The Baffling Simplicity of Black Holes

By Corey S. Powell | August 20, 2013 12:53 pm

I’ve had black holes on the brain lately.* There’s been a flurry of related research announced lately, even the discovery of black hole-like vortexes in the Atlantic Ocean, and astronomers are keenly watching as a gas cloud is ripped apart by the monster black hole at the center of our galaxy. All of which has prompted me to think about the odd simplicity of how black holes work.

Black hole

A solitary black hole betrays its presence solely through gravity, which bends and warps the light of more distant objects in this illustration. (Credit: NASA/ESA and G. Bacon/STScI)

In fact, you might say that black holes are the simplest objects in the universe. Think of all the attributes you would have to list in order to describe the Earth. There are oceans, continents, clouds, volcanoes, animals, plants, people…really, all of science except for astronomy and its cousin disciplines is dedicated to describing our planet and the varied things that exist on it or in it. Black holes, in contrast, have only three defining attributes: mass, spin, and electric charge. List those three and you can paint a complete portrait of a black hole.

The irony is that black holes are also among the most puzzling objects in the universe, because theorists understand so little about their insides. What happens to the information that is lost when objects fall across the event horizon? What happens to the laws of physics at the singularity in the center? Can black holes create wormholes across space and time? Those internal oddities are so incomprehensible that Albert Einstein did not believe that black holes were a real physical possibility. Which makes it a little confusing to read articles about how Einstein might have been wrong about how black holes work, since he didn’t think they worked at all.

But on the outside, mass, spin, and charge tell you everything there is to observe about a black hole. Practically speaking, most black holes probably have very little net charge, so you could plausibly pare the list to mass and rotation. Even rotation has little impact if you are considering a black hole from a great distance–which, for your sake, I certainly hope you are. Viewed from afar, then, a black hole is a one-attribute object. Mass is the one and only thing you need to know.

Which brings me to another profoundly strange and simple thing about black holes. For an ordinary sphere–a bowling ball, for example–the mass increases as the cube of the radius. If one bowling ball is twice the diameter of another it will weigh eight times (2 cubed) as much. The rule breaks down a bit for large objects like planets, but in a very straightforward way. Their incredible bulk compresses their insides, so as planets get more massive their interiors tend to get more dense, assuming you are making an apples-to-apples comparison of the same type of planet. Some planets around other stars have masses several times that of Jupiter, but they are similar in size because of this gravitational squishing.

Galactic center.

Panoramic view of the Sagittarius region of the sky captures the abundant stars and gas clouds visible toward the center of the Milky Way. The central black hole lurks, unseen, behind the dark lane at middle left. (Credit: ESO/S. Guisard)

Black holes do something completely different, however. Their radius increases in direct proportion to the mass. Double the mass of a black hole, and its diameter doubles as well. (I’m using the event horizon–the point-of-no-return that defines the shape of the black hole–as its “surface” in this discussion.) The math of calculating the diameter of a black hole could not be easier. A black hole with the mass of the sun has a diameter of about 6 kilometers, or 4 miles. Want to know the diameter of the black hole at the center of the Milky Way? Based on the motions of stars circling around it, the black hole has a mass of 3.6 million suns. Just multiply 4 x 3.6 million and you’ve got your answer: It is 14 million miles wide.

The direct relationship between size and mass has a funny effect. The more massive a black hole is, the less dense it is–and the dropoff happens rapidly, as the square of the radius. (Again, I’m using the event horizon to define the surface of the black hole.) A solar-mass black hole crams the sun’s entire 865,000-mile-wide bulk into that 4-mile wide sphere, corresponding to a density 18 quadrillion times the density of water. It’s a staggering number. The black hole at the center of the Milky Way has a mass of 3.6 million suns, which means its density is (3.6 million x 3.6 million) times lower. That translates to about 1,400 times the density of water–still very high, and more than 100 times the density of lead, but no longer so incomprehensible.

Other black holes are much more massive than the central one in our galaxy, though, which means they are also much puffier. The galaxy M87 contains a monster black hole that astronomers have measured as having the mass of 6.6 billion suns. Its density is about 1/3,000th the density of water. That is similar to the density of the air you are breathing right now!

Now to the most mind-blowing part. If you keep going to higher masses, the radius of the black hole keeps growing and the density keeps shrinking. Let’s examine the most extreme case: What is the radius of a black hole with the mass of the entire visible universe? Turns out that its radius is…the same as the radius of the visible universe. Almost as if the entire universe is just one huge black hole.

Sagittarius A*

Reconstructed motions of stars around the Milky Way’s central black hole (blue trails) make it possible to measure its mass. The orange swirl depicts G2, a gas cloud currently passing close to the hole’s event horizon. (Credit: ESO/S. Gillessen/MPE/Marc Schartmann)

OK, the full story is more complicated than that. The universe is not an independent object placed within a larger metric of space-time, so the comparison isn’t exactly correct. But there is a fundamental truth in there. The overall density of the universe seems to be exactly the critical value that produces an overall flat geometry. That critical density marks the boundary between space that curves in on itself and space that does not–between a closed universe and an open one. Such a balancing point is, indeed, related to the boundary point at which a mass collapses inside its event horizon and becomes a black hole. More on that here.

Want to know what life looks like inside a black hole? Look around. You’re soaking in it.

Follow me on Twitter: @coreyspowell

* If you ask my wife she’ll tell you I’ve had black holes on the brain for at least as long as we’ve been married, but that’s a whole other story.

  • Buddy199

    “It looks as if the background dark energy were ‘tuned’ to just the right density so that the remaining mass density needed to make the universe flat would be in the narrow range that could support life. Why would that be? Again, we don’t know, but perhaps the best guess is that some huge number of different possible background densities, allowed by string theory, all occur. In the universes where the wrong ones occurred, nobody is asking why.” – Mike W

    This gets to the question of a multi-verse. The problem I have with that idea is that it is by definition untestable by scientific means; the possible other universes are forever isolated from each other, unable to exchange information of any sort. Scientifically, we can only prove that a single universe exists, ours. Which makes the extremely improbable physical values it is based upon all the more baffling.

    • John McIntire

      There’s not really a mystery here, the universe isn’t “fine-tuned” because there is no tuner:
      http://thoughtsonscienceandpseudoscience.blogspot.com/2012/12/the-fine-tuning-of-universe-goldilocks.html

      • Buddy199

        Regarding fine-tuning, one could write a book just citing the arguments for it made by some of the most distinguished scientists in the world. Here is just a tiny sample, collated by physicist Gerald Schroeder, who holds a Ph.D. from the Massachusetts Institute of Technology, where he later taught physics.

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        Michael Turner, astrophysicist at the University of Chicago and Fermilab: “The precision is as if one could throw a dart across the entire universe and hit a bullseye one millimeter in diameter on the other side.”
        Paul Davies, professor of theoretical physics at Adelaide University: “The really amazing thing is not that life on Earth is balanced on a knife-edge, but that the entire universe is balanced on a knife-edge, and would be total chaos if any of the natural ‘constants’ were off even slightly.

        Roger Penrose, the Rouse Ball Professor of Mathematics at the University of Oxford, writes that the likelihood of the universe having usable energy (low entropy) at its creation is “one part out of ten to the power of ten to the power of 123.” That is “a million billion billion billion billion billion billion billion billion billion billion billion billion billion zeros.”

        Steven Weinberg, recipient of the Nobel Prize in Physics, and an anti-religious agnostic, notes that “the existence of life of any kind seems to require a cancellation between different contributions to the vacuum energy, accurate to about 120 decimal places.” As the website explains, “This means that if the energies of the Big Bang were, in arbitrary units, not:

        1000000000000000000000000000000000000000000000000000000000000000000000000000
        00000000000000000000000000000000000000000000

        But instead:

        1000000000000000000000000000000000000000000000000000000000000000000000000000
        00000000000000000000000000000000000000000001

        There would be no life of any sort in the entire universe.”

        Unless one is a closed-minded atheist (there are open-minded atheists), it is not valid on a purely scientific basis to deny that the universe is improbably fine-tuned to create life, let alone intelligent life.

        • John McIntire

          OK let’s leave alone the theistic complications derived by some people, and try to ignore that “tuning” is a loaded word. It still is valid to deny apparent fine-tuning, because how can you say “improbably” fine-tuned? There are not multiple samples to due the probabilities, the constants of nature don’t vary whenever we have looked, so speaking of probabilities seems beyond useless in this regard, at least to me.
          .
          Lots of decimal places doesn’t mean anything, really. The odds of me blinking with only one eye while standing on my left leg and clapping just this very instant in time is .00000000000000000000000000000000000000000000000001, but so what? See what I am getting at here? And what if we changed the units until the number was only accurate to the thousands, like 3700 or thereabouts, doesn’t seem so neat.
          .
          Mathematical constants are accurate not just to 120 decimal places, but to infinity. Does mathematics show fine-tuning as well?

          • Buddy199

            “Fine-tuning” needlessly anthropomorphizes the situation, although it is not a completely illogical assumption among various possibilities. Leaving that aside, from a scientific point of view, we can only be sure there is one universe. Assumptions about a multi-verse are as untestable as assumptions about angels and devils. So, in a sample of one, the various improbabilities that keep piling up (as mentioned in this article) which describe the physical state of our one universe are curious to say the least. At least they seem very improbable and baffling to minds much greater than mine, which is why they feel the need to postulate an infinite number of alternate universes as a possible mathematical explanation, rather than accept each of these oddities as just a statistically meaningless one off. But as I said, creating an infinitely large imaginary sample out of whole cloth unsupported by hard experimental evidence is more of a dodge than anything else. The real scientific question is how in a sample of just one could, not just one or two quirks, but so many improbabilities occur resulting in a stable universe, with maybe the strangest quirk of all: small beings stuck out in the astronomical boondocks who can reflect on and somewhat comprehend such vastness.

          • John McIntire

            Agree in the weirdness and sense of awe and wonder, but we really have to pick another word than “improbability” in regards to this topic, because it is not being used in its scientific sense, but it’s still being associated with scientific concepts. Therein lies the apparent mystery.
            .
            To me, the better “mystery” actually lies in the math…why does math work at all…what the hell is math…is the universe all math all the way down, or just turtles all the way down?
            .
            In any case, I like the poetic way you put it. You’re getting an upvote, good sir!

          • Buddy199

            Interesting discussion in an otherwise routine day, thank you!

  • Peter

    OK, so what about the black hole in M87? Is it possible that the stuff inside it is lumpily distributed? Could stars and planets – or even a secondary black hole – form inside it?

  • Ken Albertsen

    In effect, you sort of ‘burst my bubble’ – of thinking that all black holes were immensely dense masses. Oh well, I reckon I can wrap my head around one of the largest black holes having roughly the density of air, but give some time to do so, ok?

    • James

      To clarify: It’s still correct to think of black holes as dense objects, if one is talking about the singularity (which is where all of the mass is).

      Corey is defining density with respect to the volume inside the horizon, which is slightly confusing since all of the mass ends up in the singularity (in a finite proper time), according to classical general relativity. Since the singularity is a point–it has a radius of zero–and a non-zero mass, it has infinite density.

      So while it’s correct to say that the average density of the M87 black hole is the same as the density of air, it’s wrong to think of the black hole as a uniformly spread mass of low density. The standard picture says that it’s mostly empty space, with a very heavy point in the centre.

      • coreyspowell

        Absolutely right. But from the perspective of an external observer–ie, everyone who is not in the black hole–an object with a mass of 6.6 billion suns becomes a black hole when its density exceeds the density of ordinary air.

        That’s an important point for understanding how supermassive black holes formed in the first place. They did not necessarily require numerous supernova explosions to produce black holes that then merged and combined into a single object. They could also have formed by direct collapse of a huge gas cloud–and the disappearance of that cloud inside its event horizon happens at a much lower density than intuition would suggest.

        • ronaldmsonntag

          I have a question regarding the supposed singularity at the center of any black hole. By definition, this is a point with zero dimensions. However, I haven’t heard anyone talk about the opposing “force” called Heisenberg’s uncertainty principle. It is at the heart of the quantum foam (which is not zero dimensioned) and explains part of the Casimir effect.

          To my knowledge, I have never read that the uncertainty principle does not apply to black holes and I’m asking how do physicists account for two competing infinities? The infinitely small confinement of mass versus the infinitely large uncertainty in the momentum of that mass as required by Heisenberg’s principle? Which infinity wins?

  • Mace Norry

    I question whether Sgr A* is only twenty-five million miles wide. The Sun is one AU away from the Earth and as we all know, an AU is approximately ninety-three million miles. So at twenty-five million miles wide, the black hole at the center of the Milky Way is about 26% of an AU wide. Though black holes are dense, I would think that it’s wider than that. Then again, even at twenty-five million miles wide, it is much bigger than a stellar sized black hole.

    • coreyspowell

      The formula for the radius of a black hole is another oddly simple thing. It is just: R=2GM/c**2 (sorry, I don’t have the right fonts to do that exactly right), where G is the gravitational constant, c is the speed of light, and M is the mass of the black hole. That’s where the diameter number comes from.

      But thank you for questioning my number, because you helped me discover a dumb error in my column: I switched between miles and kilometers at one point and got my values scrambled. In fact Sgr A* is 25 million kilometers wide–even smaller than I stated.

      • Mace Norry

        Thanks for the info! I think it gets tricky because of the term supermassive. Supermassive doesn’t mean super big. After all, the mass is condensed, for instance, in a neutron star to a sphere the size of Manhattan. Black holes are even denser. Thus, they are massive, but they’re not very big. Easy to forget sometimes. Thanks again.

  • Buddy199

    “What is the radius of a black hole with the mass of the entire visible universe? Turns out that its radius is…the same as the radius of the visible universe. Almost as if the entire universe is just one huge black hole.”

    But the universe has an overall flat geometry, a black hole has a geometry where space curves in on itself. So the universe has radius / mass characteristics of a black hole but not the spacial characteristics?

    • coreyspowell

      Like I said, it’s not a perfect equivalence. But think of it this way: One of the core definitions of a black hole is that light cannot escape from it, because its escape velocity is greater than the speed of light. At critical density, the universe teeters on the edge between expanding forever and collapsing back in on itself. If it eventually collapses back in on itself, then no part of the universe–not even light–was able to escape the curvature of space determined by the mass of the universe. That is equivalent to saying that the escape velocity of the universe is…greater than the speed of light.

      If you are following current cosmology you know that dark energy seems to be causing the expansion of the universe to accelerate. Dark energy could evolve in the future, and even change so that the universe eventually collapses anyway. The universe certainly does not fit all the characteristics of a black hole–but it does have some key traits in common, including having the radius that is the same as the Schwartzchild radius for all the mass in the universe and an “escape velocity” that equals the speed of light.

  • http://blogs.discovermagazine.com Gene Partlow

    It’s worth pointing out that there may plausibly be more to the seeming coincidence that the total mass-equivalent content of our universe (ordinary luminous stuff, like stars and galaxies + dark
    matter + dark energy) and its Hubble radius actually fit the Schwarzschild criterion for a black hole. A number of theorists have speculated that black holes may generate baby universes. For example: see Cumrun Vafa, Igor Novikov among others on the archive at http://xxx.lanl.gov/find and type baby
    universe black hole in the abstract box. One way
    or another, the idea points to a possible infinity of universes, all of them having black holes through ordinary stellar evolution, with each black hole generating a baby universe essentially ‘outside’ the parent universe.

    As for plausible contact between universes, Brane
    theory requires that ‘brane universes’ would be in
    some physical contact via propagation of gravitation
    and the quantum wave function, between brane
    universes. No one knows yet what this would mean
    in a given universe, but the idea is, as Powell says,
    pretty staggering indeed.

  • alysdexia

    Nope.
    https://www.quora.com/Physics/Have-the-known-true-laws-of-physics-ever-been-broken-ever/answer/Autymn-Castleton

    (He forgot temperature.) I’m sure the black-hole univers calculation is bullshit. Masses are only estimated at the galactic limit; beyond that is the Hubble flow or expansion by dark energhy; that is, negative mass.

    • coreyspowell

      See my comment below. Dark energy contributes to the overall energy density of the universe in a positive way. But I would agree (using less, er, colorful language) that there are fundamental differences between the universe and a classical black hole with a singularity at the center.

      The visible edge of the universe is like an inverse event horizon: It is the place where space is receding from us at the speed of light. In a flat universe, the distance of the visible edge is therefore going to define a volume equivalent to the volume of an cosmic-mass black hole. But the visible edge of the universe is not an outer boundary of space-time curvature, nor is there any central location (or singularity) in the universe.

  • http://blogs.discovermagazine.com Max Julius

    So, if we’re IN a black hole, then what’s life like outside one?

  • http://blogs.discovermagazine.com Khya Mehta

    is it true that light can not escape the black hole? if so, what happens to the principle of energy of the universe remaining constant?!(considering quantum theory of light) Isn’t it a misconception that light can not escape?

  • Sankaravelyudhan nandakumar

    Blackholes with
    spinup and spin down dynamics may have reversible magneticfield as dual
    blackhole acting as Hawkings radiations after magneticfield reversal.

    hubblesite.org support: ISSUE=7545 PROJ=13

    Blackholes with spinup and spin down dynamics may have reversible
    magneticfield as dual blackhole acting as Hawkings radiations after
    magneticfield reversal. – 00086639

    There is no escape from a black hole in classical
    theory,” Hawking told Nature.
    Quantum theory, however, “enables energy and information to escape from a black
    hole”. A full explanation of the process, the physicist admits, would require a
    theory that successfully merges gravity with the other fundamental forces of
    nature. But that is a goal that has eluded physicists for nearly a century. “The
    correct treatment,” Hawking says, “remains a mystery.” In a thought experiment,
    the researchers asked what would happen to an astronaut unlucky enough to fall
    into a black hole. Event horizons are mathematically simple consequences of
    Einstein’s general theory of relativity that were first pointed out by the
    German astronomer Karl Schwarzschild in a letter he
    wrote to Einstein in late 1915, less than a month after the publication of
    the theory. In that picture, physicists had long assumed, the astronaut would
    happily pass through the event horizon, unaware of his or her impending doom,
    before gradually being pulled inwards — stretched out along the way, like
    spaghetti — and eventually crushed at the ‘singularity’, the black hole’s
    hypothetical infinitely dense core.

    But on analysing the situation in detail,
    Polchinski’s team came to the startling realization that the laws of quantum
    mechanics, which govern particles on small scales, change the situation
    completely. Quantum theory, they said, dictates that the event horizon must
    actually be transformed into a highly energetic region, or ‘firewall’, that
    would burn the astronaut to a crisp.

    Hawking’s new suggestion is that the apparent
    horizon is the real boundary. “The absence of event horizons means that there
    are no black holes — in the sense of regimes from which light can’t escape to
    infinity,” Hawking writes.

    Supermassive black holes located
    in the hearts of galaxies are spinning faster than they ever have before,
    perhaps as a result of recent merging events, report Alejo Martinez-Sansigre of
    the University of Portsmouth and Steve Rawlings of the University of Oxford.

    The experimental consensus seems
    to currently be that the black holes that power active galactic nuclei are
    spinning at close to extremality, meaning they have close to the maximum amount
    of angular momentum for their mass. One reason to expect this is that, absent
    quantum mechanics, the black hole can only generate energy to the extent that
    it is spinning. The AGN’s are generating a huge amount of energy, so something
    must be maintaining the spin, and this should be the in falling matter.

    A more recent theory suggests
    that the material falling into the black hole will come from different
    directions – chaotic accretion – and approximately half the time the black hole
    and the matter will end up rotating in the opposite direction, on average
    resulting in a spin down of the black holes.”

    Black holes with high accretion
    rates typically had low spins, while black holes with low accretion rates had a
    bimodal distribution. “About three-quarters to two-thirds had low spins,
    but the remaining one-quarter to one-third had near maximal spins. This
    suggested to us that accretion is related to low spins. “The only
    mechanism left to spin up that fraction of low-accretion black holes is merger
    events between two black holes of similar mass. If they underwent a merger, but
    afterwards there was little accretion, then they would retain the relatively
    high spins from the merger.”

    But a possible torque reversal can also be there which calls for a
    attraction and repulsion among blackhole says Sankarvelayudhan Nandakumar

    Citation:Magnetic
    field reversal and jet speed variation
    contributing a reversiable magnetic field call for an acceleration and
    deceleration in Blackholes:

    Galaxies are thought to have
    formed from matter and energy that originated in the Big Bang, the theoretical
    explosion that started the expansion of the universe. How supermassive black
    holes formed at the center of active galaxies is under lively debate. The
    dominant school of thought is that matter clumped together to form stars during
    the initial phases of the first galaxies, and some stars were so large and
    dense that they could not withstand their own gravity and imploded to create
    black holes. Another school of thought, to which Vestergaard’s research has
    contributed, is that black holes came into being first and stars formed around
    them to create galaxies.

    Sankaravelayudhan
    Nandakumar,Oford astrophysicist says that temperature difference in condensed
    matter physics of space may call for a spring type reaction calling for
    magneticfield reversal under extreme temperature variations in understanding
    blackhole dynamics as energy rewinding and future radiation of evolution. in
    galaxies forming a blackhole. BH from a low-mass galaxy but is below the escape
    velocity from the Milky Way (MW) galaxy. If central BHs were common in the
    galactic building blocks that merged to make the MW, then numerous BHs that
    were kicked out of low-mass galaxies should be freely floating in the MW halo
    today. We use a large statistical sample of possible merger tree histories for
    the MW to estimate the expected number of recoiled BH remnants present in the
    MW halo today. We find that hundreds of BHs should remain bound to the MW halo
    after leaving their parent low-mass galaxies.

    The galaxies around these early
    supermassive black holes were very young, with intense star formation. Other
    astronomers have established that the mass of a black hole and the mass of its
    galaxy are strictly correlated. These data support the theory that early black
    holes formed first and galaxies formed around them. “But we need a lot more
    data on this to know for sure if this hypothesis is correct,” says Vestergaard

    Understanding this connection
    between stars in a galaxy and the growth of a black hole, and vice-versa, is the
    key to understanding how galaxies form throughout cosmic time”If a black
    hole is spinning it drags space and time with it and that drags the accretion
    disc, containing the black hole’s food, closer towards it. This makes the black
    hole spin faster, a bit like an ice skater doing a pirouette. A
    new way to measure supermassive black hole spin in accretion disc-dominated
    active galaxies Astronomers report the exciting discovery of a new way
    to measure the mass of super massive black holes in galaxies. By measuring
    the speed with which carbon monoxide molecules orbit around such black holes,
    this new research opens the possibility of making these measurements in many
    more galaxies than ever before. Supermassive black holes are now known to
    reside at the centres of all galaxies. In the most massive galaxies in the
    Universe, they are predicted to grow through violent collisions with other
    galaxies, which trigger the formation of stars and provides food for the black
    holes to devour. These violent collisions also produce dust within the galaxies
    therefore embedding the black hole in a dusty envelope for a short period of
    time as it is being fed. Galaxies with hidden supermassive black holes tend to
    clump together in space more than the galaxies with exposed, or unobscured,
    black holes. The Herschel Space Observatory has shown galaxies with the most
    powerful, active black holes at their cores produce fewer stars than galaxies
    with less active black holes. The results are the first to demonstrate black
    holes suppressed galactic star formation when the universe was less than half
    its current age. Galaxies with massive black holes were found to have high
    rates of star formation, with some forming stars at a thousand times the rate
    of our own Milky Way galaxy today. But intriguingly, the Herschel results show
    that the fastest-growing black holes are in galaxies with very little star
    formation – once the radiation coming from close to the black hole exceeds a
    certain power, it tends to “switch off” star formation in its galaxy.

    Gas falling toward a black hole spirals inward and piles up into an accretion
    disk, where it becomes compressed and heated. Near the inner edge of the disk,
    on the threshold of the black hole’s event horizon — the point of no return –
    some of the material becomes accelerated and races outward as a pair of jets
    flowing in opposite directions along the black hole’s spin axis. These jets
    contain particles moving at nearly the speed of light, which produce gamma rays
    – the most extreme form of light — when they interact.

    The idea is that low-mass proto-galaxies with
    black holes at their center would have merged, creating a gravitational kick
    that would send the now larger black hole outward fast enough to escape the
    host dwarf galaxy, but not fast enough to leave the overall galactic halo. Hubblesite.org
    support: ISSUE=7909 PROJ=13

    A new type of combinational hydrodynamics over Bermuda
    triangle – 00132046

    hubblesite.org support: ISSUE=7955 PROJ=13

    hubblesite.org support: ISSUE=7972 PROJ=13

    http://blog.vixra.org

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Out There

Notes from the far edge of space, astronomy, and physics.

About Corey S. Powell

Corey S. Powell is DISCOVER's Editor at Large and former Editor in Chief. Previously he has sat on the board of editors of Scientific American, taught science journalism at NYU, and been fired from NASA. Corey is the author of "20 Ways the World Could End," one of the first doomsday manuals, and "God in the Equation," an examination of the spiritual impulse in modern cosmology. He lives in Brooklyn, under nearly starless skies.

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