Introduction

Einstein’s theories of relativity consistently fascinate non-physicists, and this article goes into relativity in greater detail. First, though, I shall say something about the theories which physicists work with today.

The everyday world is very well described by the theories worked out by Isaac Newton in the seventeenth century. Newton’s greatest physical realization was that undisturbed motion is a body’s natural state, and that it alters its velocity, meaning that it accelerates, only when a force acts on it. Now, it is far from obvious that the everyday running down of objects is not the natural state, but is the result of frictional forces; moreover Newton’s idea ran counter to the teaching of Aristotle. Newton’s laws go awry on the very small scale – within atoms, for instance – and on the very large scale, when things move at high speeds approaching the speed of light, or have very great mass – the ultimate example being the universe itself. The appropriate theory of the small scale is called quantum mechanics, whereas that of the large scale is relativity. Newton’s system emerges as an excellent approximation to each in the appropriate limit: quantum mechanics looks much like Newton’s when describing objects of everyday size, and relativity looks much like Newton’s mechanics at speeds much less than that of light, and in very weak gravity (including the Earth’s). But the quantum and relativistic corrections are always there, even if small. Only in the 20th century has technology evolved sufficiently to let us look at things on a small enough scale, and to move things around fast enough, for these effects to be observed. The speed of light is so great that light can travel seven times round the Earth in one second.

There should, of course, be a single theory capable of describing anything from the tiniest sub-atomic particle to the entire universe. To find it we would have to join quantum mechanics and relativity.

This enterprise has not yet been successful: at the deepest level, the two theories seem not to want to marry. Of course, unsolved problems are what scientists thrive on; this is an exciting area of research.

For the moment, though, we have to treat quantum mechanics and the full, general theory of relativity separately. Here I shall examine relativity, pure and simple. These words are used advisedly: conceptually, relativity is purer and simpler than the Newtonian system, although it is harder to do sums with.

The basis of relativity

Let me begin with an example of the strange things that relativity predicts. Suppose you are sitting on a train, traveling along the tracks at 100km/hr, and you roll a ball from the back of the carriage to the front with a speed along the carriage floor of 10km/hr. At what speed would a farmer standing by the tracks say that the ball was going past him? The train is going by at 100km/hr, and the ball is doing an extra 10km/hr on top of that, so isn’t the answer just 110km/hr? That is what Newton would have said; but even he would have been wrong. In fact, the answer is smaller, by a millionth of a millimeter every hour. The discrepancy becomes dramatically greater at larger speeds: if the rail authority were to introduce trains traveling at 3/4 of the speed of light, or 810 million km/hr, and I could bowl the ball at, say, 1/2 the speed of light, or 540 million km/hr, the answer is not 3/4 + 1/2 or 5/4 of the speed of light, but remains less – only 10/11 of the speed of light, in fact. There is nothing wrong with being startled by this, for intuition is shaped by everyday experience – which does not include trains traveling at millions of kilometers per hour. In science, though, the final court of appeal is to experiment, and (like it or not!) this does happen. If I can not only explain why speeds do not add up, but also get you thinking “of course – how much more natural”, then this essay will have achieved its aim.

Imagine two spacecraft moving apart, along a line, at a particular relative speed. Far along that line, an alien sends out a pulse of electromagnetic radiation. (Light, X-rays and radio waves are all examples of electromagnetic radiation, having differing wavelengths. The pulse, being instantaneous, has no wavelength.) What speeds do the two astronauts say the pulse goes past them at? The radiation propagates through a vacuum; and a vacuum, being nothing, must look the same to each observer. This is different from sound waves in air, where co-moving observers perceive different windspeeds. Here there is no medium to ‘do the waving’, and consequently nothing indicating that one spacecraft is in any sense privileged over the other; so we are forced to suggest that they both give the same answer. That is, they both register the same speed for the pulse, even though they themselves are in relative motion! Nor can the speed of the alien relative to either spacecraft matter, since there is no medium. This is all very different from the answers expected on Newtonian grounds, which would differ by the relative speed; yet it is what actually happens. A host of experiments, beginning with Albert Michelson and Edward Morley’s in 1887, before relativity was worked out, confirmed that the speed of light is a universal constant; and we can now understand why this result is very natural. Anything else implies that one observer is privileged over the other. In fact there is only their relative motion. Now you can see why Einstein called his theory ‘relativity’.

These ideas took over 200 years to surface from Newton’s day, for two reasons: first, the technology did not exist to run the Michelson-Morley experiment earlier, and second, no-one would have dreamed of doing it anyway. The idea that one observer is privileged made good sense then: someone sitting motionless on the Earth is different from others wandering around on its surface. It was only when people began to get a larger perspective on the cosmos that they could even contemplate these problems. Even today there is a good deal written against relativity by those who wish to resurrect the idea of an absolute frame of reference, in which a mysterious all-pervading aether has no drift velocity and carries the waves. A good deal of the better material of this sort has been written by engineers, because they develop a higher level of intuitive ‘feel’ for Newton’s everyday physics than anyone else, and never work with the speeds at which it goes badly wrong. They consequently find it harder to renounce. Nevertheless there is always a subtle mistake somewhere in their reasoning.

Now, speed is distance traveled through space divided by elapsed time; so, if speeds do not add up in the way we expect, space and time must get ‘mixed up’ in some way. Since there are three space dimensions and one time, the mix occurs in a 4-dimensional picture; hence the phrase ‘time is the fourth dimension’. The most concise description of relativity employs four dimensions in the same way that we describe space using three. One strange consequence of this is length contraction: someone moving relative to you is shorter, as measured by you, than when you measure them at rest. Likewise, since neither person is privileged over the other, your friend would also measure you as shorter! This paradoxical effect is always present, but is only pronounced when you and your friend’s relative speed approaches that of light – far beyond even the fastest airplanes.

Time dilation

Another strange effect is time dilation: if you move fast relative to someone then your watch, to them, runs slower. Likewise their watch, to you, runs slower. A good explanation of this has been given using clocks which consist of a photon – an element of light – bouncing between two parallel mirrors, the clock ticking every time it bounces. When this clock moves past you, traveling in the plane of the mirrors, the photon is also moving past you to keep between the mirrors – you see it moving on a diagonal path. You therefore register that the photon has further to go between bounces, and since its speed is unchanged it must take longer between each bounce. That means to you the clock ticks more slowly, which means time itself.

Because time is something which passes, this effect, unlike length contraction, seemingly gives rise to an inconsistency. Suppose you come back together. Your watches can be compared, and you can definitely judge which lost more. But according to relativity, doesn’t each of you say the other’s watch did? This is called the twin paradox, and like all paradoxes it is a consequence of faulty reasoning. Time dilation is only that simple for observers in uniform relative motion; for observers who move apart and back towards each other, the analysis is more complicated. Correct application of the laws shows that there is no problem. The effect has been verified using today’s incredibly accurate clocks, and in the slower decay of moving radioactive particles.

Further to this point, there is no such thing as absolute simultaneity. Events which are simultaneous according to one observer will be separated in time according to another observer who is in motion relative to the first. Moreover, the second observer may see the events happen in reverse order, depending on his direction of motion relative to the line connecting the events. Fortunately, if one event is known to have caused the other, their order remains the same to every observer.

Impartiality can only be maintained between observers if unaccelerated co-moving observers see the same laws of physics. Suppose I recognize that three physical quantities dependent on position and time, A, B and C, are related by the law A=B times C; then the A, B and C written down by you, moving relative to me, must also be related by the same law: your A=your B times your C. Of course, the values of A, B and C measured by each of us will not necessarily be the same; but the law relating the perceived values must be the same one. This criterion was understood in Newtonian physics, but the old, incorrect transformations of space and time – those giving the Newtonian result when adding velocities – had been used. An overhaul was needed in the light of the new transformations, to relate correctly my values of A, B and C to yours. The answer depends on what these quantities are: magnetic fields, electric fields, or whatever. With the notable exception of gravity, all of the laws of physics were incorporated into the scheme. Electromagnetism needed no alteration at all; before Einstein, this had been seen as a failing! Other laws, accurate at low speeds, had to be adjusted at higher speeds so as to satisfy this criterion. That could be done in more than one way, and experiments performed at high speeds were needed to pick the winner.

Einstein’s equation E=mc²

The greatest prediction of relativity was that, if a force is applied to a body, so as to accelerate it, the body does not move ever faster, but approaches the speed of light. The perceived mass of an object in relativity depends on its speed to the observer, and the nearer to the speed of light it comes, the greater its mass. It therefore becomes increasingly hard to accelerate, and can never ‘break the light barrier’ as the energy imparted to the object goes into its mass. This is the famous equation E=mc², where E is energy, m is mass and c the speed of light; in everyday life, we see only the ‘rest mass’ of objects. This is the equation which fuels the sun, converting its mass to energy. It is the equation which underpins nuclear bombs and nuclear power stations. Both were inconceivable before the 20th century, not only because E=mc² was unknown but because accessing the mass in the atomic nucleus, to turn it into energy, is technologically very difficult. But it pays off spectacularly, because mass is a very concentrated form of energy. The mass of a bus ticket is enough to power the bus round the world!

The basic theory I have outlined was put together by Albert Einstein in 1905, in a paper called Zur Elektrodynamik Bewegter Körper, or in English On the Electrodynamics of Moving Bodies. It remains one of the greatest physics papers ever; but it is fair to say that Einstein was not alone on the trail. The Dutch physicist Hendrik Lorentz, and the French mathematician Henri Poincaré, to name two, were also actively involved.

But what about gravity? General relativity

The laws of gravity presented peculiar difficulties in being incorporated into the scheme. Einstein’s greatest achievement lay in solving this problem; a measure of its difficulty is that it took the best theoretical physicist for three hundred years a full decade to crack, from 1905 to 1915. This work was based entirely on theoretical considerations, moreover, for no direct conflict with experiment then existed – in sharp contrast to the birth of quantum theory. Einstein wrote to a colleague that he would not say one word in defense of the theory, for he believed (rightly) that its elegance alone was sufficiently compelling. Following his 1905 paper, and his contributions to quantum theory, Einstein was already the most famous physicist of the age, and this latest success was trumpeted on the front pages of prominent newspapers. His 1905 paper set forth what is now called Special Relativity; with the incorporation of gravity this became General Relativity. To do it full justice remains beyond the scope even of undergraduate courses, but we can learn the basics.

So far I have considered a hypothetical family of observers, moving at different relative speeds; but said that these relative speeds aren’t changing. The observers are not mutually accelerated. The laws of physics certainly do not look the same to accelerated and unaccelerated observers: somebody waking inside a plummeting lift is in ‘free fall’, and is unaware of gravity accelerating both self and lift together towards the Earth. But a person on the ground is well aware of gravity. The equivalence, within the lift, of gravity and acceleration is called the equivalence principle, and it is the basis of general relativity. In fact it is the observer in the lift, and not the one on the ground, who sees the same physics as our observers in special relativity. Such people are said to be in an inertial frame. This is a further de-mythologising of terra firma: now it is free-fallers who are ‘more natural’. (Newton actually realized he was making an assumption about ‘absolute space’ and privileged observers, but this seemed reasonable in his time.)

Any acceleration produces a force on the accelerated body which can be felt, for example by a deep space astronaut turning his thrusters on. But to what is this acceleration relative? Special relativity dealt with observers in uniform relative motion to each other, but here there is only one observer. The answer is that the acceleration is relative to the local inertial frame, which is determined by everything in the universe. Notice, with your arms hanging loosely, that the stars appear stationary. Now execute a pirouette: your arms fly up under centrifugal force, and the stars appear to rotate. It is no coincidence that the frame in which the stars don’t rotate is the one in which there is no centrifugal force. This issue had been highlighted by the philosopher Ernst Mach.

The mysteries of special relativity are minor next to those of general relativity. Its predictions are conventionally envisaged using the idea of curved space. The local inertial frames from each small region of space-time must be patched together to make up the whole space-time, and in the presence of masses they cannot be patched onto a flat background, just as segments of orange peel cannot be pasted onto a flat surface without distortion. The explanation is that gravity accelerates light as it accelerates anything else; but the speed of light is also constant. These apparently contradictory facts are reconciled by distorting (‘curving’) space, or more accurately space-time. The point of view of general relativity is that mass (and equivalently energy) in the universe curves space-time; and curvature of space-time in turn influences the motion of masses moving in the universe.

General Relativity immediately explained an anomalous wobble in the orbit of the planet Mercury. The great test, though, would be to observe the distortion of space near a heavy object, by following the course of a light ray past it. The ray should appear to bend, on top of the bending due to the object’s gravitational pull on the energy/mass of the light; in other words, the object’s gravitational field acts as a lens! The only practicable test was on starlight passing the sun, which would be visible during an eclipse when the overwhelming light from the sun is obscured. The first eclipse subsequent to Einstein’s prediction fell on May 29th, 1919. The astrophysicist Arthur Eddington organized expeditions to Brazil and the Gulf of Guinea on behalf of the Royal Astronomical Society of London, and the results again confirmed Einstein’s theories and put him on the front pages.

With modern technology, many other tests have now been done. Radar has been bounced off planets, and super-accurate clocks have been placed in orbit; the stronger the gravity, the slower they run. The effects are tiny in the fields accessible to us, but General Relativity has always been confirmed.

Black holes and cosmology

General Relativity’s most dramatic prediction is the black hole. Black holes are bodies for which the escape velocity exceeds the speed of light. A black hole is specified by its mass, rotation (with respect to the local inertial frame in its absence), and electromagnetic energy alone. No sudden change in the local gravity is felt by the occupants of a craft as it passes the boundary of no return, known as the event horizon. Insidiously, the pull of a really large black hole at its event horizon is too weak to be easily detected. The drama is obvious to a distant observer, who sees the craft frozen on the event horizon, in very heavily redshifted light; similar effects on nearby matter might serve as a warning. No matter what it does, the hapless craft is drawn, in a finite span according to its own clock, to the center of the hole (at least, if the hole is non-rotating). At the center the force of gravity is infinite, and the craft is crushed to a point. This is called a singularity.

Black holes cannot be seen but their existence can be inferred from various unusual celestial observations. They are formed when certain stars collapse under their own gravity at the ends of their lives. Radiation pressure can no longer support the star, and – if it is sufficiently massive – nor can anything else.

The most significant implications of general relativity lie in cosmology. We know that the universe today is expanding, in the sense that light sent today from one galaxy to another will arrive after a shorter flight than light sent tomorrow. An internal description of the expansion is necessary because it is meaningless to ask what the universe is expanding into; it includes everything, by definition.

On the basis of early calculations by Karl Schwarzschild, it was suspected that the expansion started from a definite point in the past, the ‘Big Bang’. This work was made rigorous in the 1960s by the ‘singularity theorems’ of Roger Penrose and Stephen Hawking in the 1960s. (When I first heard that Penrose had won the Nobel Prize for physics in 2020, I assumed it was for this, rather than for his work on black holes which the Committee actually cited.) If general relativity is true then the universe, and even space and time, did indeed begin at a point. Asking what happened before that would be like asking what is north of the North Pole. When the mathematics is done, that is actually a fairly close analogy.

But general relativity isn’t exact, because it doesn’t take quantum theory into account. As we use general relativity in conjunction with our observations to predict what the universe looked like ever farther backwards in time, we find the universe shrinking smaller and smaller and getting hotter and hotter. Far back enough, it was so small and hot that we would need a quantum theory of gravity to continue further back with any accuracy. So our lack of understanding acts as a veil over what happened earlier still. But we can nevertheless say that, about 14 billion years ago, the universe was so small and hot that the expansion was explosive – a pretty large bang if not a big one. Speculation inevitably takes place about the earliest universe, behind the veil. Some of this speculation is more like science fiction, often presented with undue confidence; more responsible would be to say “We don’t know yet”.

What about a quantum theory of gravity?

Physics knows of four forces. There is gravity. It is always an attractive force. Then there is electromagnetism. According to our theory of electromagnetism, charges may be positive or negative, and charges of the opposite sign attract each other whereas charges of the same sign repel. We know that the atomic nucleus contains many protons, all positively charged. Gravity isn’t strong enough to hold them together against their mutual electric repulsion, so a further force must do that. We call it the strong force, since it is stronger than electromagnetism. But it gets weaker as the distance increases between two particles much faster than electromagnetism or gravity does, so we are not familiar with it in everyday life, as we are with electromagnetism and gravity. Finally, some atomic nuclei split spontaneously split in two despite the binding provided by the strong force. This is the phenomenon known as radioactivity, and to do that there must be a further force, which again is significant only inside the atomic nucleus. We call it the weak force.

We have working quantum theories of electromagnetism, the strong force and the weak force. In fact the weak force and electromagnetism have been unified into a single quantum theory, and a brilliant success it is. But we don’t have a working quantum theory of gravity. If we try to apply quantum theory to gravity, some of the answers turn out to be infinite, and that’s a warning that something has gone wrong. People hope that the infinities might go away in a fully unified theory of gravity with all the other forces, but that’s speculation and we are a long way from having such a theory today.

Einstein’s gravity theory is about space and time curving, and space and time are the arena for the other three forces. So Einstein’s theory puts gravity on a different basis from those other three. The other three forces are also what physicists call gauge theories. In a gauge theory the physics looks the same to a whole class of observers who differ from each other in some particular way. In fact a gauge theory of gravity has been worked out that predicts the same as Einstein’s theory, except for an extra interaction between electron ‘spins’ that is so weak we could not spot the difference experimentally. (Electrons have intrinsic spin, which according to this theory feels and induces twisting of space and time, as well as curving it.) This gauge theory of gravity has a lot going for it; if any one researcher can be said to be responsible for it then it is the late Tom Kibble, and a group in Cambridge has developed it better than anybody else from the late 1990s. It deserves a lot more exploration, and although it still gives infinite answers when a quantum version is sought, I reckon it is the right gravity theory to try to unify with the other forces, because all four are gauge theories. Einstein would not have liked it, though, because he was deeply committed to the geometric view of gravity; the gauge approach, which has a whiff of quantum mechanics about it, would not have been to his taste.