not exist in physical reality.”
As backing for his view, Einstein could also appeal to the internal structure of an idealized star made of matter whose density is constant throughout the stellar interior (right half of Figure 3.5). Such a star was prevented from imploding by the pressure of the gas inside it. Karl Schwarzschild had used general relativity to derive a complete mathematical description of such a star, and his formulas showed that, if one makes the star more and more compact, then in order to counteract the increased strength of its internal gravity, the star’s internal pressure must rise higher and higher. As the star’s shrinking circumference nears 
= 1.125 times its critical circumference, Schwarzschild’s formulas show the central pressure becoming infinitely large. Since no real gas can ever produce a truly infinite pressure (nor can any other kind of matter), such a star could never get as small as 1.125 times critical, Einstein believed.
Einstein’s calculations were correct, but his reading of their message was not. The message he extracted, that no object can ever become as small as the critical circumference, was determined more by Einstein’s intuitive opposition to Schwarzschild singularities (black holes) than by the calculations themselves. The correct message, we now know in retrospect, was this:
Einstein’s cluster of particles and the constant-density star could never be so compact as to form a black hole because Einstein demanded that some kind of force inside them counterbalance the squeeze of gravity: the force of gas pressure in the case of the star; the centrifugal force due to the particles’ motions in the case of the cluster. In fact, it is true that no force whatsoever can resist the squeeze of gravity when an object is very near the critical circumference. But this does not mean the object can never get so small. Rather, it means that, if the object does get that small, then gravity necessarily overwhelms all other forces inside the object, and squeezes the object into a catastrophic implosion, which forms a black hole. Since Einstein’s calculations did not include the possibility of implosion (he left it out of all his equations), he missed this message.
We are so accustomed to the idea of black holes today that it is hard not to ask, “How could Einstein have been so dumb? How could he leave out the very thing, implosion, that makes black holes?” Such a reaction displays our ignorance of the mindset of nearly everybody in the 1920s and 1930s.
General relativity’s predictions were poorly understood. Nobody realized that a sufficiently compact object must implode, and that the implosion will produce a black hole. Rather, Schwarzschild singularities (black holes) were imagined, incorrectly, to be objects that are hovering at or just inside their critical circumference, supported against gravity by some sort of internal force; Einstein therefore thought he could debunk black holes by showing that nothing supported by internal forces can be as small as the critical circumference.
If Einstein had suspected that “Schwarzschild singularities” can really exist, he might well have realized that implosion is the key to forming them and internal forces are irrelevant. But he was so firmly convinced they cannot exist (they “smelled wrong”; terribly wrong) that he had an impenetrable mental block against the truth—as did nearly all his colleagues.
In T. H. White’s epic novel The Once and Future King there is a society of ants which has the motto, “Everything not forbidden is compulsory.” That is not how the laws of physics and the real Universe work. Many of the things permitted by the laws of physics are so highly improbable that in practice they never happen. A simple and time-worn example is the spontaneous reassembly of a whole egg from fragments splattered on the floor: Take a motion picture of an egg as it falls to the floor and splatters into fragments and goo. Then run the motion picture backward, and watch the egg spontaneously regenerate itself and fly up into the air. The laws of physics permit just such a regeneration with time going forward, but it never happens in practice because it is highly improbable.
Physicists’ studies of black holes during the 1920s and 1930s, and even on into the 1940s and 1950s, dealt only with the issue of whether the laws of physics permit such objects to exist—and the answer was equivocal: At first sight, black holes seemed to be permitted; then Einstein, Eddington, and others gave (incorrect) arguments that they are forbidden. In the 1950s, when those arguments were ultimately disproved, many physicists turned to arguing that black holes might be permitted by the laws of physics, but are so highly improbable that (like the reassembling egg) they never occur in practice.
In reality, black holes, unlike the reassembling egg, are compulsory in certain common situations; but only in the late 1960s, when the evidence that they are compulsory became overwhelming, did most physicists begin to take black holes seriously. In the next three chapters I shall describe how that evidence mounted from the 1930s through the 1960s, and the widespread resistance it met.
This widespread and almost universal twentieth-century resistance to black holes is in marked contrast to the enthusiasm with which black holes were met in the eighteenth-century era of John Michell and Pierre Simon Laplace. Werner Israel, a modern-day physicist at the University of Alberta who has studied the history in depth, has speculated on the reasons for this difference.
“I am sure [that the eighteenth-century acceptance of black holes] was not just a symptom of the revolutionary fervour of the 1790s,” Israel writes. “The explanation must be that Laplacian dark stars [black holes] posed no threat to our cherished faith in the permanence and stability of matter. By contrast, twentieth-century black holes are a great threat to that faith.”
Michell and Laplace both imagined their dark stars as made from matter with about the same density as water or earth or rock or the Sun, about 1 gram per cubic centimeter. With this density, a star, to be dark (to be contained within its critical circumference), must have a mass about 140 million times greater than the Sun’s and a circumference about 3 times larger than the Earth’s orbit. Such stars, governed by Newton’s laws of physics, might be exotic, but they surely were no threat to any cherished beliefs about nature. If one wanted to see the star, one need only land on a planet near it and look at its light corpuscles as they rose in their orbits, before plummeting back to the star’s surface. If one wanted a sample of the material from which the star was made, one need only fly down to the star’s surface, scoop some up, and bring it back to Earth for laboratory study. I do not know whether Michell, Laplace, or others of their day speculated about such things, but it is clear that if they did, there was no reason for concern about the laws of nature, about the permanence and stability of matter.
The critical circumference (horizon) of a twentieth-century black hole presents quite a different challenge. At no height above the horizon can one see any emerging light. Anything that falls through the horizon can never thereafter escape; it is lost from our Universe, a loss that poses a severe challenge to physicists’ notions about the conservation of mass and energy.
“There is a curious parallel between the histories of black holes and continental drift [the relative drifting motion of the Earth’s continents],” Israel writes. “Evidence for both was already non-ignorable by 1916, but both ideas were stopped in their tracks for half a century by a resistance bordering on the irrational. I believe the underlying psychological reason was the same in both cases. Another coincidence: resistance to both began to crumble around 1960. Of course, both fields [astrophysics and geophysics] benefitted from postwar technological developments. But it is nonetheless interesting that this was the moment when the Soviet H-bomb and Sputnik swept away the notion of Western science as engraved in stone and beyond challenge, and, perhaps, instilled the suspicion that there might be more in heaven and earth than Western science was prepared to dream of.”
2. Figure 1.3, and the lessons of the tale of Mledina and Serona in Chapter 2.