Black Hole

by Marcia Bartusiak

  It’s true for smaller celestial bodies as well. Earth, for instance, is not holding onto an orbiting satellite with some phantom towline. Rather, the satellite is moving in a “straight” line—straight, that is, in its local frame of reference—the four dimensions of space-time, which are impossible for our three-dimensional minds to fully picture. But we can try with two dimensions.

  General relativity says that space-time is like a vast rubber sheet. In this two-dimensional depiction, masses such as the Earth indent this flexible mat, curving space-time and generating the force we call gravity. (Johnstone, courtesy of Wikimedia Commons)

  Think of two ancient explorers, who imagine the Earth as flat, walking directly north from the equator from separate locations. They move not one inch east or west but only push parallel to each other northward. But then they hear that they are moving closer to each other. They might conclude that some mysterious force is pushing them together. A space traveler perched high above, however, knows what’s really happening. The Earth’s surface is, of course, curved, and so the two explorers are merely following the spherical contour. In spherical space, two straight lines, originally parallel, will cross (unlike in flat space). In a similar fashion, a satellite is following the straightest route in the four-dimensional warp of space-time carved out by the Earth. As long as a heavenly body continues to exist, the indentations it creates in space-time will be part of the landscape of the cosmos. What we think of as gravity—the tendency of two objects to be drawn toward each other—can be pictured as a result of these indentations. In other words, space-time and mass-energy are the Siamese twins of the cosmos, each acting and reacting to the other. As physicist John Wheeler always liked to say, “Spacetime tells matter how to move; matter tells spacetime how to curve.”

  Consequently, Newton’s empty box was suddenly gone. Space was no longer an inert and empty arena, as imagined since the dawn of time. Einstein showed us that space-time, this new physical quantity introduced to physics, is a real-time player in the universe at large. Thinking back on this accomplishment in his autobiographical notes, Einstein wrote, “Newton, forgive me.”

  No apologies needed. Einstein didn’t completely overturn Newton’s law of gravity. Newton got us to the Moon and back just fine, thank you, for gravity is the weakest force in our everyday life. Remember that a tiny magnet can easily pick up a paperclip against the entire pull of Earth’s gravity. Newton’s equations can handily deal with gravity in that environment.

  What Einstein did was extend the law of gravity into realms formerly inaccessible, situations where gravity’s pull is so strong—so monstrously powerful—that it causes matter to fall in at near the speed of light. Newton’s laws completely break down at that point. General relativity is required when there are immense concentrations of gravitation on hand: in the world of stars, galaxies, and the universe at large, where gravity is king.

  And Einstein offered an answer to the question that Newton thought unanswerable: there is a mechanism for gravity’s effects. Every object is just following the warps that other masses indent into space-time.

  Einstein’s success in accounting for that tiny, unexplained shift in Mercury’s orbit was certainly a victory for his new general theory of relativity but not a fait accompli. That awaited confirmation that a light beam indeed bends by his predicted amount as it passes by a massive object like the Sun. Even as he was working on general relativity, Einstein in 1911 had suggested a specific test that astronomers could perform to confirm this response to space-time’s curvatures: photograph a field of stars at night, then for comparison photograph those same stars when they pass near the Sun’s limb during a solar eclipse. A beam of starlight passing right by the Sun would be gravitationally attracted to the Sun and so curve in a bit. Upon photographing that star, it would then appear to have shifted its standard position on the sky, the position it would have if the Sun wasn’t in the way.

  Three solar-eclipse expeditions were launched to carry out the test, but all were unsuccessful due to either bad weather or interference from the ongoing war in Europe. Data comparison problems plagued the results of a fourth test, an American effort led by astronomers from California’s Lick Observatory, which were never published. That was a lucky break for Einstein. The dubious Lick results went against him, and some of the other expeditions had been carried out when his theory, still in the works, was predicting a smaller, incorrect deflection.

  That’s why all eyes were on British astronomers when they announced that they would try to measure the stellar shift in 1919 during a solar eclipse whose path traversed South America and crossed over to central Africa. Arthur Eddington, a renowned astrophysicist best known for his work on stellar physics, led the government-sponsored mission to tiny Príncipe Island, off the coast of West Africa. To minimize the risk of bad weather, two other astronomers journeyed to the village of Sobral in the Amazon forests of northern Brazil. With telescopes and cameras in hand, each team was hoping to measure the extremely tiny effect. Einstein calculated that a ray of starlight just grazing the Sun’s surface should get deflected by a mere 1.7 arcseconds (one thousandth the width of the Moon). Put another way, that’s roughly the width of the graphite in a pencil seen from across an American football field.

  The same star field depicted when near the Sun and when away from the Sun. A star’s light follows a curved path as it passes close to the Sun. But with our eyes tracing the star’s light back as a straight-line path, it appears to us that the star has shifted its position in the celestial sky. (The Cosmic Times Team, NASA Goddard Space Flight Center)

  On the day of the eclipse, May 29, Eddington and his assistant took sixteen photographs, most of them ultimately useless because of intervening clouds. “We have no time to snatch a glance at [the Sun],” Eddington wrote of his adventure. “We are conscious only of the weird half-light of the landscape and the hush of nature, broken by the calls of the observers, and beat of the metronome ticking out the 302 seconds of totality.”

  Fortunately, two of their pictures had decent images of the essential stars. For several days afterward, Eddington spent the daytime hours taking the first stab at measuring those stars. He and his companion carefully compared their pictures to another photo of the same celestial region, one taken months earlier in England at night when the Sun was far below the horizon. Eddington, an early supporter of relativity, freely admitted he was unscientifically rooting for Einstein and so was elated to see that the stars near the Sun did appear to have shifted their positions by an amount that matched Einstein’s prediction, give or take several percent. It certainly didn’t match the shift calculated from Newton’s laws. Here was evidence that the streams of starlight were indeed bending around the darkened Sun, following its indentation in space-time. The Sobral expedition in Brazil, which had fine weather and many more photographs, confirmed Eddington’s finding once they were all back in Great Britain and thoroughly examined all the images.

  The results were officially announced the following November at a special joint meeting in London of the Royal Society and the Royal Astronomical Society. On the wall behind the lectern hung a picture of Isaac Newton, whose historic law of gravitation was undergoing its first big modification. The news from the meeting quickly flashed around the globe. “LIGHTS ALL ASKEW IN THE HEAVENS,” blared the headline in the New York Times. “Men of Science More or Less Agog Over Results of Eclipse Observations … Stars Not Where They Seemed or Were Calculated to Be, but Nobody Need Worry.”

  By then Einstein was forty years old, and his life in public was never the same again. His bushy mustache, helter-skelter hair, and world-weary eyes made him instantly recognizable wherever he went. Celebrities, from presidents to movie stars, clamored to wine and dine the man whose name was now synonymous with “genius.”

  In a letter to Max Born in 1920, Einstein compared himself to King Midas: “Like the man in the fairy tale, whose touch turned everything into gold, thus it is with me, with everything
turning into banner-line news: suum cuique [to each his own].” For a man of thought, who yearned for a haven of quiet to contemplate his physics, it was a state of affairs that he deemed a dazzling misery. “I really did consider flight … ,” he continued to Born. “Now I just think about purchasing a sailboat and a little cottage near Berlin by the water.”

  3

  One Would Then Find Oneself … in a Geometrical Fairyland

  While the results of the 1919 solar-eclipse expedition played a large role in bringing Einstein to the public’s attention, not to mention worldwide fame, there was an even earlier triumph for general relativity in the halls of academia. It had to do with the way in which general relativity’s equations (a set of ten, exceedingly complex to work with) could be solved. Einstein had arrived at his first predictions based on approximations of the gravitational field around the Sun. He made some simplifying assumptions about his equations in order to make them easier to manage. Only in that way could he estimate the shift in Mercury’s orbit and the amount of bending as starlight passed close by the Sun. To Einstein, an exact solution, a result that captured the entire physics and mathematics of the problem without involving approximations, appeared insurmountable. But to his surprise, that wasn’t the case at all.

  Very soon after Einstein’s final presentation before the Berlin Academy—in less than a month, in fact—the German astronomer Karl Schwarzschild arrived at the first full solution to general relativity’s equations. He sent his findings to Einstein immediately, believing that he was allowing “Mr. Einstein’s result to shine with increased purity,” as he noted in his report. It was this remarkable achievement, which both surprised and delighted Einstein, that initiated the long march toward our modern conception of the black hole.

  Both a practical astronomer and a theorist, Schwarzschild was a stand-out in a multitude of fields. He made major contributions in electrodynamics, optics, quantum theory, and stellar astronomy; he was a pioneer in substituting photographic plates for the human eye at the telescope; and he could at times be quite bold in his speculations. Fifteen years before Einstein even introduced the notion of space-time bending, Schwarzschild had pondered whether space was curved rather than flat—either turned inward like a sphere or curved outward like a hyperbola out to infinity. “We can wonder how the world would appear in a spherical or a pseudo-spherical geometry … ,” he told a meeting of German astronomers in 1900. “One would then find oneself, if one will, in a geometrical fairyland; and one does not know whether the beauty of this fairyland may not in fact be realized in nature.” No wonder Schwarzschild latched onto Einstein’s equations so quickly; he had been anticipating them (and avidly followed Einstein’s progress in developing general relativity) over the years.

  As director of the Potsdam Astrophysical Observatory, then the most esteemed position a German astronomer could attain, Schwarz-schild wanted to remove all doubt as to the uniqueness of Einstein’s results. And in aiming for that goal, he ended up devising a method that became a valuable tool for relativists for years afterward.

  Karl Schwarzschild (American Institute of Physics Emilio Segrè Archives, courtesy of Martin Schwarzschild)

  In carrying out this endeavor, Schwarzschild did what all good mathematicians do—devise a scheme that makes the mathematics of the problem simpler. For one, he used spherical coordinates, which makes it easier to map the gravitational field around a spherical mass—in this case, a nonspinning star. To see how this approach can make a complex question simpler, imagine an everyday problem: Take an airplane circling an airport from three miles (almost five kilometers) away. If you want to describe its path using the geometry of a flat grid, the result is very messy. If you designate its east-west position x and its north-south position y, then the algebraic equation that describes its entire route is x2 + y2 = 32. But let’s say you shift to a different geometry altogether: a graph with radial, or circular, coordinates. In that case you don’t have to worry about x’s and y’s at all. The plane is always three miles from the center of a circle, and the equation that describes its flight path is no more complicated than r = 3 (radius = 3). That, in a way, is what Schwarzschild did.

  But his new set of coordinates led to a whopping predicament when he looked at the very center of space-time, where his star resided. As Ralph Sampson, the astronomer royal for Scotland, remarked at the time, “The consequences … are so startling that it is difficult to believe they have any relationship to reality.” To understand this dilemma, picture what happens if all the mass of that star, say the Sun, is squeezed down to a very small size. Upon doing this, Schwarzschild discovered that, around this hypothetical point, a spherical region of space suddenly arose out of which nothing—no signal, not a glimmer of light nor bit of matter—could escape. In its day, it was called “Schwarzschild’s sphere.” Today we call this boundary the “event horizon.” That’s because no event occurring within its borders can be observed from the outside. More than an indentation, space-time in this case becomes a bottomless pit. Light and matter can go in but never come back out. It’s a point of no return. The light and matter get crushed down to a singular point, a condition of zero volume and infinite density called a “singularity.” It’s where the ordinary laws of physics completely break down.

  But I’m getting ahead of myself. That is how we currently picture such a singularity. Schwarzschild and others in his day actually viewed this situation fairly differently: watching how objects, such as light particles, approached Schwarzschild’s sphere, “they got stuck, so to speak,” explains historian Eisenstaedt. “This was then taken as bona fide evidence that all trajectories ended up or died at the [sphere], where time … stopped. … The [light’s] trajectory appeared to perpetually approach the magical sphere, as if to vanish there.” Or maybe they simply piled up on the surface of this magical ball. It was a strange and weird place. The “Schwarzschild singularity” (as it was also called) was an impenetrable sphere from their perspective.

  Arthur Eddington, in his 1926 book The Internal Constitution of the Stars, was confident that no star could possibly collapse to such a compacted state. So, why worry about it? As he fancifully put it, “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).”

  A 1924 illustration of various light beams approaching a Schwarzschild singularity (empty sphere in the center). Those that couldn’t escape just vanished at the surface where time stopped. (From Max von Laue, Die Relativitätstheorie, volume 2, 1924)

  That was one way to look at it. But despite Eddington’s imaginative description, most relativists at the time did not seriously think that space-time itself was being significantly warped and twisted around Schwarzschild’s singularity. “They realized that the spatial component might be slightly bent, that time may be a bit out of step, but nobody imagined that Schwarzschild’s solution could represent a space really different, completely different, from Newton’s,” explains Eisenstaedt. That awaited new mathematical insights in the 1960s. Relativists needed the ability to map the entire region of space-time around the singularity—a grand, calculation-intensive enterprise impossible for physicists in the 1910s and 1920s to undertake. The modern-day vision of a “black hole,” a pit in space-time, was not yet imagined.

  But, still, what was the best way to describe this unusual place? Schwarzschild used the term discontinuity. In France and Belgium it became the sphère catastrophique, for it did appear like a catastrophic place where all the laws of physics went awry. For Eddington, it was the “magic circle.” Others simply referred to it as a “frontier” or “barrier.”

  And how big would that magical sphere be? That depended on the amount of mass caught inside it. If our Sun, which is nearly 900,000 miles (1.4 million kilometers) wide, were suddenly squeezed down to a point, its magic sphere would be less than 4 miles (6 kilometers) across. But the Earth, perched some 93 million miles (150 million kilometers) away, would
not be affected at all. Indeed, all the planets would still orbit around the Sun in the same way they have for some four billion years if the Sun were that size. Although the Sun’s mass is more condensed, it exerts the same gravitational pull on us. It is only closer in that the gravitational pull of the magical sphere starts to soar.

  What happens if the amount of mass is greater, such as the equivalent of ten suns squished down to a point? In that case, the magic sphere would stretch almost forty miles (around sixty kilometers) across. The equations showed that the width of the magical sphere (that is, the event horizon) expands as more and more mass is trapped within it.

  Einstein didn’t expend a lot of energy worrying about these singularities. He figured that Schwarzschild’s novel entity was just a sign that general relativity was still incomplete, that such a hazardous outcome would disappear after he had formulated a unified theory of gravity and electromagnetism, a venture on which he spent the rest of his life with no success.

  Many viewed Schwarzschild’s sphere as merely an artifact, one that arose because of the coordinates being used but had no physical significance whatsoever. Others were not worried for practical reasons. Why be concerned about all that mass being squeezed below its event horizon, when no star was ever seen to be that small? This never happens, they said. Nature was surely providing the means to save the day. Schwarzschild himself figured that the pressure pushing back from all that squeezed matter would step in to prevent the collapse in the first place. Einstein thought so, too. He even worked out a little calculation at a Paris meeting in 1922 showing how a star’s pressure would prevent a large star from catastrophically collapsing. Moreover, no one at this time thought that densities could ever be greater than when atoms are packed as tightly as possible. But Schwarzschild wasn’t even predicting the existence of such a situation; to him, his clever theoretical setup simply enabled him to obtain an exact solution for Einstein’s equations of general relativity and more easily map the gravitational field around a star. It was all just a mathematical game. As he reported, the problem of infinite pressures at the center was “clearly not physically meaningful.”