General relativity is a theory of gravity – the best theory of gravity we have. It was the insight of a German physicist in 1915. Fifty years later, that theory was ready and able to explain the bizarre properties of such newly discovered exotic objects as black holes, pulsars and quasars
ALBERT Einstein used to tell that the unique flash of insight that set him on the path to general relativity came when he realised that a man falling from a roof – or a person trapped inside a freely falling lift – does not feel the force of gravity. People in the falling lift will float, completely weightless, able to push themselves from wall to wall or floor to ceiling with great ease.
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Of course, we have now seen people in exactly this situation – astronauts in spacecraft, falling freely in orbit around the Earth. In such “weightless” conditions, objects obey, precisely, Newton’s famous laws of motion, proceeding in straight lines unless interfered with by forces. But Einstein had to imagine all the things we have seen for ourselves on television – pencils hanging in mid air, liquids that refuse to pour, and so on. Einstein’s genius saw all this, and the important point missed by everybody else. If the acceleration of the falling lift, plunging downwards at an ever increasing speed, can precisely cancel out the force of gravity, then that force and acceleration are exactly equivalent to one another.
The power of this insight – the principle of equivalence – is clear if we imagine the lift replaced by a closed laboratory which is being accelerated through space by a constant force. Everything in the laboratory falls to the floor, and a physicist who carries out experiments inside will be unable to tell whether the downward force is due to an acceleration or to the force of gravity pulling things down. We are used to thinking of acceleration as being caused by a force, but from the point of view of the lift’s occupant, that force is caused by acceleration.
Now, said Einstein, imagine setting up an experiment inside such a laboratory to measure the behaviour of a beam of light that crosses from one side of the room to the other. In a laboratory moving at constant velocity, far from any planet or star, the light will travel in a straight line across the laboratory. But in an accelerated laboratory, the opposite wall has speeded up and moved forward relative to the light beam in the time that it takes the light to cross the room. Inside the laboratory, it will seem as if the beam of light is bent.
It looks as if there is, after all, a way to distinguish acceleration from gravity. But no, says Einstein. We must keep the principle of equivalence unless it is proved false. If the light beam is bent in an accelerated laboratory (an accelerated frame of reference) then it must also be bent by gravity, and by the exactly equivalent amount.
Since light has no mass, how can it be affected by gravity? Einstein puzzled over this from 1911 to 1915 before coming up with a mathematical theory, the general theory of relativity, that explained light bending, and much more besides.
Ten years earlier, Einstein had published three papers, at the age of 26, that would alone have ranked him among the half-dozen great pioneers of 20th century physics. Those papers dealt with the special theory of relativity, how light is quantised into packets of energy (photons) and the way tiny particles move through the air or a liquid (Brownian motion). But all three topics were in the mainstream of physics at the time. Significant though Einstein’s work was, if he had not done it then before long other researchers would have reached the same conclusions.
That is not true of the general theory of relativity. It was not a response to any observational puzzle (although it did, almost as an afterthought, account for an old mystery about the orbit of Mercury). Einstein was motivated by a deeper philosophical need, the quest for simplicity and unity in nature. If it had not been for Einstein, a comprehensive theory of gravity might not have been developed for decades, until scientists were pressed to consider the need for such a theory by the discovery of objects such as black holes, pulsars and quasars.
Relativity visualised
Bending space and time
THE NEW picture of the Universe casts aside the everyday notion of empty space and replaces it by an almost tangible continuum in four dimensions (three of space and one of time) that can be bent and distorted by the presence of material objects. It is those bends and distortions that provide the “force” of gravity, bend light beams, and deflect moving objects from straight-line trajectories, a situation summed up by the aphorism “matter tells space how to curve, space tells matter how to move”.
It is easier to visualise what is going on in terms of a two-dimensional elastic surface. Imagine a rubber sheet stretched tightly across a frame to make a flat surface. That is a “model” of Einstein’s version of empty spacetime. Now imagine dumping a heavy bowling ball in the middle of the sheet. It bends. That is Einstein’s model of the way space distorts near a large lump of matter.
When you roll a marble across the original flat sheet, it makes only a tiny indentation, and rolls in a straight line. But when you roll the marble near the bowling ball, the distortion in the rubber sheet makes it follow a curved path. That is Einstein’s model for the force of gravity. Objects are simply following a path of least resistance, a geodesic – the equivalent of a straight line through a curved portion of spacetime.
And this explains light bending. The effect is the same for a marble, a planet, or a beam of light. When it moves near a large mass-through a gravitational field of force, on the old picture – it follows a curved trajectory.
General relativity predicted exactly how much a beam of light should get bent when it passes near the Sun, in order to be exactly equivalent to the bending seen by a physicist in an accelerated frame of reference. The new theory made a clear and testable prediction, that stars observed “near” the Sun on the sky during a total eclipse (but actually, of course, much further away along the line of sight) would be displaced by a certain amount compared with their observed positions at other times. The prediction was confirmed by observations made during a total eclipse of the Sun in 1919 (see Box).
General relativity has also passed every other test applied to it. One of these concerns what is known as the precession of the orbit of the innermost planet. Mercury, the closest planet to the Sun, orbits where the gravitational field is strong (that is to say, where spacetime is strongly distorted). Astronomers already knew in 1915 that the orbit has a curious behaviour, which cannot be completely explained by Newton’s theory of gravity. As Mercury follows an elliptical orbit around the Sun, the ellipse itself shifts slightly each orbit, tracing out a pattern like a child’s drawing of the petals of a daisy.
This shift is exactly explained by general relativity. Anywhere that gravity is weak, general relativity and Newton’s famous inverse square law – which says force is proportional to the inverse square of the distance between two masses – give the same answers to the appropriate calculations. But in a strong field, according to general relativity, gravity deviates from the precise inverse square law. The size of the “post-Newtonian” effect is just big enough at the distance of Mercury from the Sun to produce the puzzling changes in orbit.
The Universe at large
Einstein’s great triumph
GENERAL RELATIVITY is a geometrical theory. It gives a well-defined physical meaning to a completely specified geometry of matter, space and time. But “completely” is a key word here. In his search for a unified description of nature, Einstein developed a theory that completely describes the Universe – and which, strictly speaking, only describes the complete Universe (or a complete universe).
When general relativity is applied to “local” problems such as calculating the orbits of planets in the Solar System, it is being used in an approximation. In practice, such approximations can be made as accurate as you like, using boundary conditions to join the equations describing a local object like the Sun on to the rest of the Universe. But the point is that Einstein did not have to expand his theory to make it capable of dealing with the whole Universe – making it a cosmological theory. General relativity, from its birth, dealt quite happily with the whole Universe.
When Einstein tried to describe the simplest possible mathematical model of the Universe using his new equations, however, he ran into a problem. At that time, in 1917, the received wisdom was that our Milky Way Galaxy was the entire Universe, a stable collection of stars. But the equations describing a complete cosmology of space, time and matter refused to produce such a picture. They insisted that the Universe must either be expanding or contracting.
The only way Einstein could hold the model universe still, to mimic the appearance of the Milky Way, was to add an extra term to the equations, called the cosmological constant. In 1917, he wrote “that term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars”. A dozen years later, observers, led by the pioneer Edwin Hubble in California, found that the Milky Way was not the entire Universe, but simply one galaxy among many millions, and that distant galaxies are all receding from each other.
The Universe is expanding, exactly as the pure equations of general relativity predicted in 1917, when Einstein himself refused to believe the evidence of his own theory. There is no need for the cosmological constant, and Einstein’s equations now provide the basis for the highly successful big bang description of the birth and evolution of the entire Universe (Inside Science Nos. 1 and 69).
Exotic phenomena
Relativity up to date
WITHIN the expanding Universe, general relativity is required to explain the workings of exotic objects where spacetime is highly distorted by the presence of matter – on the old picture, where large masses produce strong gravitational fields. The most extreme version of this, and one that has caught the popular imagination, is the phenomenon of black holes.
The concept of black holes is so familiar today, as a feature of Einstein’s masterwork, that it may come as a surprise to learn that although the name black hole was first used in an astronomical context (by John Wheeler, of Princeton University) only in 1968, the concept goes back more than two centuries, to the British polymath John Michell.
Michell realised that because the speed of light is finite, and because the speed needed to escape from an object (the escape velocity) is greater for larger bodies, there must come a point where not even light can escape from the surface of a “star”. This would be achieved by packing a hundred million Suns alongside one another in a huge sphere.
Low-density black holes with “only” the density of our Sun (1410 kilograms per cubic metre – a quarter of the Earth’s density), or even less, may indeed exist in our Universe. If so, they would trap light by their gravitational pull – or, in terms of general relativity, by bending spacetime around themselves so much that it becomes closed, pinched off from the rest of the Universe.
But there is another way to make a black hole, which was first recognised as a mathematical possibility in the 1930s. If a star keeps the same mass but shrinks inward, or stays the same size while accumulating mass, density increases. Eventually, the distortion of spacetime around it increases until, once again, a situation is reached where the object collapses and folds spacetime around itself, disappearing from all outside view. Not even light can escape from its gravitational grip, and it has become a black hole.
The notion of such stellar-mass black holes seemed no more than a mathematical trick, something that surely could not be allowed to exist in the real Universe, until 1968, and the discovery of pulsars.
Scientists now know that pulsars are the rapidly spinning remains of dead stars. They contain about as much matter as our Sun packed into a volume no bigger than that of a large mountain on Earth. Such neutron stars have roughly the density of the nucleus of an atom, and at 2×1017 kilograms per cubic metre are very close to the critical density at which gravity would overwhelm them and they would collapse into black holes. A neutron star could gain enough extra mass to do the trick, by accumulating matter from interstellar space, or by stripping gas from a companion star, by tidal forces.
The discovery of neutron stars made the possibility of black holes respectable. In the 1970s, several objects were discovered that might mark the locations of black holes.
An object which emits no light (or anything else) cannot be observed directly. But a black hole orbiting around another star, and swallowing gas that it is tearing off its companion, would be a messy eater. The gas funnelling down into the black hole will get hot, as the particles in the gas are accelerated and bash against one another. Astrophysicists calculated that the particles would get hot enough to radiate at X-ray frequencies. Scientists have now discovered X-ray sources in binary systems with the right properties to match those predicted for black holes by the equations of general relativity.
With black holes made respectable by these discoveries, they were soon invoked to explain another puzzling discovery of the 1960s, the quasars. Quasars are the energetic cores of some galaxies, which produce enormous amounts of energy from a region of space no bigger across than our Solar System.
Allowing matter to fall into a strong gravitational field – converting gravitational potential energy into heat – is the most efficient way to produce energy, apart from the annihilation of particles with their antiparticle counterparts. Dropping a mass m into a black hole from infinite distance would release almost half of its rest mass energy, mc2. If only a few per cent of this available energy is actually released when mass falls into a black hole, the energy needed to power a quasar could be provided by a big black hole which swallows just one or two times the mass of our Sun each year.
The kind of black hole invoked would contain about a hundred million times the mass of our Sun – very much the sort of object envisaged by Michell two centuries ago. This would be no more than 0.1 per cent of the mass of all the stars in the galaxy surrounding the quasar. Such a black hole could arise simply because too many stars got too close together in the core of a galaxy.
A large concentration of mass will also bend light (that is, bend spacetime so that light follows a curved path) near it, even without being a black hole. In some cases, the mass concentration can act as a lens, focusing light from a distant galaxy or quasar to produce two (or more) images on the sky. Astronomers have now found such gravitational lensing in the Universe, where multiple images of a single quasar occur as a result of lensing by an intervening cluster of galaxies.
But the most impressive and complete proof of the accuracy of Einstein’s theory come from yet another phenomenon, gravitational radiation.
The existence of gravitational radiation depends on Einstein’s concept of spacetime as a real, physical phenomenon which can be distorted by the presence of matter. The distortions are similar conceptually to the way in which a lump of matter when dropped into a pool of water makes waves on the surface of the water.
Ultimate proof
Gravity rules the waves
THE IMAGE of matter as solid lumps embedded in a stretched rubber sheet, spacetime, makes the origin of gravitational waves clear. When one of the lumps vibrates, it sends out ripples through the sheet, and these ripples set other lumps of matter vibrating. This is like the way a vibrating charged particle sends out electromagnetic waves and which shake other charged particles. But gravitational radiation is very weak; only 10-40 times as strong as electromagnetic radiation.
Researchers hope to measure the tiny ripples in spacetime produced by massive objects far from Earth in the near future (see above Box). But they already have proof that gravitational radiation exists.
A binary system with two very dense stars orbiting rapidly around one another would, according to the equations, be a powerful source of gravitational radiation. Just such a system has been found. It is called the “binary pulsar”. One of the stars in the system is a pulsar. The other is a neutron star that is not a radio source. They orbit each other every 7.75 hours.
Pulsars are superbly accurate “clocks”, keeping time by the sweep of their radio beams, like those of a lighthouse, as the neutron star rotates. Variations in the pulse rate from the binary pulsar show how the pulsar moves in its orbit – the observed pulse rate speeds up when the pulsar is moving towards us, and slows down when it is moving away. This is a version of the Doppler effect.
The period of the binary’s orbit is slowly decreasing. This means that the two neutron stars are getting closer together as time passes. The reason is that the system is losing energy, in the form of gravitational radiation. General relativity predicts that the period of the binary pulsar should decrease by 75 millionths of a second each year; observations are so precise that they show a decrease of 76±2 millionths of a second a year.
This is one of the greatest triumphs of Einstein’s general theory of relativity. That theory is now established beyond any doubt as the best one that we have to explain gravity and the Universe at large.
Evidence of the eclipse
DURING the solar eclipse of 1919, a team led by the English physicist Arthur Eddington measured the positions of several stars, shown by dots in the left diagram, which lay in nearly the same direction on the sky as the Sun (circle) at the time. Light from the distant stars passed through the region of space affected by the Sun’s gravity.
When Eddington compared these positions with the measured positions of the same stars when the Sun was on the opposite side of the sky, he found that they were apparently deflected. Each appeared to have moved by an amount which depended on the angular separation of the star from the Sun at the time of the eclipse. Light from each of those stars had been “bent” as it passed by the gravitational field of the Sun. These “deflections” (crosses in right figure) fell exactly on the curve predicted by Einstein’s theory.
Prospects for catching gravitational waves
A SYSTEM such as the binary pulsar is like an extreme version of a rotating weightlifter’s barbell. Viewed in the plane of rotation, this produces gravitational waves, which can be visualised in terms of their effect on a circular ring. Physicists call this kind of radiation “quadrupole radiation”.
Quadrupole radiation can be understood most simply in terms of radiation from electric charges. A pair of electric charges, one positive and one negative, forms a dipole, and when these two charges move (vibrating or rotating) they produce dipole electromagnetic radiation. A dipole itself is electrically neutral overall. A pair of dipoles is a quadrupole (with two positive and two negative charges), and when the charges in such an array move (for example, with one dipole orbiting around the other) they produce quadrupole radiation.
Unlike electricity, mass comes with only one “sign”, so there is no gravitational equivalent of electric dipole radiation. Two masses which rotate around one another actually behave like a pair of dipoles, producing gravitational quadrupole radiation, which can be visualised in terms of its effect on a circular ring.
As the wave passes through, the ring is stretched in one direction and squeezed in another at right angles, becoming an eclipse. Then, the pattern reverses. The pattern of alternate squeezing and stretching in two directions at right angles is the characteristic signature of quadrupole gravitational radiation.
Three test masses, placed in a right-angle “L” shape, could detect such radiation as it squeezes and stretches spacetime.
Such systems are now being constructed, with heavy masses placed in evacuated tubes several kilometres long, using laser interferometers to measure their positions to an accuracy of 10-18 metres. Researchers expect to detect gravitational radiation with such “telescopes” during the 1990s.
Further reading
The best guide to Einstein’s masterwork for the non-specialist is Was Einstein Right? by Clifford Will (Basic Books, 1986). The cosmological implications of general relativity are described in In search of the Big Bang, by John Gribbin (Corgi, 1987); Kate Charlesworth and John Gribbin tell the story of the Universe graphically in the Cartoon History of Time (Cardinal, 1990). Einstein’s overall contribution to science is assessed in Subtle is the Lord, by A. Pais (OUP, 1982), and in Einstein: A Centenary Volume, edited by A. P. French (Harvard University Press, 1980).
