For the Love of Physics

by Walter Lewin

  No neutron star has been detected to date in the remnant of SN 1987A. Some astrophysicists believe that a black hole was formed during core collapse after the initial formation of a neutron star. In 1990 I made a bet with Stan Woosley of the University of California, Santa Cruz; he is one of the world’s experts on supernovae. We made a bet whether or not a neutron star would be found within five years. I lost the hundred-dollar bet.

  There’s more that these remarkable phenomena produce. In the superhot furnace of the supernova, higher orders of nuclear fusion slam nuclei together to create elements far heavier than iron that end up in gas clouds that may eventually coalesce and collapse into new stars and planets. We humans and all animals are made of elements that were cooked in stars. Without these stellar kilns, and without these stunningly violent explosions, the first of which was the big bang itself, we would never have the richness of elements that you see in the periodic table. So maybe we can think of a core-collapse supernova as resembling a celestial forest fire (a small one, to be sure), that in burning out one star creates the conditions for the birth of new stars and planets.

  By any measure neutron stars are extreme objects. They are only a dozen miles across (smaller than some asteroids orbiting between Mars and Jupiter), about hundred thousand times smaller than the Sun, and thus about 300 billion (3 × 1014) times more dense than the average density of the Sun. A teaspoon of neutron star matter would weigh 100 million tons on Earth.

  One of the things I love about neutron stars is that simply saying or writing their name pulls together the two extremes of physics, the tiny and the immense, things so small we will never see them, in bodies so dense that they strain the capacity of our brains.

  Neutron stars rotate, some of them at astonishing rates, especially when they first come into being. Why? For the same reason that an ice skater spinning around with her arms out spins more rapidly when she pulls them in. Physicists describe this by saying that angular momentum is conserved. Explaining angular momentum in detail is a bit complicated, but the idea is simple to grasp.

  What does this have to do with neutron stars? Just this: Every object in the universe rotates. So the star that collapsed into the neutron star was rotating. It threw off most of its matter in the explosion but held on to one or two solar masses, now concentrated in an object a few thousand times smaller than the size of the core before collapse. Because angular momentum is conserved, neutron stars’ rotational frequency therefore has to go up by at least a factor of a million.

  The first two neutron stars discovered by Jocelyn Bell (see below) rotate about their axes in about 1.3 seconds. The neutron star in the Crab Nebula rotates about 30 times per second, while the fastest one that has been found so far rotates an astonishing 716 times per second! That means that the speed at the star’s equator is about 15 percent of the speed of light!

  The fact that all neutron stars rotate, and that many have substantial magnetic fields, gives rise to an important stellar phenomenon known as pulsars—short for “pulsating stars.” Pulsars are neutron stars that emit beams of radio waves from their magnetic poles, which are, as in the case of the Earth, noticeably different from the geographic poles—the points at the end of the axis around which the star rotates. The pulsar’s radio beam sweeps across the heavens as the star rotates. To an observer in the path of the beam, the star pulses at regular intervals, with the observer only seeing the beam for a brief moment. Astronomers sometimes call this the lighthouse effect, for obvious reasons. There are half a dozen known single neutron stars, not to be confused with neutron stars in binaries, which pulse over an extremely large range of the electromagnetic spectrum, including radio waves, visible light, X-rays, and gamma rays. The pulsar in the Crab Nebula is one of them.

  Jocelyn Bell discovered the first pulsar in 1967 when she was a graduate student in Cambridge, England. She and her supervisor, Antony Hewish, at first didn’t know what to make of the regularity of the pulsations, which lasted for only about 0.04 seconds and were about 1.3373 seconds apart (this is called the pulsar period). They initially called the pulsar LGM-1, for “Little Green Men,” hinting that the regular pulsations might have been the product of extraterrestrial life. A second LGM was soon discovered by Bell with a period of about 1.2 seconds, and it became clear that the pulses were not produced by extraterrestrial life—why would two completely different civilizations send signals to Earth with about the same period? Shortly after Bell and Hewish published their results, it was recognized by Thomas Gold at Cornell University that pulsars were rotating neutron stars.

  Black Holes

  I told you we’d get here. It is finally time to look directly at these bizarre objects. I understand why people might be afraid of them—if you spend a little time on YouTube, you’ll see dozens of “re-creations” of what black holes might look like, and most of them fall in the category of “death stars” or “star eaters.” In the popular imagination black holes are super-powerful cosmic sinkholes, destined to suck everything into their insatiable maws.

  But the notion that even a supermassive black hole swallows up everything in its vicinity is a complete fallacy. All kinds of objects, chiefly stars, will orbit a stellar mass black hole or even a supermassive black hole with great stability. Otherwise, our own Milky Way would have disappeared into the enormous 4-million-solar-mass black hole at its center.

  So what do we know about these strange beasts? A neutron star can only contain up to about 3 solar masses before the gravitational pull collapses it to form a black hole. If the original single nuclear-burning star was more massive than about twenty-five times the mass of the Sun, at core collapse the matter would continue to collapse rather than stopping at the neutron star stage. The result? A black hole.

  If black holes have companion stars in binary systems, we can measure their gravitational effect on their visible partners, and in some rare cases we can even determine their masses. (I talk about these systems in the next chapter.)

  Instead of a surface, a black hole has what astronomers call an event horizon, the spatial boundary at which the black hole’s gravitational power is so great that nothing, not even electromagnetic radiation, can escape the gravitational field. I realize this doesn’t make much sense, so try to imagine that the black hole is like a heavy ball resting in the middle of a rubber sheet. It causes the center to sag, right? If you don’t have a rubber sheet handy, try using an old stocking, or a pair of discarded pantyhose. Cut out as large a square as you can and put a stone in the middle. Then lift the square from the sides. You see immediately that the stone creates a funnel-like depression resembling a tornado spout. Well, you’ve just created a three-dimensional version of what happens in spacetime in four dimensions. Physicists call the depression a gravity well because it mimics the effect gravity has on spacetime. If you replace the stone with a larger rock, you’ll have made a deeper well, suggesting that a more massive object distorts spacetime even more.

  Because we can only think in three spatial dimensions, we can’t really visualize what it would mean for a massive star to make a funnel out of four-dimensional spacetime. It was Albert Einstein who taught us to think about gravity in this way, as the curvature of spacetime. Einstein converted gravity into a matter of geometry, though not the geometry you learned in high school.

  The pantyhose experiment is not ideal—I’m sure that will come as a relief to many of you—for a number of reasons, but the main one is that you can’t really imagine a marble in a stable orbit around a rock-generated gravity well. In real astronomical life, however, many objects achieve stable orbits around massive bodies for many millions, even billions of years. Think of our Moon orbiting the Earth, the Earth orbiting the Sun, and the Sun and another 100 billion stars orbiting in our own galaxy.

  On the other hand, the demonstration does help us visualize a black hole. We can, for instance, see that the more massive the object, the deeper the well and the steeper the sides, and thus the more energy it takes to climb
out of the well. Even electromagnetic radiation escaping from the gravity of a massive star has its energy reduced, which means its frequency decreases and its wavelengths become longer. You already know that we call a shift to the less energetic end of the electromagnetic spectrum a redshift. In the case of a compact star (massive and small), there is a redshift caused by gravity, which we call a gravitational redshift (which should not be confused with redshift due to Doppler shift—see chapter 2 and the next chapter).

  To escape from the surface of a planet or star, you need a minimum speed to make sure that you never fall back. We call this the escape velocity, which is about 11 kilometers per second (about 25,000 miles per hour) for the Earth. Therefore, all satellites bound to Earth can never have a speed larger than 11 kilometers per second. The higher the escape velocity, the higher the energy needed to escape, since this depends both on the escape velocity and on the mass, m, of the objects that want to escape (the required kinetic energy is 1/2 mv2).

  Perhaps you can imagine that if the gravity well becomes very, very deep, the escape velocity from the bottom of the well could become greater than the speed of light. Since this is not possible, it means that nothing can escape that very deep gravity well, not even electromagnetic radiation.

  A physicist named Karl Schwarzschild solved Einstein’s equations of general relativity and calculated what the radius of a sphere with a given mass would be that would create a well so deep that nothing could escape it—a black hole. That radius is known as the Schwarzschild radius, and its size depends on the mass of the object. This is the radius of what we call the event horizon.

  The equation itself is breathtakingly simple, but it is only valid for nonrotating black holes, often referred to as Schwarzschild black holes.* The equation involves well-known constants and the radius works out to just a little bit less than 3 kilometers per solar mass. That’s how we can calculate the size—that is to say, the radius of the event horizon—of a black hole of, for example, 10 solar masses, is about 30 kilometers. We could also calculate the radius of the event horizon of a black hole with the mass of the Earth—it would be a little less than 1 centimeter—but there’s no evidence that such black holes exist. So if the mass of our Sun were concentrated into a sphere about 6 kilometers across, would it be like a neutron star? No—under the gravitational attraction of that much mass packed into such a small sphere, the Sun’s matter would have collapsed into a black hole.

  Long before Einstein, in 1748, the English philosopher and geologist John Michell showed that there could be stars whose gravitational pull is so great that light could not escape. He used simple Newtonian mechanics (any of my freshmen can do this now in thirty seconds) and he ended up with the same result as Schwarzschild: if a star has a mass N times the mass of our Sun, and if its radius is less than 3N kilometers, light cannot escape. It is a remarkable coincidence that Einstein’s theory of general relativity gives the same result as a simple Newtonian approach.

  At the center of the spherical event horizon lies what physicists call a singularity, a point with zero volume and infinite density, something bizarre that only represents the solution to equations, not anything we can grasp. What a singularity is really like, no one has any idea, despite some fantasizing. There is no physics (yet) that can handle singularities.

  All over the web you can see animated videos of black holes, most of them at once beautiful and menacing, but nearly all immense beyond belief, hinting at destruction on a cosmic scale. So when journalists began writing about the possibility that the world’s largest accelerator, CERN’s Large Hadron Collider (LHC), near Geneva, might be able to create a black hole, they managed to stir up a good deal of concern among nonscientists that these physicists were rolling dice with the future of the planet.

  But were they really? Suppose they had accidentally created a black hole—would it have started eating up the Earth? We can figure this out fairly easily. The energy level at which opposing proton beams collided in the LHC on March 30, 2010, was 7 teraelectron volts (TeV), 7 trillion electron volts, 3.5 trillion per beam. Ultimately, the LHC scientists plan to reach collisions of 14 TeV, far beyond anything possible today. The mass of a proton is about 1.6 × 10–24 grams. Physicists often say that the mass, m, of a proton is about 1 billion electron volts, 1 GeV. Of course, GeV is energy and not mass, but since E = mc2 (c being the speed of light), E is often referred to as “the mass.” On the Massachusetts Turnpike there are signs: “Call 511 for Travel Information.” Every time I see one I think about electrons, as an electron’s mass is 511 keV.

  Assuming that all the energy of the 14 TeV collision went into creating a black hole, it would have a mass of about 14,000 times that of a proton, or about 2 × 10–20 grams. Boatloads of physicists and review committees evaluated a mountain of literature on the question, published their results, and concluded that there was simply nothing to worry about. You want to know why, right? Fair enough. OK, here’s how the arguments go.

  First, scenarios in which the LHC would have enough energy to create such tiny black holes (known as micro black holes) depend on the theory of something called large extra dimensions, which remains highly speculative, to say the least. The theory goes well beyond anything that’s been experimentally confirmed. So the likelihood even of creating micro black holes is, to begin with, exceptionally slim.

  Clearly, the concern would be that these micro black holes would somehow be stable “accretors”—objects that could gather matter, pull it into themselves, and grow—and start gobbling up nearby matter and, eventually, the Earth. But if there were such things as stable micro black holes, they would already have been created by enormously energetic cosmic rays (which do exist) smacking into neutron stars and white dwarfs—where they would have taken up residence. And since white dwarfs and neutron stars appear stable on a time scale of hundreds of millions, if not billions of years, there don’t seem to be any tiny black holes eating them up from within. In other words, stable micro black holes appear to pose zero threat.

  On the other hand, without the theory of extra dimensions, black holes with a mass smaller than 2 × 10–5 grams (called the Planck mass) could not even be created. That is to say, there is no physics (yet) that can deal with black holes of such small mass; we would need a theory of quantum gravity, which doesn’t exist. Thus the question of what the Schwarzschild radius would be for a 2 × 10–20 gram micro black hole is also meaningless.

  Stephen Hawking has shown that black holes can evaporate. The lower the mass of a black hole, the faster it will evaporate. A black hole of 30 solar masses would evaporate in about 1071 years. A supermassive black hole of 1 billion solar masses would last about 1093 years! So you may ask, how long would it take for a micro black hole of mass 2 × 10–20 grams to evaporate? It’s an excellent question, but no one knows the answer—Hawking’s theory does not work in the domain of black hole masses lower than the Planck mass. But, just for curiosity’s sake, the lifetime of a black hole of 2 × 10–5 grams is about 10–39 seconds. So it seems that they evaporate faster than the time it takes to produce them. In other words, they cannot even be produced.

  It clearly seems unnecessary to worry about possible 2 × 10–20 gram LHC micro black holes.

  I realize that this didn’t stop people from suing to prevent the LHC from starting operations. It makes me worry, however, about the distance between scientists and the rest of humanity and what a lousy job we scientists have done of explaining what we do. Even when some of the best physicists in the world studied the issue and explained why it wouldn’t pose any problems, journalists and politicians invented scenarios and fanned public fears on the basis of almost nothing. Science fiction at some level appears more powerful than science.

  There’s nothing more bizarre than a black hole, I think. At least a neutron star makes itself known by its surface. A neutron star says, in a way, “Here I am, and I can show you that I have a surface.” A black hole has no surface and emits nothing at all (apart fro
m Hawking radiation, which has never been observed).

  Why some black holes, surrounded by a flattish ring of matter known as an accretion disk (see the next chapter), shoot out extremely high energy jets of particles perpendicular to the plane of the accretion disk, though not from inside the event horizon, is one of the great unsolved mysteries. Take a look at this image: www.wired.com/wiredscience/2009/01/spectacular-new/.

  Everything about the interior of a black hole, inside the event horizon, we have to derive mathematically. After all, nothing can come out, so we receive no information from inside the black hole—what some physicists with a sense of humor call “cosmic censorship.” The black hole is hidden inside its own cave. Once you fall through the event horizon, you can never get out—you can’t even send a signal out. If you’ve fallen through the event horizon of a supermassive black hole, you wouldn’t even know that you’ve passed the event horizon. It doesn’t have a ditch, or a wall, or a ledge you need to walk over. Nothing in your local environment changes abruptly when you cross the horizon. Despite all the relativistic physics involved, if you are looking at your wristwatch you wouldn’t see it stop, or appear to go faster or slower.

  For someone watching you from a distance, the situation is very different. What they see is not you; their eyes are receiving the images of you carried by light that leaves your body and climbs its way out of the black hole’s gravity well. As you get closer and closer to the horizon, the well gets deeper and deeper. Light has to expend more of its energy climbing out of the well, and experiences more and more gravitational redshift. All emitted electromagnetic radiation shifts to longer and longer wavelengths (lower frequencies). You would look redder and redder, and then you would disappear as your emissions would move into longer and longer wavelengths, such as infrared light and then longer and longer radio waves and all wavelengths would become infinity as you cross the event horizon. So even before you crossed the threshold, to the distant observer you would have effectively disappeared.