Black Hole

by Marcia Bartusiak

  That seems like a reasonable assumption, until you make the comparison fairly drastic. The effects of Einstein’s assertion are not really noticeable unless the comparative speeds are extremely high. So, let’s do that: Consider a spaceship consistently moving away from Earth at 185,000 miles per second, just under the velocity of light. Common sense might lead you to believe that the astronauts would be going nearly as fast as any light beam passing by the Earth—that they even had a chance of overtaking the light, if they went a little faster, as Einstein once contemplated. But that’s not the case at all. The astronauts on that spaceship will still measure the velocity of that passing light beam at 186,282 miles per second, just as we do back here on Earth.

  This situation seems bizarre, but only because our commonplace notions of space and time get in the way. In our everyday lives we think like Newton and all the ancient philosophers before him did: that space is an empty box, forever the same and immovable. You are either at rest or in motion within this fixed space that surrounds us.

  Similarly, there is a universal clock, which ticks off the seconds for all the inhabitants of the cosmos in the same way. Events everywhere, from one end of the universe to the other, are in step with this grand cosmic timepiece, no matter what your position or speed.

  But Einstein ingeniously realized that wasn’t the case at all. The seeming paradox that arises for those fast-moving astronauts—how they can possibly measure the same speed of light as us—is solved by acknowledging that time is not absolute. Time is, well, relative. The very term velocity (miles per hour or meters per second) involves keeping track of time, but the astronauts and earthlings do not share the same time standard. That was Einstein’s genius. He recognized that Newton’s universal clock was a sham.

  Since nothing can travel faster in a vacuum than the speed of light, two observers set apart in different frames of reference cannot agree on what time it is. The finite speed of light prevents the two from simultaneously synchronizing their watches. Einstein discovered that observers separated by distance and movement will not agree on when events in the universe are taking place.

  This mismatch has other consequences as well. By just looking, the earthlings and astronauts will also not agree on one another’s measurements. Mass, length, and time are all adjustable, depending on one’s individual frame of reference. Look from Earth at a clock on that swiftly receding spaceship. You will see time progressing more slowly than here on Earth. You will also see the spaceship foreshortened in the direction of its motion. Those on the spaceship, who perceive no changes in themselves or in their clock’s progression, look back at their receding home planet and see the same contraction of objects and slowing of time in the earthlings’ surroundings! Each of us measures a difference in the other to the same degree. Space shrinks and time slows down when two observers are uniformly speeding either toward or away from each other. Space and time are different in each reference frame—enough to keep the speed of light the same in both environments. As soon as the astronauts start moving in relation to the Earth, they spawn (in a sense) their own “bubble” of experience, different from ours. We no longer share the same worldview. The only thing that we earthlings and the astronauts will agree on is the speed of light in a vacuum. That is the one universal constant.

  With absolute time destroyed, there was also no need for absolute space either. Our intuition that the solar system sits serenely at rest, with the spaceship speeding away in some motionless container of space, no longer works. No such entity exists. In reality, we could just as easily consider the astronauts at rest, with the Earth speeding away. That being the case, wrote Einstein, the ether becomes “superfluous.” Armed with this new viewpoint, he said, physicists no longer needed “an ‘absolutely stationary space’ provided with special properties.” The ether had once provided a unique reference frame for physicists; it marked an absolute and universal state of rest. But Einstein revealed that this ethereal substance had been a fiction all along. “For me—and many others—the exciting feature of this paper was not so much its simplicity and completeness,” said physicist Max Born during a fiftieth-anniversary celebration of the publication of special relativity, “but the audacity to challenge Isaac Newton’s established philosophy, the traditional concepts of space and time.”

  The mathematician Hermann Minkowski, who once taught Einstein, brilliantly discerned an even deeper beauty in Einstein’s new theory. With his expert mathematical know-how, Minkowski recognized that he could recast special relativity into a geometric model. He showed that Einstein was essentially making time a fourth dimension. Space and time coalesce into a single entity known as space-time. You can think of space-time as a series of snapshots stacked together, tracing changes in space over the seconds, minutes, and hours. Only now the snapshots are melded together into an unbreakable whole. Dimensionally, time acts like just another component of space. It’s a consequence of the speed of light being constant; speed is defined as distance over time. If the distance a light beam travels contracts, so too must time slow down, to keep the ratio a constant. The two are irrevocably linked. “Henceforth,” said Minkowski in a famous 1908 lecture, “space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

  Minkowski cleverly recognized that, although different observers in different situations may disagree on when and where an event occurred, they will agree on a combination of the two. From one position, an observer will measure a certain distance and time interval between two events. Perched in another frame of reference, a different observer may see more space or less time. But in both cases they will see that the total space-time separation is the same. The fundamental quantity becomes not space alone, or time alone, but rather a combination of all four dimensions at once—height, width, breadth, and time. Einstein was not an expert in mathematics and so didn’t appreciate this geometrical representation. When first acquainted with Minkowski’s idea, he declared the abstract mathematical formulation “banal” and “a superfluous learnedness.” This was because Minkowski’s novel outlook, to him, didn’t seem to offer any added value to the physics he had so meticulously constructed. But he would soon change his mind.

  Special relativity is called “special” for a reason. It covers only a very restricted type of motion: objects moving straight ahead at a constant velocity. That’s a pretty narrow range of movement. So, soon after devising this new law, Einstein was determined to extend its rules to all types of motion, things that are accelerating away from us, slowing down, twisting, or turning. But special relativity was “child’s play,” said Einstein, compared to the development of a more general theory of relativity, one that would cover all those additional dynamical situations—in particular, gravity, which involves acceleration.

  Over the ensuing years Einstein’s reputation grew and soared, which some of his former teachers found surprising. Often bored in class, he had antagonized his professors, which made it difficult for him to obtain an academic position upon graduation. Consequently, Einstein had to start his career as a junior examiner at the Swiss patent office, a job he actually found quite fulfilling, recalling his seven years there as one of the happiest times in his life. He wrote his first important papers while employed there (including the one on special relativity and a treatise on the photoelectric effect, for which he won the Nobel Prize in 1921). But once his status as a physicist vastly improved with those publications under his belt, he was able to leave the patent office in 1909 to carry out a series of university appointments in both Zurich and Prague. He attained the peak of professional recognition in 1914, when he moved to the prestigious University of Berlin as a research professor and became a member of the Prussian Academy of Sciences. It was over those many years of settling and resettling that he waged his mental battle with general relativity, amid teaching responsibilities, a failed marriage, and World War I. For nearly a decade, he struggled with th
e problem of recasting Newton’s laws of gravity in the light of relativity.

  He didn’t go straight to the equations and fiddle with them. That wasn’t his style at all. What he did at the outset was think—and think hard. Einstein knew that he first needed to establish a theoretical framework that could match what is experienced in the world around us. He carried out a variety of thought experiments in his head to see where they would lead. “Like a child who builds a toy house with blocks of various colors,” explains science historian Jean Eisenstaedt, “Einstein started with sets of principles, the conceptual blocks or theoretical elements that he could place, move, suppress, and arrange in different ways; these are the bricks he used to erect his theoretical buildings.”

  Albert Einstein around the time that he was working on general relativity in the early 1910s. (Hebrew University of Jerusalem, Albert Einstein Archives, courtesy of American Institute of Physics Emilio Segrè Visual Archives)

  Einstein first recognized that the force we feel upon a uniform acceleration and the force we feel when under the control of gravity are one and the same. In the jargon of physics, gravity and a constant acceleration are “equivalent.” There is no difference between being pulled down on the Earth by gravity or being pulled backward in an accelerating car. To arrive at this conclusion, Einstein imagined a windowless room far out in space, magically accelerating upward, moving faster and faster with each passing second. Anyone in that room would find their feet pressed against the floor. In fact, without windows to serve as a check, you couldn’t be sure you were in space. From the feel of your weight, you could as easily be standing quietly in a room on Earth. Both the magical, accelerating space elevator and the Earth, with its gravitational field keeping you in place, are equivalent systems. Einstein reasoned that the fact that the laws of physics predict exactly the same behavior for objects in the accelerating room and in Earth’s gravitational hold means that gravity and acceleration are, in some fashion, the same thing.

  These thought experiments, which Einstein carried out liberally to get a handle on his questions, led to some interesting insights. Watch someone throw a ball outward in that accelerating elevator in space and the ball’s path will appear to you, situated outside, to curve downward as the elevator moves upward. A light beam would behave in the same way. But since acceleration and gravity have identical effects, Einstein then realized that light should also be affected by gravity, being attracted (bent) when passing a massive gravitational body, such as the Sun. The nearby matter makes a light beam’s path curve.

  Driven by his powerful physical intuition, Einstein began to pursue these ideas more earnestly around 1911. At that time he was beginning to confirm that clocks would slow down in gravitational fields. Special relativity already suggested that a moving clock would run more slowly; now Einstein was stating that a stationary clock would also tick more slowly when immersed in a gravitational field, an effect never before contemplated by physicists. He was saying that a clock in space will tick faster than one weighed down by Earth’s gravity.

  He was also coming to recognize that his final equations would likely be “non-Euclidean”—that is, based on a geometry different from the one you likely learned in grade school with basic axioms set down by the famous Greek mathematician Euclid in the third century BCE. In Euclid’s world, space is entirely flat in all directions—an unchanging vista. But it was slowly dawning on Einstein that gravity would involve curvatures of space. Or to put it more correctly, curvatures of space-time, that invention by Minkowski that he had so blithely dismissed a few years earlier. Einstein was finally coming to appreciate Minkowski’s mathematical take on special relativity and its creation of that “banal” four-dimensional manifold. Without Minkowski’s earlier contribution, admitted Einstein contritely, the “general theory of relativity might have remained stuck in its diapers.” Minkowski, unfortunately, was not around to hear that apology; he had died in 1909 of appendicitis at the age of forty-four.

  By the summer of 1912, Einstein was at last eager to mold his burgeoning conjectures into proper mathematical form. Ignorant of non-Euclidean geometries, though, he joined up with mathematician Marcel Grossmann, an old college chum, to aid him in mastering the intricacies of this new mathematics. “Grossman,” cried out Einstein upon arriving at his friend’s home in Zurich, “you must help me or else I’ll go crazy.” Einstein chose well in seeking out assistance. It was Grossmann who pointed out to Einstein that his ideas would best be expressed in the specific geometric language first developed by the German mathematician Bernhard Riemann in the 1850s and later extended by German and Italian geometers. By 1914, Einstein moved to Berlin and continued on his own, inexorably amending and tweaking his solutions, but now he was additionally armed with the mathematical insights Grossman had introduced him to.

  His progress, though, was sluggish, and by the following year he was increasingly frustrated. His theory, as it then stood, could not accurately account for a particular shift in the orbit of Mercury. From his earliest days of contemplating a general theory of relativity, Einstein knew that a successful formulation of a new law of gravity would have to account for that anomaly.

  Why is that so? Let me explain. Mercury, a planet positioned about thirty-six million miles (fifty-eight million kilometers) from the Sun, slowly revolves around the Sun, as do all the other planets. Yet these planetary orbits are not perfectly circular (as Kepler discovered) but more elliptical. With that in mind, imagine Mercury’s orbit as an elongated ring. The point of this extended ring that is closest to the Sun—what is known as a planet’s perihelion—shifts around over time. For Mercury the perihelion advances about 574 arcseconds each century (about 0.04 percent of its orbital circumference). Most of this tiny shift is due to Mercury’s interaction with the other planets; their combined gravitational tugging alters the orbit’s alignment. But that can account for only 531 arcseconds. The remaining 43 arcseconds (as measured today) were left unexplained, a nagging mystery to astronomers for decades. Newton’s laws couldn’t resolve the discrepancy, at least given the known makeup of the solar system. That led some to speculate that Venus might be heavier than previously thought or that Mercury had a tiny moon. The most popular solution suggested that another planet, dubbed “Vulcan” for the Roman god of fire, was orbiting closer to the Sun than Mercury, providing an extra gravitational pull. There were even a few reports of Vulcan sightings, but none were reliable.

  Einstein wanted his general theory of relativity to explain that added little gravitational nudge, once and for all. As his equations stood in early 1915, Einstein was able to predict an extra shift in Mercury’s orbital motion of 18 arcseconds (five thousandths of a degree) per century. But he was aiming for the measured change that was around twice as large. Discouraged, he went back and reviewed his previous work over the years. It was then that he noticed a mistake in one step of the derivations he had earlier conducted with Grossmann, a tactic the two had ultimately dismissed at the time. This spurred Einstein to reconsider that approach once again. He began to modify the equations and in the process became aware of some earlier misunderstandings. His years of toil and vexation were about to end.

  His major effort took place over the course of November 1915. On each of the four Thursdays of that month he reported his incremental progress to the Prussian Academy. A breakthrough came soon after his second report on November 11. That week he was at last able to successfully calculate Mercury’s extra orbital shift. He would later write a friend that he had palpitations of the heart on seeing this result: “I was beside myself with ecstasy for days.” Here was the theory’s first experimental success, grounding it in the real world. Moreover, Einstein’s new formulation also predicted that starlight would get deflected around the Sun twice as much as he had earlier calculated (and twice the amount if Newton’s theory is used). That’s because Newton’s laws take only space into account; Einstein now understood that gravity affects both space and time alike, hence doubling th
e effect.

  Triumph arrived on November 25, the day he presented his concluding paper, titled “The Field Equations of Gravitation.” In this culminating talk he presented the final modifications to his theory, one more term added in to complete the job. At last, he no longer needed a special frame of reference. He had arrived, truly and without question, at a general theory of gravity. The month’s flurry of computational activity had been exhausting. In a letter to his longtime friend Michele Besso shortly afterward, Einstein wrote that his “boldest dreams have now been fulfilled,” signing off, “your contented, but rather worn out Albert.”

  We usually visualize gravity as a force, something pushing or pulling on us. But Einstein introduced a new way to think of gravity—not as a force but instead as an inherent response to curvatures in space-time. From this viewpoint, objects that appear to be controlled by a force are actually just following the natural, curved pathways before them. Light, as it gets bent, is following the twists and turns of the space-time highway. And Mercury, being so close to the Sun, has more of a “dip” to contend with, which partly explains the extra shift in its orbit.

  How is that so? From Einstein’s perspective, space is not simply an enormous empty expanse but instead a sort of boundless rubber sheet, a physical entity unto itself. And given that image, such a sheet can be manipulated in many ways: It can be stretched or squeezed; it can be straightened or bent; it can even be indented in spots. So, massive stars like our Sun sit in this flexible mat like cosmic bowling balls, creating depressions. The more massive the object, the deeper the indent. As a consequence, planets circle the Sun, not because they are held by invisible tendrils of force, as Newton had us think, but because they are simply caught in the natural hollow carved out by the star.