One day the sun will fail. The fuel powering its nuclear fusion will run out, the sky will grow cold and, if Earth survives at all, humankind will be plunged into perpetual winter. To stay alive, our descendants will need to make alternative arrangements. They will first exhaust the resources of Earth, then the solar system, and eventually all the stars in all the galaxies in the visible universe. With nothing left to burn, they will surely cast their gaze on the only remaining store of energy: black holes. Might they be able to harvest this energy and save civilization?
I'm here with some bad news. The plan is not going to work. The reasons come down to the physics of such exotic entities as quantum strings and that venerable science-fiction favorite: the space elevator.
False Hope
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On the face of it, extracting energy (or indeed anything at all) from a black hole sounds impossible. Black holes, after all, are shrouded by an “event horizon,” a sphere of no return where the gravitational field becomes infinite. Anything that strays inside this sphere is doomed. Hence, a wrecking ball intended to demolish a hole and release its energy would itself be wrecked, swallowed by the black hole, along with its unfortunate operator. A bomb tossed into the hole, far from destroying it, would merely enlarge it—by an amount equal to the mass of the bomb. What goes into a black hole never comes out: not asteroids, not rockets, not even light.
Or so we used to think. But then, in what is to me the most shocking and delightful physics paper ever written, in 1974 Stephen Hawking showed that we were wrong. Building on earlier ideas of Jacob D. Bekenstein, now at the Hebrew University of Jerusalem, Hawking showed that black holes leak small amounts of radiation. You are still going to die if you fall in, but although you yourself will never make it out, your energy eventually will. This is good news for would-be black hole miners: energy can escape.
The reason that energy escapes lies in the shadowy world of quantum mechanics. One of the signature phenomena of quantum physics is that it allows particles to tunnel through otherwise impassable obstacles. A particle rolling toward a tall barrier sometimes appears on the other side. Do not try this at home—fling yourself at a wall, and you are unlikely to rematerialize unscathed on the other side. But microscopic particles tunnel more readily.
Quantum tunneling is what permits an alpha particle (a helium nucleus) to escape the clutches of a radioactive uranium nucleus, and quantum tunneling is what permits “Hawking radiation” to leak from a black hole. Particles escape the event horizon's infinite gravitational field not by blasting past but by tunneling through. (No one has ever seen a black hole leak, of course. But it is such a compelling mathematical consequence of applying quantum mechanics to curved spacetime that no one doubts it.)
Because black holes leak, we may hope to feast off their energy. But the devil is in the details. No matter how we try to extract this energy, we will see that we encounter problems.
One simple approach would be to just wait. After enough time, the black hole should disgorge its energy, photon by photon, back into the universe and into our waiting hands. With each bit of energy lost, the black hole shrinks, until eventually it dwindles away to nothing. In that sense, a black hole is like a delicious cup of coffee whose surface you are forbidden to touch on pain of gravitational dismemberment. There is still a way to consume the cataclysmic coffee: wait for it to evaporate and breathe in the fumes.
There is a catch. Although waiting is simple, it is also achingly slow. Black holes are extremely dim—one with the mass of the sun glows at 60 nanokelvins; until the 1980s we did not even know how to make something that cold in a laboratory. To evaporate a solar-mass black hole takes 1057 times the current age of the universe, a stupendously long time. In general, the lifetime of a black hole is the cube of its mass, m
3. Our shivering descendants will be motivated to speed things up.
Initial cause for optimism on their behalf is that not every Hawking particle that escapes the event horizon goes on to escape to infinity. In fact, practically none of them do. Almost every particle that tunnels past the event horizon is later recaptured by the gravitational field and reclaimed by the black hole. If we could somehow pry these photons from the black hole's grasp, rescuing them after they have escaped the horizon but before they were recaptured, then maybe we could gather the energy of black holes faster.
To understand how we can liberate these photons, we must first investigate the extreme forces at work near a black hole. The reason most particles get recaptured is that they are not emitted straight out. Imagine shining a laser from just outside the horizon. You must aim directly overhead for the light to escape; the closer you are to the horizon, the more carefully you have to aim. The gravitational field is so strong that even if you are just a little bit off vertical, the light will circle around and fall back in.
It might seem strange that rotational velocity can hurt a particle's escape prospects. After all, it is precisely orbital velocity that keeps the International Space Station aloft—it provides the centrifugal repulsion that counteracts gravity. Get too close to a black hole, however, and the situation reverses—rotational velocity impedes escape. This effect is a consequence of general relativity, which states that all mass and energy are subject to gravity—not just an object's rest mass but also its orbital kinetic energy. Close to a black hole (more precisely, within one and a half times the event horizon radius), the gravitational attraction of the orbital kinetic energy is stronger than the centrifugal repulsion. Inside this radius, more angular velocity makes particles fall faster.
This effect means that if you slowly rappel down toward a black hole horizon, you will soon become very hot. You will be bathed not just by the photons that would have escaped to infinity as Hawking radiation but also by those that would never have made it. The black hole has a “thermal atmosphere”; the closer you get to the event horizon, the hotter it gets. This heat carries energy.
The fact that energy is stored outside of the event horizon has given rise to the clever proposal that we can “mine” a black hole by reaching in, grabbing the thermal atmosphere and carting it off. Dangle a box close to, but not over, a black hole horizon, fill the box with hot gas, and then drag it out. Some of the contents would have made it out unaided, as conventional Hawking radiation, but most of the gas, had we not intervened, was destined to fall back in. (Once the gas is out of the near vicinity of the event horizon, transporting it the rest of the way to Earth is relatively easy: simply pack it onto a rocket and fly it home or convert the gas into a laser and beam it back.)
This strategy is like blowing on our delicious but dangerous coffee. Unassisted, most of the water vapor emitted falls back in, but blowing across the surface removes the freshly escaped vapor before it has a chance to be recaptured. The conjecture is that by stripping a black hole's thermal atmosphere, we can rapidly devour the hole in an amount of time that scales not like the m
3 needed for evaporation but like the considerably faster m.
In recent work, however, I showed that this conjecture is false. The problem does not come from any elevated musing on quantum mechanics or on quantum gravity. Instead it arises from the most unsophisticated of considerations: you cannot find a strong enough rope. To mine the thermal atmosphere, you need to be able to dangle a rope near a black hole—you need to create a space elevator. But, I discovered, constructing an effective space elevator near a black hole is impossible.
Elevator to the Sky
A space elevator (sometimes known as a sky hook) is a futuristic structure, made famous by science-fiction author Arthur C. Clarke in his 1979 novel The Fountains of Paradise. He imagined a rope that dangles from outer space down toward the surface of Earth. It is held up not with a push from below (as in a skyscraper, where each floor supports the floors above) but with a pull from above (each segment of rope supports the one below). The far end of the rope is moored to a huge, slowly orbiting mass way out beyond geostationary orbit that tugs the rope outward, keeping the whole thing aloft. The near end of the rope dangles down to just above the planet's surface, where it stops—the balancing of the various forces ensures that it just floats there, as if by magic (magic, Clarke once remarked, being indistinguishable from sufficiently advanced technology).
The point of this advanced technology is that with the rope in place, putting cargo in orbit becomes much easier. No longer do we need the danger, inefficiency or waste of a rocket, which, for the first part of its journey, mainly lifts fuel. Instead we would attach to the rope an electrically powered elevator. Once the marginal cost of moving cargo to low-Earth orbit is just the cost of the electricity, getting a kilogram into space drops from the tens of thousands of dollars that the Space Shuttle charged to a couple of bucks—a journey to space for less than a subway ride.
The technological obstacles to building a space elevator are formidable, and the greatest of them is finding a suitable material for the rope. The ideal material needs to be both strong and light—strong so that it does not stretch or break under the strain and light so that it does not unduly burden the rope above.
Steel is not strong enough, not even nearly. In addition to the weight of everything beneath, a segment of steel must also bear its own weight, so the cable must get thicker and thicker the higher up you go. Because steel is so heavy, compared with its strength, near Earth the cable must double in thickness every few kilometers. Long before it reaches the geostationary point, it has become impractically thick. Building a space elevator around Earth with 19th-century building materials just will not work. But 21st-century materials are already showing promise. Carbon nanotubes, long ribbons of carbon arranged in a hexagonal honeycomb lattice, are 1,000 times stronger than steel. Carbon nanotubes are excellent candidates to build an extraterrestrial space elevator.
It would cost many billions of dollars, be by far the largest megaproject ever undertaken, require figuring out how to spin the nanotubes into threads tens of thousands of kilometers long, and face many other obstacles besides. But for a theoretical physicist like me, once you have decided that a proposed structure does not actually violate the known laws of physics, everything else is just engineering. (By this measure, the problem of building a fusion power plant is also “solved,” even though there is a conspicuous absence of fusion power plants fueling our civilization, with the commendable exception of the sun.)
Black Hole Elevator
Around a black hole, of course, the problem is much harder. The gravitational field is more intense, and what works around Earth is pathetically inadequate for the task.
It is possible to show that even using the much vaunted strength of carbon nanotubes, a hypothetical space elevator that reached down close to a black hole horizon would either have to be so thin near the black hole that a single Hawking photon would break it or so thick far out from the black hole that the rope itself would collapse under its own gravity and become a black hole of its own.
These limitations rule out carbon nanotubes. But just as the Iron Age followed the Bronze Age, and just as carbon nanotubes will some day follow steel, so, too, might we expect that materials scientists will invent stronger and stronger and lighter and lighter materials. And so they might. But the progress cannot go on indefinitely. There is a limit to progress, a limit to engineering, a limit to the tensile-strength-to-weight ratio of any material—a limit imposed by the laws of nature themselves. This limit is a surprising consequence of Albert Einstein's famous formula E = mc
2.
The tension in a rope tells you how much energy you must expend to make it longer: the tenser the rope, the more energy it costs to lengthen. An elastic band has tension because to make it longer you must spend energy rearranging its molecules: when the molecules are easy (energetically cheap) to rearrange, the tension is small; when the molecules are expensive to rearrange, the tension is large. But rather than just rearranging bits of the existing rope, we could always just manufacture a whole new section of rope and stick it on the end. The energy cost to extend a rope in this way is equal to the energy contained in the mass of the new rope segment and is given by the formula E = mc
2—the mass (m) of the new segment of rope times the speed of light squared (c2).
This is a very energy-expensive way to lengthen a rope, but it is also a fail-safe way. It provides an upper limit on the energy cost of extending a rope and thus a limit on the tension of a rope. The tension can never be greater than the mass per unit length times c
2. (You might think that two ropes woven together would be twice as strong as one. But they are also twice as heavy and so will not improve the strength-to-weight ratio.)
This fundamental limit on the strength of materials leaves a lot of room for technological progress. This limit is hundreds of billions of times stronger than steel and, pound for pound, still hundreds of millions of times stronger than carbon nanotubes. Still, it means we cannot improve our materials indefinitely. Just as our efforts to propel ourselves ever faster must end at the speed of light, so our efforts to build stronger materials must end at E = mc
2.
There is a hypothetical rope material that precisely reaches the limit—that is as strong as any material can be. This material has never been seen in a lab, and some physicists doubt whether it even exists, but others have devoted their life to its study. The strongest rope in nature may never have been seen, yet it already has a name: a string. Those who study strings—string theorists—hope they are the fundamental constituents of matter. For our purposes, what matters is not their fundamentalness but their strength.
Strings are strong. A section of rope made of strings the same length and weight as a shoelace can suspend Mount Everest. Because the toughest engineering challenges call for the toughest materials, if we want to build a space elevator around a black hole our best shot is to use strings; where nanotubes failed, perhaps fundamental strings will succeed. If anything can do it, strings can; conversely, if strings cannot, black holes are safe.
It turns out that while strings are strong, they are not quite strong enough. Rather they are tantalizingly on the edge of being strong enough. Any stronger, and it would be easy to construct a space elevator even around a black hole; any weaker, and the project would be hopeless—the string itself would break under its own weight. Strings are exactly marginal in that whereas a rope that is made of strings dangling toward the surface of a black hole does have just enough strength to support its own weight, it has none left over to support the elevator's cargo. The rope supports itself but only at the expense of dropping the box.
This, then, is what shields black holes from prying. The laws of nature themselves limit our building materials, which means that although a rope can reach the dense thermal atmosphere of a black hole, it cannot expeditiously plunder it. Because the strength of a string is borderline, we would be able to extract a limited amount of energy from the rarified upper atmosphere using a shorter rope.
But this thin and insubstantial diet is not much better than just waiting: the lifetime of the black hole still scales like m
3, the same as the unaided evaporation lifetime. By poaching the occasional photon here and there, we may be able to shorten the lifetime of a black hole by some small factor, but we will not achieve the kind of industrial extraction required to feed a hungry civilization.
In this particular, the finite speed of light is our constant enemy. Because we cannot travel faster than light, we cannot escape a black hole's event horizon. Because we cannot extract more than mc
2 worth of energy from our fuel, we are destined to cast our gaze on black holes. And because a rope can never be stronger than the speed of light squared times its mass per unit length, we will not be able to feast off the hole's contents.
With the sun gone, we will be living in perpetual winter. We may look to the great trove of energy in the thermal atmosphere of a black hole, but we will grasp for it at our peril. Reach too eagerly or too deep, and rather than our box robbing the black hole of its radiation, the black hole will instead rob us of our box. It is going to be a cold winter.