To measure something, you need to prod it. For instance, imagine that you are measuring the length of a pencil. You could run your fingers along it and measure how long it is; however, you’ll probably give the pencil a nudge, disturbing the pencil’s velocity slightly. A better way would be to place a ruler gently next to the pencil, but in fact, comparing the lengths of the two objects also changes the pencil’s speed a tiny bit. You can only look at the pencil when light is bouncing off it; though the disturbance is very slight, the photons that carom off the pencil nudge it ever so gently, changing the pencil’s velocity a tiny bit. No matter what way you think of to measure the pencil, you will give it a tiny nudge in the process. Heisenberg’s uncertainty principle shows that there is no possible way to measure the pencil’s length—or an electron’s position—and its velocity with perfect accuracy at the same time. In fact, the better you know a particle’s position, the less you know about its velocity, and vice versa. If you measure an electron’s position with zero error—you know exactly where it is at a given moment—you must have zero information about how fast it is going. And if you know a particle’s velocity with infinite precision—zero error—you have infinite error when you measure its position; you know nothing at all about where it is.* You can never know both at the same time, and if you have some information about one, you must have some uncertainty about the other. It’s another unbreakable law.
Heisenberg’s uncertainty principle applies to more than just measurements performed by humans. Like the laws of thermodynamics, the principle applies to nature itself. Uncertainty makes the universe seethe with infinite energy. Imagine an extremely tiny volume in space, like a really small box. If we analyze what is going on inside that box, we can make some assumptions. For instance, we know, with some precision, the position of the particles inside. After all, they can’t be outside the box; we know that they are restricted to a certain volume, because if they were outside the box, we would not be looking at them. Because we have some information about the particles’ position, the Heisenberg uncertainty principle implies that we must have some uncertainty about the particles’ velocity—their energy. As we make that box smaller and smaller, we know less and less about the particles’ energy.
This argument holds everywhere in the universe—in the center of the earth and in the deepest vacuum of space. This means that in a sufficiently small volume, even in a vacuum, we have some uncertainty about the amount of energy inside. But uncertainty about the energy in a vacuum sounds ridiculous. The vacuum, by definition, has nothing in it—no particles, no light, nothing. Thus, the vacuum should have no energy at all. Yet according to Heisenberg’s principle, we cannot know how much energy there is in a volume of the vacuum at any given time. The energy in a tiny volume of vacuum must be fluctuating constantly.
How could the vacuum, which has nothing in it, have any energy at all? The answer comes from another equation: Einstein’s famous E = mc2. This simple formula relates mass and energy: the mass of an object is equivalent to a certain amount of energy. (In fact, particle physicists don’t measure the mass of the electron, say, in kilograms or pounds or any of the usual units of mass or weight. They say that the electron’s rest mass is 0.511 MeV [million electron volts]—a lump of energy.) The fluctuation in the energy in the vacuum is the same thing as a fluctuation in the amount of mass. Particles are constantly winking in and out of existence, like tiny Cheshire cats. The vacuum is never truly empty. Instead, it is seething with these virtual particles; at every point in space, an infinite number are happily popping up and disappearing. This is the zero-point energy, an infinity in the formulas of quantum theory. Interpreted strictly, the zero-point energy is limitless. According to the equations of quantum mechanics, more power than is stored in all the coal mines, oil fields, and nuclear weapons in the world is sitting in the space inside your toaster.
When an equation has an infinity in it, physicists usually assume that there is something wrong; the infinity has no physical meaning. The zero-point energy is no different; most scientists ignore it completely. They simply pretend that the zero-point energy is zero, even though they know it is infinite. It’s a convenient fiction, and it usually doesn’t matter. However, sometimes it does. In 1948 two Dutch physicists, Hendrick B. G. Casimir and Dik Polder, first realized that the zero-point energy can’t always be ignored. The two scientists were studying the forces between atoms when they realized that their measurements didn’t match the forces that had been predicted. In a search for an explanation, Casimir realized that he had felt the force of nothing.
The secret to the Casimir force lies with the nature of waves. In ancient Greece, Pythagoras saw the peculiar behavior of waves that traveled up and down a plucked string—how certain notes were allowed and others were forbidden. When Pythagoras strummed a string, the string sounded a clear note, the tone known as the fundamental. When he gently placed his finger in the middle of the string and plucked again, he got another nice, clear note, this time one octave above the fundamental. One-third of the way down yielded another nice tone. But Pythagoras realized that not all notes are allowed. When he placed his finger randomly on the string, he seldom got a clear note. Only certain notes can be played on the string; most are excluded (Figure 48).
Matter waves are not so different from string waves. Just as a guitar string of a given size is not capable of playing every possible note—some waves are “forbidden” from appearing on the string—some particle waves are forbidden from being inside a box. Put two metal plates close together, for instance, and you can’t fit every sort of particle inside. Only those whose waves match the size of the box are allowed in (Figure 49).
Figure 48: Forbidden notes on a guitar string
Casimir realized that the forbidden particle waves would affect the zero-point energy of the vacuum, since particles are everywhere winking in and out of existence. If you put two metal plates close together and some of those particles aren’t allowed between the plates, then there are more particles on the outside of the plates than on the inside. The undiminished zoo of particles presses on the outside of the plates, and without the full complement on the inside, the plates are crushed together, even in the deepest vacuum. This is the force of the vacuum, a force produced by nothing at all. This is the Casimir effect.
Figure 49: The Casimir effect
Though the Casimir force—a mysterious, phantom force exerted by nothing at all—seems like science fiction, it exists. It is a tiny force and very difficult to measure, but in 1995 the physicist Steven Lamoreaux measured the Casimir effect directly. By putting two gold-covered plates on a sensitive twist-measuring device, he determined how much force it took to counteract the Casimir force between them. The answer—about the weight of one slice of an ant that’s been chopped into 30,000 pieces—agreed with Casimir’s theory. Lamoreaux had measured the force exerted by empty space.
The Relativistic Zero: The Black Hole
[The star,] like the Cheshire cat, fades from view. One leaves behind only its grin, the other, only its gravitational attraction.
—JOHN WHEELER
Zero in quantum mechanics invests the vacuum with infinite energy. A zero in the other great modern theory—relativity—creates another paradox: the infinite nothing of the black hole.
Like quantum mechanics, the theory of relativity was born in light; this time it was the speed of light that caused the trouble. Most objects in the universe don’t have a speed that every observer can agree on. For instance, imagine a small boy who is throwing stones in all directions. For an observer approaching the boy, the stones seem to be going faster than for an observer who is running away; the velocity of the stones seems to depend on your direction and speed. In the same way, the speed of light should depend on whether you are running toward or running away from the lightbulb that’s shining on you. In 1887 the American physicists Albert Michelson and Edward Morley tried to measure this effect. They were baffled when they found no difference; the
speed of light was the same in every direction. How could this be?
Again, it was the young Einstein who had the answer in 1905. And again, very simple assumptions would have enormous consequences.
The first assumption Einstein made seems fairly obvious. Einstein stated that if a number of people watch the same phenomenon—say, the flight of a raven toward a tree, the laws of physics are the same for each observer. If you compare the notes of a person on the ground and a person on a train moving parallel to the raven, they would disagree about the speed of the raven and the tree. But the eventual outcome of the flight is the same: after a few seconds, the raven arrives at the tree. Both observers agree on the final result, though they might disagree about some of the details. This is the principle of relativity. (In the special theory of relativity, which we are discussing here, there are restrictions on the kind of motion that is allowed. Each observer must be moving with constant velocity in a straight line. In other words, they can’t feel an acceleration. With the general theory of relativity, the restrictions are removed.)
The second assumption is a little more troubling, especially since it seems to contradict the principle of relativity. Einstein assumed that everybody—no matter at what speed they are traveling—agrees about the speed of light in a vacuum: about 300 million meters per second, a constant denoted by the letter c. If someone shines a flashlight at you, the light rushes at you at a speed of c. It doesn’t matter whether the person holding the flashlight is standing still, running toward you, or running away; the beam of light always travels at a speed of c from your point of view—and everybody else’s.
This assumption challenged everything physicists had assumed about the motion of objects. If the raven were acting like a photon, then an observer on the train and the person standing still would have to agree on the value for the raven’s speed. That would mean that the two observers would disagree about when the raven meets the tree (Figure 50). Einstein realized that there is one way around this: the flow of time changes, depending on an observer’s speed. The clock on the train must tick more slowly than the stationary clock. Ten seconds for the observer on the ground might seem like only five seconds for somebody on the train. It’s the same thing for a person who zooms away at great speed. Every tick of his stopwatch takes more than a second from a stationary observer’s point of view. If an astronaut took a 20-year journey (according to his pocket watch) at nine-tenths of the speed of light, he would come back to Earth having aged 20 years, as expected. But everyone who stayed behind would have aged 46 years.
Figure 50: The raven’s constant speed means that time must be relative.
Not only does time change with speed, so do length and mass. As objects speed up, they get shorter and heavier. At nine-tenths of the speed of light, for instance, a yardstick would only be 0.44 yards long, and a one-pound bag of sugar would weigh nearly 2.3 pounds—from a stationary observer’s point of view. (Of course, this doesn’t mean that you would be able to bake more cookies with the same bag of sugar. From the bag’s point of view, its weight stays the same.)
This variability in the flow of time might be hard to believe, but it has been observed. When a subatomic particle travels very fast, it survives longer than expected before it decays, because its clock is slow. Also, a very precise clock has been observed to slow down ever so slightly when flown in an airplane at great speed. Einstein’s theory works. There was a potential problem though: zero.
When a spaceship approaches the speed of light, time slows down more and more and more. If the ship were to travel at the speed of light, every tick of the clock on board would equal infinite seconds on the ground. In less than a fraction of a second, billions and billions of years would pass; the universe would have already met its ultimate fate and burned itself out. For an astronaut aboard the spaceship, time stops. The flow of time is multiplied by zero.
Luckily, it is not so easy to stop time. As the spaceship goes ever faster, time slows down more and more, but at the same time, the spaceship’s mass gets greater and greater. It is like pushing a baby carriage where the baby grows and grows. Pretty soon you are pushing a sumo wrestler—not so easy. If you manage to push the carriage even faster, the baby becomes as massive as a car…and then a battleship…and then a planet…and then a star…and then a galaxy. As the baby gets more massive, your push has less and less effect. In the same way, you can take a spaceship and accelerate it, getting it closer and closer to the speed of light. But after a while, it gets too massive to push any longer. The spaceship—or for that matter any other object with mass—never quite reaches the speed of light. The speed of light is the ultimate speed limit; you cannot reach it, much less exceed it. Nature has defended itself from an unruly zero.
However, zero is too powerful even for nature. When Einstein extended the theory of relativity to include gravity, he did not suspect that his new equations—the general theory of relativity—would describe the ultimate zero and the worst infinity of them all: the black hole.
Einstein’s equations treat time and space as different aspects of the same thing. We are already used to the idea that if you accelerate, you change the way you move through space; you can speed up or slow down. What Einstein’s equations showed was that just as acceleration changes the way you move through space, it changes the way you move through time. It can speed up the way time flows or slow it down. Thus, when you accelerate an object—when you subject it to any force, be it gravity or be it the push of a gigantic cosmic elephant—you change its motion through space and through time: through space-time.
It’s a difficult concept to grasp, but the easiest way to approach space-time is through an analogy: space and time are like a gigantic rubber sheet. Planets, stars, and everything else sit on that sheet, distorting it slightly. That distortion—the curvature caused by objects sitting on the sheet—is gravity. The more massive the object that is sitting on the sheet, the more the sheet gets distorted, and the larger the dimple around that object. The pull of gravity is just like the tendency of objects to roll into the dimple.
The curvature of the rubber sheet is not only a curvature of space, but a curvature of time as well. Just as space gets distorted close to a massive object, time does, too. It gets slower and slower as the curvature gets greater and greater. The same thing happens with mass. As you get into greatly curved regions of space, bodies’ masses effectively increase, a phenomenon known as mass inflation.
This analogy explains the orbits of the planets; Earth is simply rolling around in the dimple that the sun makes in the rubber sheet. Light doesn’t go in a straight line, but in a curved path around stars—an effect that the British astronomer Sir Arthur Eddington went on an expedition in 1919 to observe. Eddington measured the position of a star during a solar eclipse and spotted the curvature that Einstein had predicted (Figure 51).
Einstein’s equations also predicted something much more sinister: the black hole, a star so dense that nothing can escape its grasp, not even light.
Figure 51: Gravity bends light around the sun.
A black hole begins, like all stars, as a big ball of hot gas—mostly hydrogen. If left to its own devices, a sufficiently large ball of gas would collapse under the weight of its own gravity; it would crush itself into a tiny lump. Luckily for us, stars don’t collapse because there is another force at work: nuclear fusion. As a cloud of gas collapses, it gets hotter and denser, and hydrogen atoms slam into one another with increasing force. Eventually, the star gets so hot and dense that the hydrogen atoms stick to one another and fuse, creating helium and releasing large quantities of energy. This energy shoots out from the center of the star, causing it to expand a little bit. During most of its life, a star is in an uneasy equilibrium: the propensity to collapse under its own gravity is balanced by the energy that comes from the fusing hydrogen in its center.
This equilibrium cannot last forever; the star has only a limited amount of hydrogen fuel to burn. After a while, the fusion reaction
dims, and the equilibrium is upset. (How long this process takes depends on how big the star is. Ironically, the bigger the star—the more hydrogen it has—the shorter its life, because it burns much more violently. The sun has about five billion years of fuel left, but don’t let that make you complacent. The sun’s temperature will increase gradually before that, boiling off the oceans and turning Earth into an uninhabitable desert like Venus. We should count ourselves lucky if we have a mere billion years left of life on Earth.) After a drawn-out series of death throes—the precise sequence of events depends, again, on the mass of the star—the star’s fusion engine fails, and the star begins to collapse under its own gravity.
A quantum-mechanical law called the Pauli exclusion principle keeps matter from squishing itself into a point. Discovered in the mid-1920s by German physicist Wolfgang Pauli, the exclusion principle states, roughly, that no two things can be in the same place at the same time. In particular, no two electrons of the same quantum state can be forced into the same spot. In 1933, the Indian physicist Subrahmanyan Chandrasekhar realized that the Pauli exclusion principle had only a limited ability to fight against the squeeze of gravity.
As pressure in the star increases, the exclusion principle states that electrons inside must move faster and faster to avoid one another. But there’s a speed limit: electrons cannot move faster than the speed of light, so if you put enough pressure on a lump of matter, electrons cannot move fast enough to stop the matter from collapsing. Chandrasekhar showed that a collapsing star that has about 1.4 times the mass of our sun will have enough gravity to overwhelm the Pauli exclusion principle. Above this Chandrasekhar limit a star’s gravity will pull on itself so strongly that electrons can’t stop its collapse. The force of gravity is so great that the star’s electrons give up their struggle once and for all; the electrons smash into the star’s protons, creating neutrons. The massive star winds up being a gigantic ball of neutrons: a neutron star.
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