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.

    • Sid

      The business of a star needing to be twenty times the mass of our Sun to collapse into a black hole – is that threshold purely a function of how stellar evolution works? Or is it a more general physical limitation, so that I couldn’t really create a “small” black hole out of, say, a smushed Earth no matter the process used?

      Thanks for the great post, James.

      • Jim Stein

        Hi Sid,

        That’s the Chandrasekhar limit (shameless plug — it gets a chapter of its own in “Cosmic Numbers”). I’m not an astrophysicist, but my understanding is that extreme gravity squishes everything down to a neutron star (all particles in it are neutrons), which is prevented from collapsing further by a quantum-mechanical principle called electron degeneracy. When gravity is strong enough to overcome that, we get a black hole.

    • Balkenstein

      Here is an interesting thought. What would the density of a blackhole be with a Shwartzchild radius the size of our Universe? Would it be higher or lower then the relative density of hydrogen in the known Universe?

      • Balkenstein

        What I am really asking is, are we infact living in something resembling a blackhole?

        • Robertj941

          I think these Black holes ARE the universe and the things we see are the flotsam and jetson of material gathering around the various black holes (gravity wells) Everything outside the black holes are limited by speed of light and electron motion. We put too much importance on ourselves as if what we see is all there is, perhaps we are just living on the outside, unable to look in?–RJ

        • Chris Long

          That’s exactly what I was thinking. lol.

      • shortfingers125

        Who knows, because since the universe is constantly expanding – and thus CREATING more space to be filled, we cannot measure that boundary because we couldn’t get outside of it!

        Plus, it’s impossible to measure the given mass of something in which all other “noticeable” matter exists in.

    • Stan Holbrook

      Can a massive black hole explode into a big bang?

    • Anonymous

      Could you reduce a Golf ball sized Earth to the schwarzschild radius by dipping it in a very large amount of liquid Helium?

      • shortfingers125

        The S. radius of the earth IS app. a golf ball, so how could you reduce the size of an object with infinite mass?

        • shortfingers125

          *Infinite density

    • Ursacava

      Our Universe does have a definite boundary, beyond which we can never pass. The idea that our Universe could be considered as a black hole, which nothing can escape from, was discussed in the book and PBS miniseries “Cosmos”.

      • shortfingers125


    • Lance Frank

      Good article. You get an A. Were it not for the mis spelling of “solider” in the 5th paragraph I would give you an A+. Consider yourself lucky. In Hollywood that can make the difference between existence and non-existence.

      • Anonymous

        Thanks! It’s been corrected.

    • Jimsteinjr

      Balkenstein must be on the same wavelength as I am! When I originally wrote this post, and when I wrote “Cosmic Numbers”, I included the fact that, given the current calculated mass of the Universe, it was probably within its Schwartzschild radius. When this initially occurred to me, I tried to find confirmation — and if you look at the wikipedia article on this subject, you’ll find that it agrees with us.

      The reason that this idea didn’t appear in this post is that the editor felt that it was somewhat speculative, and I agreed with her. Nonetheless, here’s a question: if we are inside the Schwartzschild radius AND the Universe is expanding, what happens when the Universe hits its Schwartzschild radius? Does it bounce off it, or barrel through it, or what? It certainly makes me want to live another few billion years to find out!

      • Hakuin Suso

        Just read Nikodem J. Poplawski’s “Mass of the universe in a black hole” out of the Cornell University Library. It was submitted 23 Oct 2011 and states that if certain conditions apply sigularities cannot form inside a black hole. As a result the interior of every black hole is a new universe.

      • Jas Sahota

        I really don’t know much on the subject, but trying to learn.

        From what you wrote in the article, the Schwartzschild radius is related to the mass of an object. From what I understand of the law of conservation of mass, since this Universe inside the black hole is isolated from outside mass (being inside the black hole, past the event horizon), the mass in the system would remain constant. Therefore the Universe would never actually hit it’s Schwartzschild radius, correct?

    • Rr4ever

      I have often wondered what the result would be of two “black holes” coalescing if one of them was composed of Anti-matter. Could there be any observable difference from two “ordinary” black holes coalescing?

    • Harold Kozak

      How would you distinguish between a Schwarzchild radius and a schwarzchild length? Do you think it will ever be possible for us to use these concepts to develop a provable theory of quantum gravity?

    • Mike Cox

      In discussions about black holes, I’m put off a bit about the imagery of gravity being so strong that even something so light and fast as light – a photon – can’t escape gravity’s tug, as in:

      “…light emitted from that mass can still escape the intense gravitational pull.”

      But: photons have no mass, they can’t be slowed down by their “weight,” they do not become heavy and ponderous like we would feel on Jupiter – instead they are trapped by the distortion of space time created by the black hole.

      • shortfingers125

        No no no photons do have mass. Not enough for us to notice, obviously, but they do. Otherwise they wouldn’t be able to produce heat because they couldn’t “vibrate” in order to release the energy necessary to make us warm when we’re in the sun.

        • Mats

          No no no, photons have no mass. If they did they would not have the speed of light…

    • Arnab Roy

      Thanx Sir..I’ve Just started out readin Your BLOG..I m amazed !!!! Written in so simple terms..

    • Justcurious

      Hmm..the possibility of the universe being one super-super massive blackhole would make sense with the Schwarzschild radius. If the density of a super massive blackhole in center of a galaxy is 1/1000th the density of earth’s air, how much less dense would a blackhole the mass of the universe be?

    • shortfingers125

      What is the proportion of the mass of an object to its Schwarzschild radius?
      And, thus, why is the density of a larger black hole less than that of a smaller one if the Schwarzschild radius of an object is directly proportional to the mass of any given object? Why do black holes that differ in size have Schwarzschild radii of different densities if the Schwarzschild radius changes to suit an object’s mass? I wouldn’t assume that the size of the object matters at all, so the S. radius would theoretically keep the same density no matter the object’s mass and simply change in size in order to fit all the more or less matter into the space provided by the S. radius of that object.

    • its me

      The reason a larger black hole has less mass,because someone is using classical physics. Photons travel in bent spacetime that is why light bends.In a black hole time is at right angles to the observer and so is light?

    • Vinod Kancherla

      i am really confused when some one say photon have no mass….. because stars are loosing billions of tonnes of mass every sec where that mass will go if photon has no mass…………

      • wacg

        The photons are created in the sun by a process called fusion. Fusion converts MASS into energy (photons). I’ll let you puzzle out the rest for yourself.

    • WACG

      If a matter black hole coalesced with an equal mass black hole of ANTI matter would there be any observable difference with what would happen if two matter based black holes coalesced?

      • Fraize

        They would instantly annihilate each other, utterly and completely – probably not the interesting interaction you’d want or expect. However, there is a massive amount of matter and energy spiraling towards both objects as they merge. With no gravity well holding all that hot material in ever-shrinking whirlpools, they would fling themselves outward in all directions. To anybody outside the event, it would look like a massive explosion, but really, it’s just condensed matter and energy flinging itself out from what had been a prison just moments before.