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The Schwarzschild Radius: Nature's Breaking Point

How far can you compress something before you reach nature’s ultimate breaking point—that is, before you create a black hole? Inspired by Einstein’s theory of general relativity and its novel vision of gravity, the German physicist Karl Schwarzschild took on this question in 1916. His work revealed the limit at which gravity triumphs over the other physical forces, creating a black hole. Today, we call this number the Schwarzschild radius. The Schwarzschild radius is the ultimate boundary: We can receive no information from the black hole that lies within it. It is as if a portion of our universe has been cut off. However, there is a lot more to the black hole story, which actually starts in the late 1700s with a little-known scientist named John Michell. Michell devised the torsion balance, a piece of equipment which enables the strength of forces to be computed quite accurately. He gave his torsion balance to Henry Cavendish, who used it to obtain the first accurate measurement of the weight of the Earth. Charles Augustin de Coulomb later used a torsion balance to establish the strength of electrical attraction and repulsion, and high-tech torsion balances are still an important measurement tool today. Michell was the first person to conceive of the possibility of a gravitational mass so large that light could not escape from it, and was then able to come up with an estimate of how large such a body must be. Though Michell’s calculation did not produce the right answer—after all, he was working with Newton’s laws, not Einstein’s, and the speed of light was not known to high accuracy at the time—he deserves great credit for being the first to imagine the cosmic beasts we now know as black holes. More than a century later, Karl Schwarzschild would be the first to correctly analyze the relation between the size of a black hole and its mass. It was 1916, and he was a soldier stationed on the Russian front. But he was not your typical soldier. A distinguished professor specializing in astrophysics, he enlisted in the German army when he was more than forty years old. His reading matter at the front was also different from the reading matter preferred by the ordinary soldier. Albert Einstein had just published his General Theory of Relativity, and Schwarzschild not only managed to obtain a copy (probably no mean feat in itself, considering the circumstances), but was able to do significant research in the thick of a war zone. Although Schwarzschild survived the hazards of battle, he sadly fell victim to pemphigus, a disease which ravaged his immune system, and died within a year—but not before he discovered the number that now bears his name. Schwarzschild showed that any mass could become a black hole if that mass were compressed into a sufficiently small sphere—a sphere with a radius R, which we now call the Schwarzschild radius. To calculate the Schwarzschild radius of any object—a planet, a galaxy, even an apple—all you need to know is the mass to be compressed. The Schwarzschild radius for the Earth is approximately one inch, meaning that you could squish the entire mass of the Earth into a sphere the size of a basketball and still not have a black hole: light emitted from that mass can still escape the intense gravitational pull. However, if you squeeze the mass of the Earth into a sphere the size of a ping-pong ball, it becomes a black hole. To Schwarzschild, black holes were merely a theoretical possibility, not a physical reality. It wasn’t until later in the twentieth century that it was shown that any star with a mass larger than twenty times that of the Sun would eventually collapse and become a black hole—a number much smaller than Michell’s original calculation. Does the Schwarzschild radius define the “size” of a black hole? The answer is both yes and no. On one hand, theorists believe that all the “stuff” inside a black hole collapses into a singularity, an infinitely small and infinitely dense point well inside the boundary defined by the Schwarzschild radius. If you could visit a black hole, you wouldn’t perceive a physical boundary along the surface defined by the Schwarzschild radius. However, you would in fact be at a very special location: You would be traversing the “event horizon” of the black hole, the point-of-no-return from which nothing, not even light, can escape. The Schwarzschild radius also suggests a second way to think about the density of the black hole. Though the density of the singularity is infinite, the density of a black hole can also be defined as the black hole’s mass divided by the volume of a sphere with the Schwarzschild radius. By this accounting, Earth-mass black hole is dense beyond belief. After all, a ping-pong ball has a volume of a few cubic inches and the mass of the Earth is six sextillion tons (give or take a few quintillion), so the density of an Earth-mass black hole is on the order of sextillion tons per cubic inch. One surprising quirk of this mathematics, however, is that the larger the mass, the lower the density of the black hole. That’s because the Schwarzschild radius increases in proportion to the amount of mass—an object with twice as much mass as the Earth will have a Schwarzschild radius that is twice as large as the Earth’s. But density is mass divided by volume, and the volume of a sphere increases as the cube of its radius. If you double the size of the Schwarzschild radius, thus accommodating twice as much mass in the black hole, you increase the volume by a factor of 2 x 2 x 2 = 8. The density of the larger black hole will only be one-quarter of the density of the smaller one. So every time you double the mass in a Schwarzschild radius black hole, thus doubling the radius, the density decreases by a factor of 4. This has a simple but rather surprising consequence. The Schwarzschild radius of a black hole whose mass is equal to that of a galaxy is so large that the density of that black hole is less than one one-thousandth the density of air on the surface of the Earth! That probably isn’t what you picture when you think of a black hole. In fact, thanks to modern computer graphics, we all share a vision of a black hole as an ominous, totally black sphere surrounded by swirling stars and planets, with those nearby spiraling into eventual annihilation. And yet our very first picture of these bizarre objects came neither from an artist’s pen or a telescope’s lens: It came from mathematics, and from one number that traced the perimeter of physics itself.

The Chandrasekhar Limit: The Threshold That Makes Life Possible

There is a thin line between a bang and a whimper. For stars, this line is called the Chandrasekhar Limit, and it is the difference between dying in a blaze of glory and going out in a slow fade to black. For our universe, this line means much more: Only by exceeding it can stars sow the seeds of life throughout the cosmos. The Chandrasekhar Limit is named for Subrahmanyan Chandrasekhar, one of the great child prodigies. Chandrasekhar graduated with a degree in physics before reaching his twentieth birthday. He was awarded a Government of India scholarship to study at Cambridge, and in the fall of 1930 boarded a ship to travel to England. While aboard the ship—still before reaching his twentieth birthday—he did the bulk of the work for which he would later be awarded a Nobel Prize. By the 1920s—a decade before Chandrasekhar began his journey to England—astronomers had realized that Sirius B, the white dwarf companion to the bright star Sirius, had an astoundingly high density—more than a million times the density of the sun. An object of this density could only exist if the atoms comprising the star were so tightly compressed that they were no longer individual atoms. Gravitational pressure would compress atoms so much that the star would consist of positively-charged ions surrounded by a sea of electrons. Prior to the discovery of quantum mechanics, physicists knew of no force capable of supporting any star against such gravitational pressure. Quantum mechanics, though, suggested a new way for a star to hold itself up against the force of gravity. According to the rules of quantum mechanics, no two electrons can be in the exact same state. Inside an extremely dense star like Sirius B, this means that some electrons are forced out of low energy states into higher ones, generating a pressure called electron degeneracy pressure that resists the gravitational force. This makes it possible for a star like Sirius B to achieve such extreme density without collapsing in on itself. This discovery was made by Ralph Fowler, who would later become Chandrasekhar’s graduate supervisor. But Chandrasekhar realized what Fowler had missed: The high-energy electrons inside the white dwarf would have to be traveling at velocities near the speed of light, invoking a set of bizarre relativistic effects. When Chandrasekhar took these relativistic effects into account, something spectacular happened. He found a firm upper limit for the mass of any body which could be supported by electron degeneracy pressure. Once this limit—the Chandraskehar limit—was exceeded, the object could no longer resist the force of gravity, and it would begin to collapse. When Chandrasekhar published these results in 1931, he set off a battle with one of the greatest astrophysicists of the era, Sir Arthur Eddington, who believed that the white dwarf state was the eventual fate of every star. At a conference in 1935, Eddington told his audience that Chandrasehkar’s work “was almost a reduction ad absurdum of the relativistic degeneracy formula. Various accidents may intervene to save a star, but I want more protection than that. I think there should be a law of Nature to prevent a star from behaving in this absurd way!” Chandrasekhar was deeply hurt by Eddington’s reaction, but colleagues can disagree profoundly and still remain friends. Chandrasekhar and Eddington remained friends, went to the Wimbledon tennis tournament together and went for bicycle rides in the English countryside. When Eddington passed away in 1944, Chandrasekhar spoke at his funeral, saying “I believe that anyone who has known Eddington will agree that he was a man of the highest integrity and character. I do not believe, for example, that he ever thought harshly of anyone. That was why it was so easy to disagree with him on scientific matters. You can always be certain he would never misjudge you or think ill of you on that account.” Vindication would eventually come to Chandrasekhar when he was awarded the Nobel Prize in 1983 for his work. The Chandrasekhar Limit is now accepted to be approximately 1.4 times the mass of the sun; any white dwarf with less than this mass will stay a white dwarf forever, while a star that exceeds this mass is destined to end its life in that most violent of explosions: a supernova. In so doing, the star itself dies but furthers the growth process of the universe—it both generates and distributes the elements on which life depends. The life of a star is characterized by thermonuclear fusion; hydrogen fuses to helium, helium to carbon, and so on, creating heavier and heavier elements. However, thermonuclear fusion cannot create elements heavier than iron. Only a supernova explosion can create copper, silver, gold, and the “trace elements” that are important for the processes of life. Lighter elements like carbon, oxygen, and nitrogen are also essential to life, but without supernova explosions, they would remain forever locked up in stars. Being heavier than the hydrogen and helium that comprise most of the initial mass of the stars, they sink to form the central core of the star—just as most of the iron on Earth is locked up in its core. If stars are, as Eddington believed, destined to become white dwarfs, those elements would remain confined to the stellar interior, or at best be delivered in relatively minute quantities to the universe as a whole via stellar winds. Life as we know it requires rocky planets to form, and there simply is no way to get enough rocky material out into the universe unless stars can deliver that material in wholesale quantities. And supernovae do just that. The Chandrasekhar Limit is therefore not just as upper limit to the maximum mass of an ideal white dwarf, but also a threshold. A star surpassing this threshold no longer hoards its precious cargo of heavy elements. Instead, it delivers them to the universe at large in a supernova that marks its own death but makes it possible for living beings to exist. Go Deeper Editor’s picks for further reading BBC: <a href="http://www.bbc.co.uk/radio4/science/testtubesandtantrums.shtml"> Test Tubes and Tantrums: Arthur Stanley Eddington and Subrahmanyan Chandrasekhar </a> In this radio program, discover the history of one of the nastiest disagreements in astrophysics. FQXi: <a href="http://www.fqxi.org/community/forum/topic/1256"> Exploding the Supernova Paradigm </a> In this blog post, Zeeya Merali investigates gaps in our understanding of supernova explosions. Nobelprize.org: <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/1983/chandrasekhar-autobio.html#"> Subramanyan Chandrasekhar – Autobiobraphy </a>

Planck's Constant: The Number That Rules Technology, Reality, and Life

In 1878—before Einstein was born, before quantum mechanics, before we knew that our galaxy was one among many—a well-known physicist named Phillip von Jolly told young Max Planck, a student aspiring to a career in physics, “In this field, almost everything is already discovered, and all that remains is to fill a few unimportant holes.” Little did von Jolly realize how seriously he had underestimated the depth and quantity of those “unimportant holes,” and he certainly had no idea that Planck was to play a vital role in helping to fill them. Fortunately for us, Planck was not turned off by Jolly’s remark, and replied that he was not so much interested in discovering new things as in understanding what was known. This might sound unusual, as most scientists are motivated by a combination of two things: a desire to understand, coupled with the urge to discover. Discovery and understanding go hand-in-hand; together they move science forward, and as science moves forward, the quality of our lives improves. Planck’s career was ultimately characterized by the discovery of something truly new, something which would lead to a deeper understanding of perhaps one of the great questions in all science: how the universe enables life to exist. Chemistry tells us that the smallest amount of water is a water molecule, and any container of water consists of a staggering number of identical water molecules. In order to resolve an underlying problem in the theory of energy distribution, Planck wondered, What if energy worked the same way? What if there were a smallest unit of energy, just as there is a smallest unit of water? The idea that energy could be expressed in discrete units, or “quantized,” was fundamental to the development of quantum theory. Indeed, you might say that Planck put the “quanta” in quantum mechanics. So what is this smallest unit of energy? Planck hypothesized the existence of a constant, now known as Planck’s constant, or h , which links a wave or particle’s frequency with its total energy. Today, we know that h = 6.6262 x 10 -34 Joule⋅second Planck’s constant has had profound ramifications in three important areas: our technology, our understanding of reality, and our understanding of life itself. Of the universal constants—the cosmic numbers which define our Universe—the speed of light gets all the publicity (partially because of its starring role in Einstein’s iconic equation E = mc 2 ), but Planck’s constant is every bit as important. Planck’s constant has also enabled the construction of the transistors, integrated circuits, and chips that have revolutionized our lives. More fundamentally, the discovery of Planck’s constant advanced the realization that, when we probe the deepest levels of the structure of matter, we are no longer looking at “things” in the conventional meaning of the word. A “thing”—like a moving car—has a definite location and velocity; a car may be 30 miles south of Los Angeles heading east at 40 miles per hour. The concepts of location, velocity, and even existence itself blur at the atomic and subatomic level. Electrons do not exist in the sense that cars do, they are, bizarrely, everywhere at once, but much more likely to be in some places than in others. Reconciling the probabilistic subatomic world with the macroscopic everyday world is one of the great unsolved problems in physics—a not-so-unimportant hole that even von Jolly would have recognized as such. Finally, Planck’s constant tells us how the universe is numerically fine-tuned to permit life to exist. Carl Sagan, one of the great popularizers of science, was fond of saying that “We are all star stuff”—the chemicals which form our bodies are produced in the explosions of supernovas. The fundamental nuclear reaction eventually leading to the explosion of a supernova is the fusion of four hydrogen atoms to produce a single atom of helium. In the process, approximately 0.7% of the mass is converted to energy via E=mc 2 . That’s not much, but there is so much hydrogen in the Sun that it has been radiating enough energy to warm our planet for more than four billion years—even from a distance of 93,000,000 miles—and will continue to do so for another five billion years. This 0.7% is known as the efficiency of hydrogen fusion, and our understanding of it is one of the consequences of Planck’s investigations. It requires a great deal of heat to enable hydrogen to fuse to helium, and the hydrogen atoms in the sun are moving at different speeds, much like cars on a freeway move at different speeds. The slower-moving hydrogen atoms just bounce off each other; they are insufficiently hot to fuse. Higher speeds, though, mean higher temperatures, and there is a small fraction of hydrogen atoms moving at sufficiently high speeds to fuse to helium. The 0.7% efficiency of hydrogen fusion is what is sometimes referred to as a “Goldilocks number.” Like the porridge that Goldilocks eventually ate, which was neither too hot nor too cold, but just right, the 0.7% efficiency of hydrogen fusion is “just right” to permit the emergence of life as we know it. The process of hydrogen fusion is an intricate high-speed, high-temperature ballet. The first step of this reaction produces deuterium, an isotope of hydrogen whose nucleus consists of one proton and one neutron. In this process, two protons slam into one another, causing one of the protons to shed its electrical charge and metamorphose into a neutron. If the efficiency of hydrogen fusion were as low as 0.6%, the neutron and proton would not bond to each other to form a deuterium atom. In this case, we’d still have stars—huge glowing balls of hydrogen—but no star stuff would ever form because the porridge would be too cold to create helium, the first step on the road to creating the elements necessary for life. On the other hand, if hydrogen fusion had an efficiency of 0.8%, it would be much too easy for helium to form. The hydrogen in the stars would become helium so quickly that there wouldn’t be much hydrogen left to form the molecule most essential for life—water. Star stuff would be produced, but without water life as we know it would not exist. Maybe something else would take the place of water, and maybe life could evolve—but not ours. Planck’s quantization of energy was an essential step on the road to the theory of quantum mechanics, which is critical to our understanding of stellar evolution. Science hasn’t filled in all the pieces of the puzzle of how life actually evolved, but quantum mechanics did begin to answer the question of how the pieces got there in the first place, and probably even Philipp von Jolly would recognize that as an important hole in our knowledge of the universe that desperately needed to be filled. But perhaps the greater lesson is this: The very moment when it feels like “almost everything is already discovered” may be the moment that the universe is about to yield up its biggest surprises—if you’re not afraid to dig in to a few holes.

James Stein

Articles

The Schwarzschild Radius: Nature's Breaking Point

The Chandrasekhar Limit: The Threshold That Makes Life Possible

Planck's Constant: The Number That Rules Technology, Reality, and Life

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