Gravity / The Cosmos


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.

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James Stein

    James D. Stein is a past member of the Institute of Advanced Studies and is currently a professor of Mathematics at California State University (Long Beach). His list of publications includes: How to Shoot from the Hip Without Getting Shot in the Foot (with Herbert L. Stone and Charles V. Harlow); How Math Explains the World (a Scientific American Book Club selection); The Right Decision (also a Scientific American Book Club selection); and How Math Can Save Your Life. He has been a guest blogger for Psychology Today and his work has been featured in the Los Angeles Times. His latest book is Cosmic Numbers: The Numbers That Define Our Universe. He lives in Redondo Beach, California.