Quantum Theory Cannot Hurt You - Marcus Chown (2007)
Part II. BIG THINGS
Chapter 9. THE FORCE OF GRAVITY DOES NOT EXIST
HOW WE DISCOVERED THE TRUTH ABOUT GRAVITY AND CAME FACE TO FACE WITH BLACK HOLES, WORMHOLES, AND TIME MACHINES
The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he does not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity.
They are 20-year-old twin sisters. They work in the same skyscraper in Manhattan. One is an assistant in a boutique at street level, the other a waitress at the High Roost restaurant on the 52nd floor. It’s 8:30 a.m. They come through the revolving doors into the foyer and go their separate ways. One heads across the marble expanse to the ground-floor shopping mall; the other sprints into the mouth of the high-speed elevator just before the doors swish shut.
The hands of the clock above the elevator spin around. Now it’s 5:30 p.m. On the ground floor the shop-assistant twin stares up at the big red indicator light as it counts down the floors. With a “ding,” the doors burst open and out comes her waitress sister… an 85-year-old bent figure clutching a silver zimmer frame!
If you think this scenario is pure fantasy, think again. It’s an exaggeration, granted, but it’s an exaggeration of the truth. You really do age more slowly on the ground floor of a building than on the top floor. It’s an effect of Einstein’s “general” theory of relativity, the framework he came up with in 1915 to fix the shortcomings of his special theory of relativity.
The problem with the special theory of relativity is that, well, it is special. It relates what one person sees when looking at another person moving at constant speed relative to them, revealing that the moving person appears to shrink in the direction of their motion while their time slows down, effects that become ever more marked as they approach the speed of light. But motion at constant speed is of a very special kind. Bodies in general change their speed with time—for instance, a car accelerates away from traffic lights or NASA’s space shuttle slows when it reenters Earth’s atmosphere.
The question Einstein therefore set out to answer after he published his special theory of relativity in 1905 was: What does one person see when looking at another person accelerating relative to them? The answer, which took him more than a decade to obtain, was contained in the “general” theory of relativity, arguably the greatest contribution to science by a single human mind.
When Einstein embarked on his quest, one problem in particular worried him: what to do about Newton’s law of gravity. Although it had stood unchallenged for almost 250 years, it was clear to Einstein that it was fundamentally incompatible with the special theory of relativity. According to Newton, every massive body tugs on every other massive body with an attractive force called gravity. For instance, there is a gravitational pull between Earth and each and every one of us; it keeps our feet glued firmly to the ground. There is a gravitational pull between the Sun and Earth, which keeps Earth trapped in orbit around the Sun. Einstein did not object to this idea. His difficulty was with the speed of gravity.
Newton assumed that the force of gravity acts instantaneously—that is, the Sun’s gravity reaches out across space to Earth and Earth feels the tug of that gravity without any delay. Consequently, if the Sun were to vanish at this very moment—an unlikely scenario!—Earth would notice the absence of the Sun’s gravity instantly and promptly fly off into interstellar space.
An influence that can cross the gulf between the Sun and Earth in no time at all must travel infinitely fast—instantaneous travel and infinite speed are completely equivalent. However, as Einstein discovered, nothing—and that necessarily includes gravity—can travel faster than light. Since light takes just over eight minutes to travel between the Sun and Earth, it follows that, if the Sun were to vanish suddenly, Earth would continue merrily in its orbit for at least eight and a bit minutes before spinning off to the stars.
Newton’s tacit assumption that gravity reaches out across space at infinite speed is not the only serious flaw in his theory of gravity. He also assumed that the force of gravity is generated by mass. Einstein, however, discovered that all forms of energy have an effective mass, or weigh something. Consequently, all forms of energy—not just mass-energy—must be sources of gravity.
The challenge facing Einstein was, therefore, to incorporate the ideas of the special theory of relativity into a new theory of gravity and, at the same time, to generalise the special theory of relativity to describe what the world looked like to an accelerated person. It was as he contemplated these gargantuan challenges that a lightbulb lit up in Einstein’s head. He realised, to his surprise and delight, that the two tasks were one and the same.
THE ODD THING ABOUT GRAVITY
To understand the connection it is necessary to appreciate a peculiar property of gravity. All bodies, irrespective of their mass, fall at the same rate. A peanut, for instance, picks up speed just as quickly as a person. This behaviour was first noticed by the 17th-century Italian scientist Galileo. In fact, Galileo is reputed to have demonstrated the effect by taking a light object and a heavy object and dropping them together from the top of the Leaning Tower of Pisa. Reportedly, they hit the ground at the same time.
On Earth the effect is obscured because objects with a large surface area are preferentially slowed by their passage through the air. Nevertheless, Galileo’s experiment can be carried out in a place where there is no air resistance to mess things up—the Moon. In 1972, Apollo 15 commander Dave Scott dropped a hammer and a feather together. Sure enough, they hit the lunar soil at exactly the same time.
What is peculiar about this phenomenon is that, usually, the way in which a body moves in response to a force depends on its mass. Imagine a wooden stool and a loaded refrigerator standing on an ice rink, where there is no friction to confuse things. Now imagine that someone pushes the refrigerator and the stool with exactly the same force. The stool, being less massive than the refrigerator will obviously budge more easily and pick up speed more quickly.
What happens, however, if the stool and the refrigerator are acted on by the force of gravity? Say someone tips them both off the roof of a 10-story building? In this case, as Galileo himself would have predicted, the stool will not pick up speed faster than the refrigerator. Despite their wildly different masses, the stool and the refrigerator will accelerate towards the ground at exactly the same rate.
Now, perhaps you appreciate the central peculiarity of gravity. A big mass experiences a bigger force of gravity than a small mass, and that force is in direct proportion to its mass, so the big mass accelerates at exactly the same rate as the small mass. But how does gravity adjust itself to the mass it is acting on? It was Einstein’s genius to realise that it does so in an incredibly simple and natural way—a way, furthermore, that has profound implications for our picture of gravity.
THE EQUIVALENCE OF GRAVITY AND ACCELERATION
Say an astronaut is in a room accelerating upwards at 9.8 metres per second per second, which is the acceleration gravity imparts to falling bodies near Earth’s surface. Think of the room as a cabin in a spacecraft whose rocket engines have just started firing. Now, say the astronaut takes a hammer and a feather, holds them out from him at the same height above the floor of the cabin, then lets them go simultaneously. What happens? Well, the hammer and feather meet the floor of course. How this event is interpreted, though, depends entirely on the particular viewpoint.
Assuming the spacecraft is far away from the gravity of any big masses like planets, the hammer and the feather are weightless. So if we look into the spacecraft from outside with some kind of X-ray vision, we see the two objects hanging motionless. However, because the spacecraft is accelerating upward, we see the floor of the cabin racing up to meet the hammer and the feather. When it strikes them, furthermore, it strikes them both simultaneously.
Say the astronaut has amnesia and has entirely forgotten he is in a spacecraft. The portholes, in addition, are blacked out so there is nothing to tell him where he is. How does he interpret what he sees?
Well, the astronaut maintains that the hammer and the feather have fallen under gravity. After all, they have done the one thing a hammer and a feather experiencing gravity would do—they have fallen at the same rate and hit the ground at the same time (ignoring air resistance of course). The astronaut is further convinced that gravity is responsible for what he has seen by the fact that his feet appear to be glued to the floor just as they would be if he was in a room on Earth’s surface. In fact, everything the astronaut experiences turns out to be indistinguishable from what he would experience if he was on Earth’s surface.
Of course, it could be a coincidence. Einstein, however, was convinced he had stumbled onto a deep truth about nature. Gravity is indeed indistinguishable from acceleration, and the reason for that could not be simpler. Gravity is acceleration! This realisation, which Einstein later called “the happiest thought of my life,” convinced him that the search for a theory of gravity and for a theory that described accelerated motion were one and the same thing.
Einstein elevated the indistinguishability of gravity and acceleration to a grand principle of physics, which he christened the principle of equivalence. The principle of equivalence recognises that gravity is not like other forces. In fact, it is not even a real force. We are all like the amnesiac astronaut in the blacked-out spacecraft. We do not realise that our surroundings are accelerating and so have to find some other way to explain away the fact that rivers flow downhill and apples fall from trees. The only way is to invent a fictitious force—gravity.
THE FORCE OF GRAVITY DOES NOT EXIST!
The idea that gravity is a fictitious force may sound a little far-fetched. However, in other everyday situations, we are perfectly happy to invent forces to make sense of what happens to us. Say you are a passenger in a car that is racing round a sharp corner in the road. You appear to be flung outward and, to explain why, you invent a force—centrifugal force. In reality, however, no such force exists.
All massive bodies, once set in motion, have a tendency to keep travelling at constant speed in a straight line.1 Because of this property, known as inertia, unrestrained objects inside the car, including a passenger like you, continue to travel in the same direction the car was travelling before it rounded the bend. The path followed by the car door however, is a curve. It should be no surprise, then, that you find yourself jammed up against a door. But the car door has merely come to meet you in the same way that the floor of the accelerating spacecraft came up to meet the hammer and feather.2 There is no force.
Centrifugal force is known as an inertial force. We invent it to explain our motion because we choose to ignore the truth—that our surroundings are moving relative to us. But, really, our motion is just a result of our inertia, our natural tendency to keep moving in a straight line. It was Einstein’s great insight to realise that gravity too is an inertial force. “Can gravitation and inertia be identical?” asked Einstein. “This question leads directly to my theory of gravity.”
According to Einstein, we concoct the force of gravity to explain away the motion of apples falling from trees and planets circling the Sun because we ignore the truth—that our surroundings are accelerating relative to us. In reality, things move merely as a result of their inertia. The force of gravity does not exist!
But wait a minute. If the motion we attribute to the force of gravity is actually just the result of inertia, that must mean that bodies like Earth are really just flying through space at constant speed in straight lines. That’s patently ridiculous! Earth is circling the Sun and not flying in a straight line, right? Not necessarily. It all depends on how you define a straight line.
GRAVITY IS WARPED SPACE
A straight line is the shortest path between two points. This is certainly true on a flat piece of paper. But what about on a curved surface—for instance, the surface of Earth? Think of a plane flying the shortest route between London and New York. What path does it take? To someone looking down from space, it is obvious—a curved path. Think of a hiker trekking between two points in a hilly landscape. What path does the hiker take? To someone looking down on the hiker from a vantage point so high that the undulations of the landscape cannot be seen, the path of the hiker wiggles back and forth in the most tortuous manner.
Contrary to expectations, then, the shortest path between two points is not always a straight line. In fact, it is only a straight line on a very special kind of surface—a flat one. On a curved surface like Earth’s, the shortest route between two points is always a curve. In light of this point, mathematicians have generalised the concept of a straight line to include curved surfaces. They define a geodesic to be the shortest path between two points on any surface, not just a flat one.
What has all this got to do with gravity? The connection, it turns out, is light. It is a characteristic property of light that it always takes the shortest route between two points. For instance, it takes the shortest path from these words you are reading to your eyes.
Now think back to the amnesiac astronaut in his accelerating, blacked-out spacecraft. Bored of experimenting with a hammer and feather, he gets out a laser and places it on a shelf on the left-hand wall of his cabin, at a height of say 1.5 metres. He then crosses to the right-hand wall of the cabin and, with a marker pen, draws a red line, also at a height of 1.5 metres. Finally, the astronaut turns on the laser so that its beam stabs horizontally across the cabin. Where does it strike the right-hand wall?
It stands to reason that, since the astronaut has fired the beam horizontally, it will hit the wall exactly on the red line. So does it? The answer is no!
While the light is in flight across the cabin, the floor of the spacecraft is all the time being boosted by the rocket motors. Consequently, the floor is moving steadily upward to meet the beam. As the light gets closer and closer to the right-hand wall, the floor gets closer and closer to the light. Or from the point of view of the astronaut, the light gets closer and closer to the floor. Clearly, when the beam hits the right-hand wall, it hits it below the red line. The astronaut sees the light beam curving steadily downward as it crosses the cabin.
Now light, remember, always takes the shortest path between two points. The shortest path on something that is flat is a straight line, whereas the shortest path on something that is curved is a curve. What then are we to make of the fact that the light beam follows a curved trajectory across the spacecraft cabin? There is only one possible inter-pretation: The space inside the cabin is in some sense curved.
Now, you can argue that this is just an illusion caused by the accelerating spacecraft. The crucial point, however, is that the astronaut has no way of knowing that he is in an accelerating spacecraft. He could just as well be experiencing gravity in a room on Earth’s surface. Acceleration and gravity are indistinguishable. This is the principle of equivalence. What the experiment with the laser beam is actually demonstrating—and this shows the tremendous power of the principle of equivalence—is that light in the presence of gravity follows a curved trajectory. Or to put it another way, gravity bends the path of light.
Gravity bends light because space, in the presence of gravity, is somehow curved. In fact, this is all gravity turns out to be—curved space.
What exactly do we mean by curved space? It is easy to visualise a curved surface like the surface of Earth. But that is because it has only two directions, or dimensions—north-south and east-west. Space is a bit more complicated than that. In addition to three space dimensions—north-south, east-west, and up-down—there is one time dimension—past-future. As Einstein showed, however, space and time are really just aspects of the same thing, so it is more accurate to think of there being four “space-time” dimensions.
Four-dimensional space-time is impossible for us to visualise since we live in a world of three-dimensional objects. This means that the curvature, or warpage, of four-dimensional space-time is doubly impossible to visualise. But that’s what gravity is: the warpage of four-dimensional space-time.
Fortunately, we can get some idea of what this means. Imagine a race of ants that spends its entire existence on the two-dimensional surface of a taut trampoline. The ants can only see what happens on the surface and have no concept whatsoever of the space above and below the trampoline—the third dimension. Now imagine that you or I—mischievous beings from the third dimension—put a cannonball on the trampoline. The ants discover that when they wander near the cannonball their paths are mysteriously bent towards it. Quite reasonably, they explain their motion by saying that the cannonball is exerting a force of attraction on them. Perhaps they even call the force gravity.
However, from the God-like vantage point of the third dimension, it is clear the ants are mistaken. There is no force attracting them to the cannonball. Instead, the cannonball has made a valleylike de-pression in the trampoline, and this is the reason the paths of the ants are bent towards it.
Einstein’s genius was to realise that we are in a remarkably similar position to the ants on the trampoline. The path of Earth as it travels through space is constantly bent towards the Sun, so much so that the planet traces out a near-circular orbit. Quite reasonably, we explain away this motion by saying that the Sun exerts a force of attraction on Earth—the force of gravity. However, we are mistaken. If we could see things from the God-like perspective of the fourth dimension—something that is as impossible for us to do as it is for the ants to see things from the third dimension—we would see there is no such force. Instead, the Sun has created a valleylike depression in the four-dimensional space-time in its vicinity, and the reason Earth follows a near-circular path around it is because this is the shortest possible path through the warped space.
There is no force of gravity. Earth is merely following the straightest possible line through space-time. It is because space-time near the Sun is warped that that line happens to be a near-circular orbit. According to physicists Raymond Chiao and Achilles Speliotopoulos: “In general relativity, no ‘gravitational force’ exists. What we normally associate with the force of gravity on a particle is not a force at all: The particle is simply travelling along the ‘straightest’ possible path in curved space-time.”
A body travelling along the “straightest” possible path through space-time is in free fall. And, since it is in free fall, it experiences no gravity. Earth is in free fall around the Sun. Consequently, we do not feel the Sun’s gravity on Earth. The astronauts on the International Space Station are in free fall around Earth. Consequently, they do not feel Earth’s gravity.3
Gravity arises only when a body is prevented from following its natural motion. Our natural motion is free fall towards the centre of Earth. The ground thwarts us, however, so we feel its force on our bodies, which we interpret as gravity. Just as centrifugal force is what we feel when a cornering car prevents us from following the natural motion in a straight line, the force of gravity is what we feel when our surroundings prevent us from following our natural motion along a geodesic.
Probably, it seems unnecessarily complicated to view massive bodies as moving under their own inertia through warped space-time rather than simply moving under the influence of a universal force of gravitational attraction. However, the two pictures are not equivalent. Einstein’s is superior. For a start, the thing that is warped is not merely space but the space-time of special relativity. The picture, therefore, automatically incorporates the peculiar interplay between space and time necessary to keep the speed of light a constant. Einstein’s picture also predicts new things.
Think of those ants on the trampoline. There are more things you can do with the material of the trampoline than merely depress it with a heavy mass like a cannonball. For instance, you could shake one corner up and down. This would cause ripples in the fabric to spread outwards across the trampoline like ripples on the surface of a pond. In the same way, the vibration of a large mass like a black hole out in space can generate ripples in the “fabric” of space-time. Such gravitational waves have yet to be detected directly, but their existence is a unique prediction of Einstein’s theory.
The fact that waves can ripple through space-time suggests that space is not the empty, passive medium imagined by Newton. Instead, it is an active medium with real properties. Matter does not simply pull on other matter across empty space, as Newton imagined. Matter distorts space-time, and it is this distorted space-time that in turn affects other matter. As John Wheeler put it: “Mass tells spacetime how to warp and warped space-time tells mass how to move.”
The distortion of space-time caused by a massive body takes time to propagate to another mass, just as the distortion of the trampoline by another cannonball takes time to reach the corners of the trampoline. Because of this, gravity—warped space-time—acts only after a delay, in perfect accord with the cosmic speed limit set by the speed of light.
The fact that space-time has some of the qualities of a real medium like air or water has implications for large bodies like planets and stars. When they rotate on their axes, they actually drag spacetime around with them. NASA has measured the effect, known as frame dragging, with an orbiting space experiment called Gravity Probe B. Frame dragging is tiny in the case of Earth but overwhelming in the case of a rapidly spinning black hole. Such a body sits at the eye of a great tornado of spinning space-time. Anyone falling into the black hole would be whirled around with the tornado, which no power in the Universe could defy.
THE RECIPE OF GENERAL RELATIVITY
Einstein’s novel take on gravity is now clear. Masses—for instance, stars like the Sun—warp the space-time around them. Other masses—for instance, planets like Earth—then fly freely under their own inertia through the warped space-time. The paths they follow are curved because these are the shortest possible paths in warped space. This is it. This is the general theory of relativity.
The devil, however, is in the details. We know how a massive body like a planet moves in warped space. It takes the shortest possible path. But how precisely does a mass like the Sun warp the space-time around it? It took Einstein more than a decade to find out, and the details would fill a textbook as big as a phone directory. However, Einstein’s starting point for the general theory of relativity is not difficult to appreciate. It is none other than the principle of equivalence.
Recall again the hammer and the feather in the blacked-out spacecraft. To the astronaut, they appeared to fall to the floor under gravity. To someone watching the experiment from outside the spacecraft, however, it was obvious that the hammer and the feather were hanging in midair and that the floor of the cabin was accelerating upwards to meet them. They were completely weightless.
This observation is of key importance. A body falling freely in gravity feels no gravity. Imagine you are in an elevator and someone cuts the cable. As it falls, you are weightless; you feel no gravity.
“The breakthrough came suddenly one day,” Einstein wrote in 1907. “I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he does not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity.”
What is the significance of a freely falling body feeling no gravity? Well, if it experiences no gravity—or acceleration, since the two are the same—then its behaviour is described entirely by Einstein’s special theory of relativity. Here then is the crucial point of contact—the all-important bridge—between the special theory of relativity and the theory of gravity sought by Einstein.
The observation that a freely falling body does not feel its weight and is therefore described by special relativity suggests a crude way to extend special relativity to a body experiencing gravity. Think of a friend standing on Earth and very obviously experiencing gravity pressing his or her feet to the ground. You can observe your friend from any point of view you like—from hanging upside down from a nearby tree or from an aeroplane flying past. But one point of view provides a big payoff. If you imagine things from a point of view that is in free fall, then you will be weightless, subject to no acceleration. Since you feel no acceleration, you are justified in using the special theory of relativity to describe your friend.
But special relativity relates what the world looks like to people moving at constant speed relative to each other and your friend is accelerating upwards relative to you. That’s true. But if you do not mind a lot of laborious calculation, you can imagine your friend travelling at constant speed, a second, say then at a slightly higher constant speed for the next second, and so on. It’s not perfect, but you can approximate your friend’s acceleration as a series of rapid steps up in speed. For each speed you simply use special relativity to tell you what is happening to the space and time of your friend.
According to special relativity, time slows down for a moving observer. It therefore follows that time slows down for your friend since your friend is moving relative to you. But wait. Your friend is moving relative to you because he or she is experiencing gravity. It follows that gravity must slow down time! This should not be too much of a surprise. After all, if gravity is simply the warpage of space-time, it stands to reason that if we are experiencing gravity, our space and our time must be distorted in some way.
The other thing that follows from thinking about your friend standing on Earth’s surface is that if gravity were stronger—say your friend was standing on a more massive planet—his or her speed relative to you in free fall would get faster quicker. According to special relativity, the faster someone moves, the more their time slows down. Consequently, the stronger the gravity someone is experiencing, the more their time slows down. What this means is that if you work on the ground floor of an office building, you age more slowly than your colleagues who work on the top floor. Why? Because, being closer to Earth, you experience a more powerful pull, and time slows down in stronger gravity.
Earth’s gravity, however, is very weak. After all, you can hold your arm out in front of you and not even the gravity of the entire Earth can force you to drop it. The weakness of Earth’s gravity means that the difference in the flow rate of time between the ground and top floors of even the tallest building is nearly impossible to measure. The opening scene, with the twin sisters aging at vastly different rates in their skyscraper workplace, is therefore a gross exaggeration. Nevertheless, there are places in the Universe with far stronger gravity.
One place is the surface of a white dwarf star, where the gravity is much stronger even than the Sun’s. Einstein’s theory of gravity predicts that time for these stars should pass slightly slower than for us. Testing such a prediction might seem impossible. However, nature has very conveniently provided us with “clocks” on the surfaces of white dwarfs. The clocks are actually atoms.
Atoms give out light. Light is actually a wave that undulates up and down like a wave on water, and atoms of particular elements such as sodium or hydrogen give out light that is unique to the element, undulating a characteristic number of times a second. These undulations can be thought of as the ticks of a clock. (In fact, the second is defined in terms of the undulations of light given out by a particular type of atom.)
How does this property of atoms help us see the effect of gravity on time? Well, with our telescopes we can pick up the light from atoms on white dwarfs. We can then compare the number of undulations per second of the light from, say, hydrogen on a white dwarf, with the number of undulations per second of hydrogen on Earth. What we find is that there are fewer undulations per second in the light from a white dwarf. Light is more sluggish. Time runs slower!4 We are seeing a direct confirmation of Einstein’s general theory of relativity.
And there are stars known as neutron stars with even stronger gravity than that of white dwarfs. As a result of the strong gravity, time on the surface of a neutron star progresses one and a half times more slowly than on Earth.
THE CONSEQUENCES OF GENERAL RELATIVITY
Time dilation is only one of the novel predictions of Einstein’s general theory of relativity. Another, already touched on, is the existence of gravitational waves. We know they exist because astronomers have observed pairs of stars, which include at least one neutron star, losing energy as they spiral in towards each other. This puzzling loss of energy can be explained only if it is being carried away by gravitational waves.
The race is now on to detect gravitational waves directly. As they pass by, they should alternately stretch and squeeze space. Experiments designed to detect them therefore use giant “rulers,” many kilometres long. The rulers are made of light, but the idea is simple—to detect the change in length of the rulers as a gravitational wave ripples past.
Another prediction of Einstein’s theory, so far passed over without comment, is the bending of light by gravity. The reason for this bending, of course, is that light must negotiate the warped terrain of four-dimensional space-time. Although Newton’s law of gravity predicts no such effect, it does when combined with the special relativistic idea that all forms of energy—including light—have an effective mass. As light passes a massive body like the Sun, it therefore feels the tug of gravity and is bent slightly from its course.
Of course, special relativity is incompatible with Newton’s law of gravity, so this light-bending prediction has to be taken with a pinch of salt. In fact, the correct theory—general relativity—predicts that the path of light will be bent by twice as much.
This extra factor of two serves to highlight something subtle about the principle of equivalence. Recall the experiment in which the astronaut fired the laser horizontally across his spacecraft and noticed that the beam was bent downwards. Because there was no way he could know he was not experiencing gravity in a room on Earth’s surface, it was possible to deduce that gravity bends the path of light. Well, there is a little lie in here. You see, it turns out that it is possible for the astronaut to tell whether he is in a rocket or on Earth’s surface.
In the accelerating rocket, the force that pins the astronaut’s feet to the floor pulls him vertically downwards—wherever he stands in the cabin. On Earth’s surface, however, it matters where you stand because gravity always pulls things towards the centre of Earth. Consequently, gravity pulls in one direction in England but in the opposite direction in New Zealand—to the English, the New Zealanders are upside down, and vice versa. Now, the direction of the pull of gravity does not change too much from one side of a room to another. Nevertheless, with sensitive-enough measuring instruments, our astronaut could always detect the change and tell whether he was in a rocket accelerating out in space or on Earth’s surface.
Surely, this invalidates the principle of equivalence and brings the whole edifice of general relativity tumbling down? Well, you might think so. However, to construct a theory of gravity it is sufficient only that the principle of equivalence apply in tiny volumes of space, and in extremely tiny, localised volumes of space you can never detect changes in the direction of gravity.
What has this got to do with Einstein’s theory predicting twice the light deflection of Newton’s? Well, we have established that the laser beam will be bent downwards as it traverses a room on Earth’s surface, and this amount turns out to be roughly what Newtonian gravity predicts. Now imagine that the room is in free fall—say it has been dropped from an aeroplane—and the astronaut carries out the same experiment. In free fall, remember, there is no gravity. So the light beam should travel horizontally across the room and not be bent at all. But not all parts of the room are in a perfect state of free fall. Because Earth’s gravity pulls in one direction from one corner of the room and from a different direction from the other corner, gravity is not perfectly cancelled out as the room falls through the air. Because of this, what the astronaut actually sees is the light beam bent downwards by roughly the same amount as in the room on Earth’s surface. The two effects add together to give twice the light bending predicted by Newton’s theory of gravity plus special relativity.
So if the light from a distant star passes close to the Sun on its way to Earth, its trajectory should be bent about twice as sharply as Newton would have predicted. Such an effect would cause the position of a star to shift slightly relative to other stars. Though impossible to see in the glare of daylight, it is observable during a total eclipse when the Moon blots out the bright solar disc. Such an eclipse was due to occur on May 29, 1919, and the English astronomer Arthur Eddington travelled to the island of Principe off the coast of West Africa to see it. His photographs confirmed that starlight was indeed deflected by the Sun’s gravity by exactly the amount predicted by the general theory of relativity.
Eddington’s observations made Einstein’s reputation as “the man who proved Newton wrong.” But it was not the end of general relativity’s successful predictions. Newton had demonstrated theoretically that the planets orbited the Sun not in circles but in ellipses—squashed circles. He proved that this was a direct consequence of the fact that the force of gravity drops off in strength with a so-called inverse-square law. In other words, when you are twice as far away from the Sun, the force of gravity is four times as weak; three times as far away, it is nine times as weak; and so on.
Relativity changes everything. For a start, all forms of energy, not just mass-energy, generate gravity. Now gravity itself is a form of energy. Think of a warped trampoline and how much elastic energy that contains. Since gravity is a form of energy, the gravity of the Sun itself creates gravity! It’s a tiny effect and most of the Sun’s gravity still comes from its mass. Nevertheless, close in to the Sun, where gravity is strong, there is a small extra contribution from gravity itself. Consequently, any body orbiting there feels a gravitational tug greater than expected from the inverse square law.
Now—and this is the point—planets follow elliptical orbits only if they are being tugged by a force obeying an inverse-square law of force. This was Newton’s discovery. Relativity predicts that the force does not obey an inverse-square law. In fact, there are other effects that also cause a departure from Newtonian gravity, like the fact that gravity takes time to travel across space. The gravity that a moving planet feels at any moment therefore depends on its position at an earlier time and, because of this, is not directed towards the dead centre of the Sun. The upshot is that planets do not follow elliptical paths that repeat but rather elliptical paths which gradually change their orientation in space, tracing out a rosette-like pattern. This is not noticeable far from the Sun. The biggest effect is close in, where gravity is strongest.
Sure enough, there is something odd about the orbit of the innermost planet, Mercury. For some time before Einstein published his theory of gravity in 1915, astronomers had been puzzled by the fact that Mercury’s orbit gradually traces out a rosette pattern in space. Most of this effect is due to the gravitational pull of Venus and Jupiter. The odd thing, however, is that Mercury’s orbit would still be tracing out a rosette pattern even if Venus and Jupiter were not there. It is a tiny effect. Although Mercury orbits the Sun once every 88 days, a rosette is traced out only once every 3 million years. Remarkably, this is exactly what Einstein’s theory predicted. Using general relativity, he could explain every last detail of Mercury’s orbit. With yet another successful prediction under its belt, there could be no doubt that Einstein had discovered the correct theory of gravity.5
THE PECULIARITIES OF GENERAL RELATIVITY
General relativity is a fantastically elegant theory. Nevertheless, it is tremendously difficult to apply to real situations—for instance, to find the warpage of space-time caused by a given distribution of mass. The reason is that the theory is rather circular. Matter tells space-time how to warp. Then warped space-time tells matter how to move. The matter, which has just moved, tells space-time how to change its warpage. And so on, ad infinitum. There’s a kind of chicken-and-egg paradox at the heart of the theory. Physicists call it nonlinearity, and nonlinearity is a tough nut for theorists to crack.
One manifestation of nonlinearity already mentioned is the fact that gravity is a source of gravity. Well, if gravity can make more gravity, that extra gravity can make a little more gravity, and so on. Fortunately, gravity is so weak that this is not normally a runaway process and the gravity generated by a massive body is usually well behaved—usually, but not always.
Some very massive stars end their lives in a spectacular way. Usually, a star is prevented from being crushed by its own gravity by the pressure of the hot gas in its interior pushing outwards. But this outward pressure only exists while the star is generating heat. When it runs out of all possible fuels, it shrinks. Usually, some other form of pressure intervenes to make a white dwarf or a neutron star, superdense stellar embers. However, if the star is very massive and its gravity is very strong, nothing can stop the star from shrinking down to a point. As far as physicists know, such stars literally vanish from existence. However, they leave something behind: their gravity.
What we are talking about here are black holes, perhaps the most bizarre of all the predictions of general relativity. A black hole is a region of space-time where gravity is so strong that not even light can escape it—hence its blackness. And “region of space-time” is the operative phrase, for the mass of the star has gone.
How can you have gravity without mass? Well, gravity arises not just from mass but from all forms of energy. In the case of the black hole, its own gravity creates more gravity and that extra gravity creates more gravity… so the hole regenerates itself like a man holding himself in midair by his boot straps. From the space-time point of view, a black hole is literally a hole. Whereas a star like the Sun creates a mere dimple in the surrounding space-time, a black hole produces a bottomless well into which matter falls but can never escape again.
As Nobel Prize-winning physicist Subrahmanyan Chandrasekhar observed: “The black holes of nature are the most perfect macroscopic objects there are in the universe: The only elements in their construction are our concepts of space and time.”6
Because of their ultrastrong gravity, black holes reveal the most dramatic effects of general relativity. Surrounding them is a surface known as an event horizon. This marks the point of no return for objects straying too close to the black hole. If you moved in close to the event horizon, you could see the back of your head since light from behind you would be bent all the way around the hole before reaching your eyes. If you could somehow hover just outside the event horizon, time would flow so slowly for you that you could in theory watch the entire future of the Universe flash past you like a movie in fast-forward!
The fact that time runs far more slowly in the strong gravity of a black hole than elsewhere in the Universe has an intriguing consequence. Imagine you are far away from a black hole and you have a friend lingering close to it. Because of the marked difference in the flow of time for both of you, while you go from Monday to Friday, your friend progresses only from Monday to Tuesday. This means that, if you could find some way to spirit yourself over to your friend’s location, you could go from Friday back to Tuesday. You could travel back in time!
It turns out that there is in fact a way to spirit yourself from one location to another. Einstein’s theory of relativity permits the existence of “wormholes,” tunnel-like shortcuts through space-time. By entering one mouth of such a wormhole and exiting a mouth near your friend, it would indeed be possible to go back in time from Friday to Tuesday.
The trouble with wormholes is that they snap shut in an instant unless held open by matter with repulsive gravity. Nobody knows whether such “exotic matter” exists in the Universe. Nevertheless, the extraordinary fact remains that Einstein’s theory of gravity does not rule out the possibility of time travel.
There are a few differences, however, between the kind of “time machine” permitted by general relativity and the type described by science fiction writers like H. G. Wells. For one thing, you have to travel a distance through space to travel a distance through time. You cannot simply sit still in a time machine, pull a lever, and find yourself in 1066. And a second important difference is that you cannot go back to a time before your time machine was built. So if you want to go on a dinosaur safari, building a time machine today will not help. You will have to find one built and abandoned by extraterrestrials (or some very smart dinosaurs) 65 million years ago!
To theorists the possibility of time machines is very unsettling. If time travel is possible, all sorts of impossible situations, or “paradoxes,” raise their ugly heads. The most famous is the grandfather paradox in which a man goes back in time and shoots his grandfather before he conceives the man’s mother. The problem is, if he shoots his grandfather, how can he ever be born to go back in time and do the dirty deed?!
Embarrassing questions like this have prompted the English physicist Stephen Hawking to propose the Chronology protection conjecture. Basically, it’s just a fancy name for an outright ban on time travel. According to Hawking, some as-yet-unknown law of physics must intervene to prevent time travel. He has no cast-iron evidence of such a law but simply asks: “Where are the tourists from the future?”
Einstein himself did not believe that time travel was possible, despite the fact that his theory of gravity predicted it. He was wrong, however, about two other predictions of his theory. He did not believe that black holes were possible, and today we have compelling evidence that they exist. And he did not believe what his theory was trying to tell him about the origin of the Universe—that it began in a Big Bang.
1 This is not at all obvious on Earth, where frictional forces act to slow a moving body. However, it is apparent in the empty vacuum of space.
2 It is worth pointing out that acceleration does not just mean a change in speed. It can also mean a change in direction. So a car travelling around a bend—even at constant speed—is accelerating.
3 Most people assume that astronauts orbiting Earth are weightless because there is no gravity in space. However, at the 500-kilometre-or-so height of the International Space Station, gravity is only about 15 per cent weaker than on Earth’s surface. The real reason astronauts are weightless is that they and their spacecraft are in free fall just as surely as someone in an elevator when the cable breaks. The difference is that they never hit the ground. Why? Because Earth is round and, as fast as they fall toward the surface, the surface curves away from them. They, therefore, fall forever in a circle.
4 For technical reasons, this effect is known as the gravitational red shift.
5 Or at least a workable theory for the time being, since even general relativity is not thought to be the last word on gravity.
6 The term “black hole” was coined by John Wheeler in 1965. Before 1965 there were very few scientific papers on such objects. Afterward, the field exploded. The term has even entered everyday language. People often talk about things disappearing down a bureaucratic black hole. The term is a perfect illustration of the importance of getting the right words to describe a phenomenon in science. If they paint a vivid picture in people’s minds, researchers are attracted to the subject.