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January/February 2015

Science
Oh, you're a science fiction reader. You know the drill. So there you are, boldly going where no one has gone before, when you realize that your ship is falling. That is—the inertial navigation unit on your spacecraft informs you that you are accelerating, as if caught in a very strong gravitational field. Yet you can see nothing that would be exerting such a field. Being a resourceful explorer, you turn on your ship's radar, which sends out a microwave pulse. If there were an object exerting the pull that your ship is experiencing, you figure the pulse would reflect and your instruments would detect the reflected pulse. But that doesn't happen. Nothing comes back. Your instruments do detect a slight increase in the number of Xrays coming from the place you are falling toward. And the stars in that area of the sky appear to be shifting their position as you fall. You consult the ship's computer. In a preternaturally calm female voice, the computer makes your predicament clear. You are falling into a black hole. Bad news for you.
The computer explains, in its usual tone of competent superiority, that you can't see the black hole because light can't escape it. Light, in this case, includes visible light and all other wavelengths in the electromagnetic spectrum. And light is what carries information to your eyes and your instruments. So you see nothing where the black hole is. The computer informs you that the Xrays you noticed were produced by gas and dust as it was falling into the black hole. And the stars appear to be moving because the starlight is bent by the gravity of the black hole on its way to your eyes. Just as a bonus, the computer gives you a little history lesson about that bending starlight. Back in 1915, Einstein's general theory of relativity predicted the bending of light by gravity. And in 1919, Sir Arthur Eddington tested this prediction by experiment—he photographed stars behind the Sun during a total solar eclipse and found that the light from those stars bent as it passed around the Sun. When Eddington measured the bending, it was consistent with Einstein's prediction, twice as much as would be expected from Newton's prediction. The gravity of a normal star like the Sun can warp the space around it. A black hole is no normal star and it can warp space to the max, distorting the appearance of the star field behind it.
That's all great to know. But as your spacecraft continues to fall toward the black hole, we figure you might be more interested in finding out how to escape your predicament. This would be a good time to discuss "escape velocity." We all know the classic phrase, "What goes up must come down." As a science fiction reader, you probably realize that this classic phrase is incorrect. Something that goes up from the surface of the Earth doesn't necessarily come down. If something goes up fast enough, it will reach escape velocity, the speed at which it can escape the Earth. Viewed from the perspective of someone watching a launch, for instance, a rocket goes up and never comes down. For every object that exerts a gravitational pull, there is an escape velocity that could overcome that pull. Maybe you can escape the gravitational clutches of the black hole by blasting your spacecraft's engines and reaching escape velocity. That would involve calculating escape velocity, of course (a task that some might prefer to delegate to the ship's computer). For the mathematically inclined, we'll note that the equation for escape velocity from the surface of a planet is v = (2GM/R)^{0.5}. In this equation, v is escape velocity, M is the planet's mass, G is the gravitational constant, and R is the planet's radius. The larger a planet's mass, the greater the escape velocity. And the smaller a planet's radius, the greater the escape velocity. In short, the smaller and more massive a planet is, the faster you'll have to move to escape. From the surface of the Earth, escape velocity is about eleven km/s (or seven miles per second), about thirtythree times the speed of sound at the surface of the Earth. To put the escape velocity from Earth in perspective, consider that the escape velocity from Comet ChuryumovGerasimenko, which is two trillion times less massive than the Earth and has a radius three thousand times smaller, is a mere 0.5 m/s, about one mph. You could easily escape the surface of this comet with a simple jump. Compare that to the velocity needed to escape the surface of the Sun, which has a mass 300,000 times that of Earth. Though the Sun has a large radius, it also has a high escape velocity—617 km/s, 56 times greater than the escape velocity from the surface of Earth. If you are close enough to an object to experience its gravitational pull, even if you are above the surface, you can calculate the escape velocity required to escape that pull. You just put your distance from the center of the object in place of the radius in the equation. The escape velocity from the Sun starting at the distance of Earth's orbit, for example, is fortytwo km/s. To leave our solar system, a spaceship has to achieve this speed. If you were as far away from a solarmass black hole as the Earth is from the Sun, your escape velocity would be exactly the same as the escape velocity at that distance from the Sun itself, fortytwo km/s. At large distances from a black hole, the laws of gravity discovered by Isaac Newton work very well. And at that distance, current rocket technology is adequate to the task of escaping a black hole. But if you were closer or if the black hole were more massive, the situation would change. Your ability to escape the black hole will depend on how fast your spacecraft can go, how massive the black hole is, and how far you are from it. More on all that in a bit, but first some more information that you might find useful.
Since you are a science fiction reader, we are confident that you remember the fastest speed an object can attain. That would be the speed of light—abbreviated as c. The speed of light is about 300,000 kilometers per second—500 times the escape velocity from the surface of the Sun. The next question that naturally comes to mind (at least to our minds, twisted as they are) is this: what happens when an object has a mass large enough and a radius small enough to have an escape velocity equal to the speed of light? Obviously, light couldn't escape such an object. And since nothing can travel faster than light, nothing else can escape either. And that, of course, is why a black hole is black. Light can't escape from its "surface." Light falls into it and doesn't come out. That's what happened to the pulse of microwaves that your ship sent out. The pulse went into the black hole and never bounced back. We've been talking about the surface of the black hole as if it were something as solid as the surface of the Earth. It isn't. The surface of a black hole is defined as the radius from which light cannot escape. Known as the Schwarzschild radius, this distance also marks the black hole's "event horizon." As we said earlier, nothing can leave the surface of a black hole—that is, nothing can pass the Schwarzschild radius. Since information can't travel faster than the speed of light, it can't escape from inside the Schwarzschild radius. For an experimental physicist like Paul, this is a disaster. The inside of a black hole is an experimentalist's worst nightmare. It's impossible to perform experiments inside a black hole and then get the information out of the black hole. (To Pat, on the other hand, this is a gift. Since experimental physicists are locked out, the inside of a black hole is fair game for fictional experiments of all kinds.) Calculating the Schwarzschild radius isn't all that difficult, if you're mathematically inclined. (If you're not, you are free to skip this paragraph.) All you need to do is turn the equation for escape velocity around. You plug in the speed of light (c) for the escape velocity and solve for the radius, R of a given mass, M. The solution is R = 2GM/c^{2}, where G is the universal gravitational constant. Suppose we solve for the Schwarzschild radius of the Sun. Right now, the Sun has a radius of 700,000 km. To become a black hole, the Sun's mass would have to be squeezed down to fit inside a radius of about three km or two miles, shrinking by a factor of 200,000 or so. The density of the resulting solarmass black hole would be greater than that of an atomic nucleus. We are talking extremely dense. A piece of this material the size of the last joint of your little finger would weigh billions of tons. Closer to home, we can also solve for the Schwarzschild radius of the Earth. To become a black hole, the entire planet would have to shrink to a sphere less than one cm in radius—about the size of a large marble, and smaller than a human eyeball.
Suppose we have a black hole with the mass of the Sun—squeezed down to a sphere just three km in radius. Using the Newtonian inverse square law of gravity, we can estimate how strong the gravity would be near the event horizon. The black hole would have a gravitational force that's four trillion (or 4 x 10^{12}) times greater than the force of gravity at the surface of the Earth. (Actually using general relativity produces an even stronger gravitational acceleration at the event horizon. But let's just stick to our Newtonian estimate.) If you are a savvy science fiction reader, you may be thinking that you'll be fine as long as you continue to fall into the black hole. After all, when you're falling toward Earth, you are in free fall. You don't feel the gravity. And the fall isn't the problem. It's the abrupt stop at the end of the fall. If there's no hard stop, maybe you'll be okay. That's a lovely and optimistic thought. But you are failing to take into account the effect of the tidal forces caused by that gravity. Let's back up and talk about tides you may have seen. If you've ever spent the day at the beach, you've probably watched the tide come in or go out. As the tide comes in, you have to move your beach towel away from the waves. The ocean waters rise in response to the gravitational pull of the Moon. So here's what's going on: The Moon pulls more strongly on stuff that's on the side of the Earth that's nearer the Moon. This gravitational pull drags the ocean toward the moon, making a tidal bulge. But that's not all. The Moon also pulls more strongly on the center of the Earth than it does on the stuff on the side of the Earth that's farthest from it. This pulls the Earth away from the ocean water on the other side of the planet, creating a second tidal bulge. These two bulges cause the gentle rise and fall of ocean waters that you observe from your beach towel. Now consider the tidal forces on your body as you approach a black hole. In this drama, the black hole is playing the role of the Moon and you are the Earth. For simplicity's sake (and dramatic effect), we'll describe the effect on your body, and let you imagine the effect on the rest of your spacecraft. Let's say you are falling into the black hole feet first. The pull of gravity on your feet will be greater—much, much greater—than the pull on your head. Your head and your feet will both become tidal bulges as the pull causes your body to lengthen into a long strand and you suffer what has been aptly named spaghettification. Such a vividly unpleasant name. You cannot resist spaghettification. When you are 100 km away from a black hole with the mass of the Sun, the difference in gravitational pull between your feet and your head will be 50,000 gravities. When you get six km from a solarmass black hole, the tidal force between your head and your feet will pull you apart with 150 million gravities. Each of your feet would be pulled away from your body with a force of over 100,000 tons. As the tidal forces pull your head and feet in opposite directions, your middle gets squeezed together, the ultimate irresistible corsetry. Gas and dust falling into the black hole are squeezed together in just this way—and that's the source of the Xrays you detected. As the squeezed gas falls, it collides with other dust and gas. These collisions heat the gas and dust to temperatures of millions of degrees Kelvin (although when we talk millions of degrees, the difference between Celsius and Kelvin is negligible). These hot gases give off radiation in the Xray region of the spectrum. This is how astronomers spot black holes. The black holes themselves are invisible, but the gas falling into them is bright when viewed in Xrays.
Perhaps, as you think about being stretched and squeezed, you have come up with the same question that Paul's students sometimes ask when he teaches about black holes: can an object be lowered on a wire into a black hole and then pulled back out? Or, to make the question a bit more personal, what if a ship that's farther away from the black hole could toss you a lifeline—some sort of incredibly strong wire cable. Could that ship haul your stretched and squeezed remains back out of the black hole? Sorry to say that the answer is no. Forces between atoms that hold the wire together are carried by the electromagnetic force, and changes in the electromagnetic force travel at the speed of light. The increase in the gravitational force on an atom inside the event horizon will need a corresponding increase in the electrical force, but the information that the force has increased will never reach the atom in the wire just outside the event horizon, and so the wire will break. No solid object can resist the forces involved in crossing the event horizon. No material is strong enough to pull something back out of a black hole. By now, we are confident that we have convinced you that you really would rather not fall into the black hole. So you'll be turning on your rockets in an attempt to escape—or at least shift your fall into a circular orbit about the black hole. Orbiting the black hole could save you—if you've acted quickly enough. You'd better blast your jets before you reach a distance from the center of the black hole that is three times the Schwarzschild radius. Once you get closer than this distance, there is no stable circular orbit. This close to the black hole we must switch from Newtonian mechanics (which allows circular orbits) to general relativity (which does not). So if you've ventured too close, there is no stable orbit possible. You have to fire your rockets continuously or you are doomed to spiral into the black hole.
If you and your spaceship do end up falling into the black hole, you will help it grow. Maybe, being the philosophical type, you wonder how much the contribution of your mass will increase its size. The answer to that question, like so many other answers related to black holes, is a little weird. If you have a threekm radius solarmass black hole and collide it with a second threekm radius black hole, what will you get? When the holes combine, the mass doubles. According to our calculation of the Schwarzchild radius, that means the radius also doubles, and you get a sixkm radius black hole. This seems fine. After all, 1+1 = 2. Until you think about it more deeply. Consider how this works with balls of clay. Take two balls of clay, each one centimeter in diameter. Squeeze them together to make a new, larger ball of clay. The diameter of this new ball will NOT be two cm. Instead, it will be about 1.25 cm (the cube root of two). This is because the clay is made from atoms that are in contact with each other and are incompressible. This means that clay has a constant density. And if you combine two objects with constant density, their total volume is the sum of each of their volumes. And to double the volume, the radius only has to increase by the cube root of two. Now you see how weird black holes really are. When you add two of them together, the mass doubles and the radius doubles. That means the black hole becomes twice as thick, twice as wide, and twice as high. Rather than simply doubling in volume (like the balls of clay), the volume increases by a factor of eight (that's 2x2x2). The density is the mass divided by the volume. So when two threekilometer black holes collide, the density of the combined black holes is less than the density of the original black holes. Double the mass and multiply the volume by eight and the density decreases to 2/8, or onequarter of the original density. As black holes gobble up gas, dust, stars, and each other (and any passing spacecraft), they can grow to become supermassive black holes. And each time a black hole adds mass, it becomes less dense. So a supermassive black hole is supermassive, but it's also super big and has a very low density. A supermassive black hole can actually be less dense than water. Theoretically, it would float in water, except of course the water would be dragged through the event horizon and added to the black hole itself. The largest supermassive black holes found to date have masses over 10 billion times the mass of the Sun. That's an entire galaxy of mass inside the event horizon. But the average density of such a supermassive black hole is around 10^{9} Kg/m^{3}. That's a trillionth the density of water and a billionth the density or air. In fact, it's a density that would be considered an excellent vacuum in any Earthbased laboratory. At the center of our Milky Way galaxy, astronomers have found a supermassive black hole called Sagittarius A*. (No, that asterisk isn't a footnote; it's part of the name, which is read as Sagittarius Astar.) We know about this supermassive black hole because astronomers can see stars orbiting the center of the galaxy. In fact they have even traced one star, known as S2, through its entire fifteenyear orbit about the black hole. Applying Kepler's laws to the orbit of this star, they calculated the mass of this black hole to be four million times the mass of the Sun. This means it has a Schwarzschild radius of four million times the three km radius of a solarmass black hole, or twelve million km. That's about seventeen times the radius of the Sun. (Paul wonders what Kepler would think about applying his discoveries to find the mass of a supermassive black hole at the center of the Milky Way Galaxy.) Right now, scientists are waiting for a gas cloud named G2 with at least three Earth masses of material, to make a close approach to the black hole. The point of closest approach to the Earth is called the perigee, to the Sun the perihelion, and to a black hole the perinigricon. Scientists are watching this gas cloud closely to see what happens as it approaches perinigricon, which it is predicted to do in 2014.
Now, we don't really recommend falling into a black hole. But if you must fall into a black hole and you have a choice about which one, we recommend you choose to fall into a supermassive black hole. Because of their low density and large radius, supermassive black holes have reduced tidal forces at the event horizon. In fact, the tidal forces at the Schwarzschild radius of a black hole with ten million solar masses is about the same as the tidal force on a person in free fall near the surface of the Earth. So you could fall into a supermassive black hole without being spaghettified. (That doesn't mean you get off easy—the local Xray dose might fry you. But given the choice, we'd prefer to avoid spaghettification.) You might be able to fall through the Schwarzschild radius of a supermassive black hole and see for yourself what is inside. The only problem is you will never be able to send information back to the world outside the black hole. You will not be able to publish your findings, and truthfully you are very likely to perish inside the black hole. Unfortunately, yours would be an unfinished science fiction story. Those of us outside will never know your fate for sure.
Paul Doherty works at The Exploratorium, San Francisco's museum of science, art, and human perception—where science and science fiction meet. For more on Paul's work and his latest adventures, visit www.exo.net/~pauld. Pat Murphy is a science writer, a science fiction writer, and occasionally a trouble maker. You can learn more about what she's up to at www.brazenhussies.net/Murphy.
 

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