7
The Fabric
Spacetime tells matter how to move; matter tells spacetime how to curve.
John Wheeler1
Einstein’s special relativity was actually referred to as the ‘theory of relativity’ for a few years following its first appearance in 1905. It became ‘special’ when it was acknowledged that the theory deals only with systems involving frames of reference moving at constant relative velocities. It doesn’t cope with frames of reference undergoing acceleration. And, because Newton’s force of universal gravitation is supposed to act instantaneously on gravitating bodies no matter how far apart they might be, this classical conception of gravity is at odds with special relativity, which denies that the influence of any force can be transmitted faster than the ultimate speed, c (and, by any measure, ‘instantaneous’ will always be faster). So, special relativity can’t describe objects undergoing acceleration nor can it describe objects experiencing Newton’s force of gravity.
Now, there are moments in the history of science when the interested onlooker can do nothing more than stare idiotically, jaw dropped and mouth agape, at the sheer audacity that we have come to associate with genius. Such a moment happened to Einstein in November 1907, on an otherwise perfectly ordinary day at the Patent Office in Bern. He had by this time received a promotion, to ‘Technical Expert, Second Class’. As he later recounted: ‘I was sitting in a chair in my patent office at Bern. Suddenly a thought struck me: If a man falls freely, he would not feel his weight.’2
Today we are so familiar with images and film clips of astronauts in zero-gravity environments that it may be difficult to grasp the immediate significance of Einstein’s insight. But this seemingly innocent thought contains the seed of the solution that would unlock the entire mystery of Newton’s force of gravity. Special relativity doesn’t deal with acceleration or gravity, and Einstein now realized that these are not two problems to be solved, but one.
How come? Imagine you climb into an elevator at the top of the Empire State Building in New York City. You press the button to descend to the ground floor. Unknown to you, the elevator is, in fact, a disguised interstellar transport capsule built by an advanced alien civilization. Without knowing it, you are transported instantaneously* into deep space, far from any planetary body or star. There is no gravity here. Now weightless, you begin to float helplessly above the floor of the elevator.
What goes through your mind? You have no way of knowing that you’re now in deep space. As far as you’re concerned, you’re still in an elevator descending from the top of the Empire State Building. Your sensation of weightlessness suggests to you that the elevator hoist cables have been suddenly cut in some horrible accident. You’re free-falling to the ground.
The alien intelligence observing your reactions does not want to alarm you unduly. They reach out with their minds, grasp the elevator/capsule in an invisible force field and gently accelerate it upwards. Inside the elevator, you fall to the floor. Relief washes over you. You conclude that the safety brakes must have engaged, and you have ground to a halt. You know this because, as far as you can tell, you’re once more experiencing the force of gravity.
Einstein called it the ‘equivalence principle’. The local experiences of gravity and of acceleration are the same. They are one and the same thing. He called it his ‘happiest thought’.3
But what does it mean? At first, Einstein wasn’t entirely sure. It would take him another five years to figure out that the equivalence principle implies another extraordinary connection, between gravity and geometry. Despite how it might appear to us, the geometry of spacetime isn’t ‘flat’ and rigid. It can bend and sag in places.
In school geometry classes, we learn that the angles of a triangle add up to 180°. We learn that the circumference of a circle is 2π times its radius, and that parallel lines never meet. Did you ever wonder why? These kinds of features (and many others besides) describe what mathematicians call a ‘flat space’, or ‘Euclidean space’, named for the famed geometer Euclid of Alexandria. This is the familiar three-dimensional space of everyday experience, on which we inscribe x, y, and z co-ordinate axes. When we combine this kind of space with the fourth dimension of time, such as Minkowski proposed, then the spacetime we get is a flat spacetime.
In a flat spacetime the shortest distance between two points is obviously the straight line we can draw between them. But what is the shortest distance between London, England, and Sydney, Australia? We can look up the answer: 10,553 miles. But this distance is not, in fact, a straight line. The surface of the Earth is curved, and the shortest distance between two points on such a surface is a curved path called an arc of a great circle or a geodesic. If you’ve ever tracked your progress on a long-haul flight, then this is the kind of path you would have been following.
Draw a triangle on the surface of the Earth (say by drawing lines between Reykjavik and Singapore, Singapore and San Francisco, San Francisco and Reykjavik) and you’ll find its angles add up to more than 180° (see Figure 6). The circumference of a circle drawn on this surface is no longer equal to 2π times its radius. Lines of longitude are parallel at the equator but they meet at the poles.
Figure 6. The angles of a triangle drawn on a sphere add up to more than 180°.
Newton’s first law of motion insists that an object will continue in its state of rest or uniform motion in a straight line unless acted on by an external force. In a flat spacetime all lines are straight, so Newton’s force of gravity is obliged to act instantaneously, and at a distance. But, Einstein now realized, if spacetime is instead curved like an arc of a great circle, then an object moving along such a path will ‘fall’. And as it falls, it accelerates.
In a curved spacetime it is no longer necessary to ‘impress’ the force of gravity on an object—the object quite happily slides down the curve and accelerates all on its own. All we need to do now is suppose that an object such as a star or a planet with a large mass-energy will curve the spacetime around it, much as a child curves the stretchy fabric of a trampoline as she bounces up and down. Other objects such as planets or moons that stray too close then follow the shortest path determined by this curvature. The acceleration associated with free-fall along this shortest path is then entirely equivalent to an acceleration due to the ‘force’ of gravity (see Figure 7).
Figure 7. An object with a large mass-energy, such as the Earth, curves the spacetime around it. The effects of this curvature were studied by Gravity Probe B, a satellite mission which was launched in April 2004. The results were announced in May 2011, and provided a powerful vindication of the general theory of relativity.
American physicist John Wheeler summarized the situation rather succinctly some years later: ‘Spacetime tells matter how to move; matter tells spacetime how to curve.’4 Through this insight Einstein saw that he might now be able to account for both acceleration and gravity in what would become known as the general theory of relativity. What this theory suggests is that there is actually no such thing as the ‘force’ of gravity. Mass-energy does generate a gravitational field, but the field is spacetime itself.
But, hold on. If gravity is the result of the curvature of spacetime, and gravity is very much a part of our everyday experience here on Earth, then shouldn’t we be able to perceive this curvature? Alas, the answer is no. The spacetime curvature caused by the mass-energy of the Earth is very slight, and subtle. The archetypal ‘level playing field’ on which we watch the game being played on a Sunday appears flat to us even though we know it sits on the surface of an Earth that is curved. In much the same way, our experience of spacetime is shaped by our local horizon. From our local perspective we perceive it to be flat even though we know it is gently curved. This is why we’re still taught Euclidean geometry in school.
Now, Euclidean geometry is complicated enough, so when we add a fourth dimension of time we expect that things get more complicated still. It will come as no surprise to learn that the mathematics of curved spacetime involve an even higher level of abstraction.
We should note in passing that it is common to suppose that Einstein’s genius extended to his ability in mathematics, and anyone flicking through a textbook on general relativity will marvel at the complexity arrayed in its pages. But Einstein was not particularly adept in maths. His maths teacher at the Zurich Polytechnic—Minkowski—declared him to be a ‘lazy dog’ and was greatly (but pleasantly) surprised by what he saw in Einstein’s 1905 papers. Fortunately, as Einstein began to grapple with the abstract mathematics of curved spacetime, help was on hand in the form of a long-time friend and colleague, Marcel Grossman. ‘… you must help me, or else I’ll go crazy’, Einstein begged him.5
We can anticipate that Einstein’s formulation of the general theory of relativity should be an equation (actually, a set of equations) of the kind which connect the curvature of spacetime—let’s say on the left-hand side—with the distribution and flow of mass-energy on the right-hand side. Given a certain sizeable amount of mass-energy we need to be able to work out the extent to which the spacetime around it will curve and this, in turn, tells us how another quantity of mass-energy will be accelerated in response.
From his moment of inspiration in 1907 it took Einstein a further eight years to formulate his theory, with many false trails and dead ends. He eventually presented the field equations of general relativity to the Prussian Academy of Sciences in Berlin on 25 November 1915, just over 100 years ago. He later declared: ‘The theory is beautiful beyond comparison. However, only one colleague has really been able to understand it… .’6 The colleague in question was German mathematician David Hilbert, who was in hot pursuit of the general theory of relativity independently of Einstein.
The resulting field equations were so complicated that Einstein judged them impossible to solve without making simplifying assumptions or approximations. And yet within a year the German mathematician Karl Schwarzschild had worked out a set of solutions. These are solutions for the specific case of a gravitational field outside a large, uncharged, non-rotating spherical body, which serves as a useful approximation for slowly rotating objects such as stars and planets.*
One of the more startling features of the Schwarzschild solutions is a fundamental boundary—called the Schwarzschild radius. Imagine a large spherical object, such as a star or planet. To escape the influence of the object’s gravity, a rocket on the surface must be propelled at a speed which exceeds the object’s escape velocity.† Further imagine that this object is compressed to a volume with a radius smaller than the Schwarzschild radius. To put this into some kind of perspective, note that the Schwarzschild radius of the Earth is about nine millimetres. Now the escape velocity is so large it exceeds the speed of light. Nothing—not even light itself—can escape the pull of the object’s gravitational field. It’s as though the spacetime is so distorted it has curved back on itself. The result is a black hole.*
Do black holes really exist in the universe? Although they are obviously difficult to detect directly, there is plenty of indirect evidence to suggest that black holes are fairly ubiquitous, and supermassive black holes are likely to sit at the centres of every galaxy.
From the beginning of his eight-year journey to the field equations of general relativity, Einstein was aware of four potential empirical tests. Actually, the first is not so much a test, more the resolution of a mystery. Newton’s theory of universal gravitation predicts that planets should describe elliptical orbits around the Sun. In this description, the planet’s point of closest approach to the Sun (called the perihelion) is a fixed point in space. The planet orbits the Sun and the perihelion is always at the same place.
However, observations of the orbits of the planets in the solar system show clearly that these points are not fixed. With each orbit the perihelion shifts slightly, or precesses. Anyone old enough to have played with a Spirograph drawing set as a child will appreciate the kinds of patterns that can result.
Of course, the Sun is not the only body in the solar system that generates a gravitational field. Much of the observed precession is caused by the cumulative gravitational pull of all the other planets. This contribution can be predicted using Newton’s gravity. Collectively, it accounts for a precession in the perihelion of the planet Mercury of about 532 arc-seconds per century.† However, the observed precession is rather more, about 574 arc-seconds per century, a difference of 42 arc-seconds.
Newton’s gravity can’t account for this difference and other explanations—such as the existence of another planet, closer to the Sun than Mercury (named Vulcan)—were suggested. But Vulcan could not be found. Einstein was delighted to discover that general relativity predicts a further contribution, of about 43 arc-seconds per century, due to the curvature of spacetime in the vicinity of Mercury.*
Perhaps the most famous prediction of general relativity concerns the bending of starlight passing close to the Sun. Now I’m not sure that everybody appreciates that Newton’s gravity also predicts this phenomenon. After all, Newton originally conceived light in the form of tiny particles, each presumed to possess a small mass. Newtonian estimates suggest that light grazing the surface of the Sun should bend through 0.85 arc-seconds caused by the Sun’s gravity.
The curvature of spacetime predicted by general relativity effectively doubles this, giving a total shift of 1.7 arc-seconds, and so providing a direct test. This prediction was famously borne out by a team led by British astrophysicist Arthur Eddington in May 1919. The team carried out observations of the light from a number of stars that grazed the Sun on its way to Earth.
Obviously, such starlight is usually obscured by the scattering of bright sunlight by the Earth’s atmosphere. The light from distant stars passing close to the Sun can therefore only be observed during a total solar eclipse. Eddington’s team recorded simultaneous observations in the cities of Sobral in Brazil and in São Tomé and Príncipe on the west coast of Africa. The apparent positions of the stars were then compared with similar observations made in a clear night sky.
Although few could really understand the implications (and fewer still—even within the community of professional physicists—could follow the abstract mathematical arguments), the notion of curved spacetime captured the public’s imagination and Einstein became an overnight sensation.
Incidentally, here’s another answer to the question why we can’t ‘see’ the curvature of spacetime. The mass of the Sun is about 330,000 times larger than the mass of the Earth, and yet the Sun curves spacetime only very slightly. I’d suggest a line that curves by only 1.7 arc-seconds—about 0.5 thousandths of a degree—would look exactly like a straight line to you or me.
General relativity also predicts effects arising from curved spacetime that are similar in some ways to the effects of special relativity. Einstein worked out the details in 1911, and they can be quickly identified using the Schwarzschild solutions.7 Basically, to a stationary observer, time is measured to slow down (and distances are measured to contract) close to a gravitating object where the curvature of spacetime is strongest. A standard clock on Earth will run more slowly than a clock placed in orbit around the Earth.
Imagine a light wave emitted from a laboratory on Earth. The light wave is characterized by its wavelength—the distance covered by one complete up-and-down cycle. What happens to this wave as it moves away from the Earth and the effects of spacetime curvature reduce?
Distances are shortened close to the Earth’s surface where gravity is strong. Consequently, distances lengthen as we move away from Earth into outer space. A centimetre as measured on Earth’s surface will be measured to be longer in space. The wavelength of a light wave will be measured to lengthen along with the distance. So, what happens is that as the light moves further and further away its wavelength is measured to get longer and longer.
A longer wavelength means that the light is measured to be ‘redder’ than it was. What we see as orange or yellow light on Earth, for example, will be measured to be red by the time it has reached a certain distance from Earth where gravity is much weaker. Physicists call this a ‘redshift’ or, more specifically, a gravitational redshift.
These are effects with some very practical consequences. A plane carrying an atomic clock from London to Washington DC loses 16 billionths of a second relative to a stationary clock left behind at the UK’s National Physical Laboratory, due to time dilation associated with the speed of the aircraft. This is an effect of special relativity.
But the clock gains 53 billionths of a second due to the fact that gravity is weaker (spacetime is less curved) at a height of 10 kilometres above sea level. In this experiment, the net gain is therefore predicted to be about 40 billionths of a second. When these measurements were actually performed in 2005, the measured gain was reported to be 39±2 billionths of a second.8
Does this kind of accuracy really matter to anyone? The answer depends on the extent to which you rely on the Global Positioning System (GPS) used by one or more of your smartphone apps or your car’s satellite navigation system. Without the corrections required by special and general relativity, you’d have a real hard time navigating to your destination.
Gravitational redshift happens as light moves away from a gravitating object. The opposite effect is possible. Light travelling towards a gravitating object will be blueshifted (shorter wavelengths) as the effects of gravity and spacetime curvature grow stronger.
The American physicists Robert Pound and Glen Rebka were the first to provide a practical Earth-bound test, at Harvard University in 1959. Electromagnetic waves emitted in the decay of radioactive iron atoms at the top of the Jefferson Physical Laboratory tower were found to be blueshifted by the time they reached the bottom, 22.5 metres below. The extent of the blueshift was found to be that predicted by general relativity, to within an accuracy of about ten per cent, reduced to one per cent in experiments conducted five years later.
There is one last prediction to consider. In June 1916, Einstein suggested that some kind of turbulent event involving a large gravitating object would produce ‘ripples’ in spacetime itself, much as we might feel the vibrations at the edge of a trampoline on which a child is bouncing. These ripples are called gravitational waves.
Despite how it might seem to us Earth-bound humans living daily with the force of gravity, this is actually nature’s weakest force. Place a paper clip on the table in front of you. Gradually lower a handy-size magnet above it. At some point, the paper clip will be pulled upwards from the table by the magnetic force of attraction and will stick to the bottom of the magnet. You’ve just demonstrated to yourself that in the tussle for the paper clip, the magnetic force generated by the small magnet in your hand is stronger than the gravitational pull of the entire Earth.
This has important consequences. A spectacular event such as two black holes coalescing somewhere in the universe is predicted to produce gravitational waves which cause the spacetime nearby to fluctuate like a tsunami. But after travelling a long distance across the universe these waves are not only not noticeable (or we would have known about them long ago), they are barely detectable. Einstein was later quite dubious about them and it was not until the 1950s and 1960s that physicists thought they might stand a chance of actually detecting them.
The physicists have had to learn to be patient. But in September 2015 their patience was finally rewarded. About a billion years ago two black holes that had been orbiting each other rather warily in a so-called binary system finally gave up the ghost. They spiralled in towards each other and merged to form one supermassive black hole. The black holes had masses equivalent to twenty-nine times and thirty-six times the mass of the Sun, merging to form a new black hole with a mass sixty-two times that of the Sun. This event took place about 12 billion billion kilometres away in the direction of the Southern Celestial Hemisphere.
How do we know? We know because on 15 September 2015 the gravitational waves generated by this event were detected by an experimental collaboration called LIGO, which stands for Laser Interferometry Gravitational-wave Observatory. LIGO actually involves two observatories, one in Livingston, Louisiana and another at Hanford, near Richland in Washington, essentially on opposite sides of the continental United States.
Each observatory houses an L-shaped interferometer, with each arm measuring four kilometres in length. The principle is much the same as that described in Chapter 5, Figure 3 for the Michelson–Morley experiment, but on a much grander scale. The interferometers are set up in such a way that light interference causes a dark fringe at the detector, so that in a ‘baseline’ measurement no laser light is visible. Then, as the subtle ‘movements’ in spacetime caused by gravitational waves ripple through the interferometer, the lengths of the interferometer arms change: one is lengthened slightly and the other becomes shorter. The interference pattern shifts slightly towards a bright fringe, and the laser light reaching the detector becomes visible for a time. The changes in distances we’re talking about here are breathtakingly small, measured in attometres (10−18 metres).
LIGO actually became operational in 2002 but was insufficiently sensitive. Operations were halted in 2010 and the observatories were upgraded with much more sensitive detectors. The enhanced observatories had only been operational for a couple of days when, quite by coincidence, the characteristic ‘chirp’ of gravitational waves from the black hole merger event passed through (Figure 8). This first successful detection of gravitational waves was announced at a press conference on 11 February 2016.
Figure 8. The LIGO observatory recorded its first gravitational-wave event on 14 September 2015. Signals detected at Hanford (H1) are shown on the left and Livingston (L1) on the right. Times are shown relative to 14 September 14 2015 at 09:50:45 UTC. Top row: the signals arriving first at L1 and then at H1 a few milliseconds later (for comparison, the H1 data are shown also on the right, shifted in time and inverted to account for the detectors’ relative orientations). Second row: the signals as predicted by a model based on a black hole merger event. Third row: the difference between measured and predicted signals.
Of course, we were already pretty confident that general relativity is broadly correct, so the detection of gravitational waves wasn’t needed to prove this. But this success is much, much more than a ‘nice-to-know’. Now that we can detect gravitational waves we have a new window on events in distant parts of the universe, one that doesn’t rely on light or other forms of electromagnetic radiation to tell us what’s going on.
Let’s reflect on all this before moving on. Einstein’s gravitational field equations connect spacetime on the left-hand side with mass-energy and momentum on the right. They tell us how to calculate the shape of the gravitational field, or the ‘gravitational well’ that forms in spacetime around the Sun or the Earth, for example. From this shape we can determine how other objects in its vicinity will be obliged to move.
But, of course, the field equations do not tell us exactly how this is supposed to work. They provide us with a recipe for performing calculations, but they’re rather vague on explanation. It might be best to think about it this way. Special relativity establishes two fundamental relationships between aspects of our physical reality that we had previously considered unconnected and independent of one another. Space and time meld into spacetime, and mass and energy into mass-energy. In general relativity, Einstein demonstrated that these too are in turn connected—spacetime and mass-energy are interdependent.
I like to think of it as the ‘fabric’ of our physical reality. This is a fabric of space, time, mass, and energy—all blurring somewhat around the edges—on which we attempt to weave a system of physics. Matter is energy, and only through approximation can it be separated from the space in which is sits, and the time in which it is.