Relativity: The Special and General Theory

Take a tape measure to a rectangular table, the kitchen table will do. Measure the short side and the long side, and write the two numbers down. Now stretch the tape from one corner to the opposite corner, across the diagonal. Before you read the number, predict it with the one theorem everybody remembers from school, the theorem of Pythagoras: a² + b² = c², where a and b are the two sides of a right angle and c is the diagonal that closes the triangle. Suppose the sides were 60 centimetres and 80 centimetres; then 60 squared is 3600, 80 squared is 6400, the sum is 10 000, and the square root of 10 000 is 100 centimetres. You look at the tape, and there it is, 100 centimetres, perhaps a millimetre more or less.

What exactly have you just confirmed? Most people would say they have checked that Pythagoras was right, that the theorem is true. Albert Einstein opens his little book by stopping us precisely here and asking, with some mischief, what we even mean when we call such a statement true. He reminds us that most of us learned Euclidean geometry as children, climbing its staircases under the eye of a conscientious teacher, and that we remember it with more respect than affection. Out of loyalty to that schooling we would scorn anyone who doubted the humblest of its theorems. And yet, he says, our proud certainty may evaporate the moment someone asks what "true" is supposed to mean.

Here is how geometry is built. It starts from a few basic notions, such as plane, point and straight line, which we picture more or less clearly, and from a few simple statements about them, the axioms, which we are willing to accept because the pictures make them feel obvious. Every other statement is then derived from the axioms by logical steps whose validity we feel obliged to grant; that derivation is what we call a proof. So within geometry a theorem is correct, or "true", when it has been properly derived from the axioms, and nothing more. The whole question of truth is thereby pushed back onto the axioms themselves. And there it dies, because, as Einstein says, the question whether an axiom is true cannot be answered by geometry, and worse, it has no meaning at all.

Consider the axiom that through two points passes exactly one straight line. You cannot ask whether that is true. All you can say is that Euclidean geometry deals with certain objects it chooses to call straight lines, and that it assigns them the property of being fixed by any two of their points. The word "true" simply does not fit statements of pure geometry, because when we say "true" we always mean, in the end, agreement with some real object, and pure geometry says nothing about real objects; it speaks only of how its concepts hang together logically. A modern comparison: the rule that a bishop in chess moves diagonally is not true or false, it defines the game. A theorem of pure geometry is like a statement about chess: correct within the rules, but silent about the world.

Why, then, do we so stubbornly feel that geometry is true? Einstein's answer is that each geometrical concept corresponds, more or less exactly, to something in nature, and that those natural things are without doubt the sole reason the concepts ever arose. Geometry may disown that origin in order to keep its logical structure clean, but the habit sits deep in us. When we think of a segment, we think of two marks scratched on a practically rigid body, a ruler, a steel bar, a table edge. And we are used to saying that three places lie on a straight line when we can find a spot from which, looking with one eye, their apparent positions fall on top of one another. A surveyor sighting through a theodolite, or a builder squinting along a laser line, is doing exactly that.

Now comes the decisive move, and it is a single added sentence. To the theorems of Euclidean geometry we add just this: two points on a practically rigid body always correspond to the same distance, the same segment, no matter how we move or turn the body. That is all it takes. With this one addition the theorems of geometry stop being statements about ideal concepts and become statements about the possible relative placement of real rigid bodies. Geometry, so extended, is a branch of physics, and now the question of truth is legitimate, because we can ask whether those statements hold for the real things we have assigned to the concepts. Somewhat loosely, Einstein says, the "truth" of a geometrical theorem in this sense means that it checks out in a construction with compass and ruler.

Your tape measure on the table was such a construction. The tape is a practically rigid body; you assumed its marks kept the same distance apart whether it lay along the side or across the diagonal; and under that assumption Pythagoras made a prediction that the world either honours or refuses. It honoured it, to within a millimetre. A GPS receiver does the same thing on a grand scale: it takes its distances from several satellites and combines them as sides of Euclidean figures to compute where you stand, which is Pythagoras in three dimensions, trusted with your life on the motorway. Said once more, in other words: pure geometry can only be consistent, never true; it becomes true or false only when its points and lines are tied to marks on rigid bodies and to lines of sight.

But notice what kind of foundation that is. Our conviction that geometry, so interpreted, is true rests entirely on experience, and on rather imperfect experience at that, on tapes and threads and sightings, each good to a millimetre or a fraction of an arc second, never to infinity. Einstein is explicit: we will assume this truth for now, as a working hypothesis, and only in the last part of the book, when we reach the general theory of relativity, will we see that it has limits, and exactly where. The GPS system will turn out to be a fine witness there too, because its clocks and signals have to be corrected for effects that only that later theory predicts. For the moment, keep the question alive: which of your certainties about space come from logic, and which from a tape measure?

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Take a pencil, lay it along the edge of your kitchen table, and mark where its tip ends with a fingernail. Now slide the pencil forward so its back end sits at the mark, and do it again, and again, until you reach the far corner. You expect, perhaps, that this is a crude substitute for a proper tape measure. What actually happens is that you have just performed the complete, fundamental act of length measurement, and nothing a tape measure does is different in kind. A tape is simply a rigid body on which somebody else has already laid the pencil end to end a thousand times and printed the count.

That is exactly how the author grounds the distance between two points of a rigid body. You choose, once and for all, one segment, a little rod he calls S, the unit measure. Given two points A and B on a rigid body, you first construct the straight line joining them, by the rules of geometry, say a taut thread or a line of sight, as we saw. Then you lay the rod along that line starting at A, end to end, until you arrive at B. The number of times you had to lay it down is the measure of the segment AB. If a table edge takes the pencil eight times with a little left over, and that little bit is half a pencil, the edge measures eight and a half pencils, and the fraction is itself found by laying down a smaller rod that fits the pencil twice. The author says all measuring of lengths rests on this, and he means it quite literally: the metre, the light-year, the width of an atom, all are counts of a unit rod, however cleverly the counting is hidden.

Now a different question: where is something? Every spatial statement of the place of an event or object, the author says, comes down to naming a point of a rigid body, the reference body, with which the thing coincides. Not only in science, but in everyday life. Take his own example, "in Berlin, at Potsdamer Platz". Unpack it and you find a rigid body, the surface of the Earth, on which a particular point has been marked and given a name, and the event is simply said to coincide spatially with that marked point. Your home address works the same way, and so does the dot on a map in your phone: a marked point on the rigid crust of the Earth.

But this primitive way of giving a place has two weaknesses. It knows only places on the surface of rigid bodies, and it depends on there being distinguishable, named points on that surface. Watch how both limits fall away without the idea of location changing at all. Suppose a cloud hangs over Potsdamer Platz. You could fix its place relative to the Earth by planting a vertical pole on the square, tall enough to touch the cloud. The length of the pole, measured by laying off the unit rod, together with the place of its foot, is a complete statement of where the cloud is.

Three refinements are hiding in that pole, and the author draws them out one by one. First, you extended the rigid body: the Earth plus the pole is a new, larger rigid body that reaches the thing you wanted to locate. Second, you described the place by a number, the measured length of the pole, instead of by a named landmark. Third, and this is the boldest step, you may speak of the height of the cloud even when no pole has been built. In his day, one photographed the cloud from several spots on the ground and, using what is known about how light travels, worked out how long the pole would have to be. Here is your modern instance in full: a GPS receiver never touches a pole or a satellite; it times radio signals from several satellites, applies the known propagation of light, and announces three numbers that tell how long the imaginary rods from the Earth's reference frame to you would be. The number stands in for the rod; the rod is never built; the meaning is still the rod.

So it pays, for describing places, to free oneself from named landmarks altogether and use measured numbers. Measuring physics does this with the Cartesian coordinate system, named after Descartes. Picture three rigid, flat walls, each perpendicular to the other two, like the floor and two adjoining walls in the corner of a room, all fastened to one rigid body. The place of any happening is given, essentially, by the lengths of the three perpendiculars dropped from it onto those three walls, the three coordinates written (x, y, z). What it says in words: go so many units out from one wall, so many from the second, so many from the third, and you are at the event. Where it comes from: it is the pole over Potsdamer Platz three times over, one imaginary pole for each wall. One calculation aloud: a lamp hanging in your room is 2 metres from the left wall, 3 metres from the back wall, and 2.5 metres above the floor, so its coordinates are (2, 3, 2.5), and each of those three numbers is a count of a metre rod laid end to end along a perpendicular. The author stresses that these three lengths are found by a sequence of manipulations with rigid rods, manipulations prescribed by the laws and methods of Euclidean geometry, which is what guarantees that the perpendiculars exist and that one count will match another.

In practice, of course, the three rigid walls are almost never built, and the coordinates are almost never obtained by actually laying rods along perpendiculars. GPS satellites carry no walls, and an astronomer measuring the position of a star drops no plumb lines. Yet the author insists that the physical meaning of every statement of place must always be traced back to this picture of rods and walls, otherwise the results of physics and astronomy dissolve, in his words, into vagueness. Whatever indirect trick produces the number, the number is a promise: had you been able to build the rigid frame and lay the rod, this is the count you would have got.

Pull the threads together and two things stand out. Every spatial description of happenings makes use of a rigid body to which the happenings are referred. And that referring quietly assumes that segments, meaning pairs of marks on a rigid body, obey the laws of Euclidean geometry: that perpendiculars can be dropped, that a rod keeps its length when moved from one wall to another, that counts add up the way school geometry says. Hold onto that assumption, because the whole book will eventually turn on asking whether it is really safe.

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Picture yourself in a railway carriage running smoothly and steadily along a straight stretch of track, standing at an open window. You hold a small stone outside the window and simply open your fingers, giving it no push in any direction. Air resistance aside, what do you see? The stone drops straight down, a vertical line from your hand to the embankment, exactly as it would if the train were standing in the station. Now put a second person on the footpath beside the track, watching the same stone. That person sees something different: the stone leaves your hand already moving forward with the train, keeps that forward speed while gravity pulls it down, and so traces a curve, a piece of a parabola, until it hits the ground.

Both observers are honest and both have good eyes, yet one reports a straight line and the other a curved arc. So Einstein asks the uncomfortable question: which of them is right, where did the stone "really" go, along a line or along a parabola? He admits that the usual statement of what mechanics does, namely that it describes how bodies change their place in space with time, quietly commits what he calls deadly sins against the spirit of clarity. The words "place" and "space" sound familiar, but the stone shows that they carry no clear meaning on their own. If "space" cannot tell us whether the path is straight or curved, then "space" is not doing any work in the sentence.

The way out is the one we built in the previous part. We drop the obscure word "space" altogether, and in its place we say "motion relative to a practically rigid reference body". A place was defined as a spatial coincidence with a point of such a body, and here two bodies are on offer: the carriage and the ground. Relative to the carriage, the stone passes a sequence of points that lie on one vertical line. Relative to the ground, it passes a sequence of points that lie on a parabola. Neither statement contradicts the other, because each is a statement about a different reference body.

For mathematical work it is convenient to replace the phrase "reference body" by "coordinate system", meaning the three rigid perpendicular walls fixed to that body, as we set them up before. Then the result reads cleanly: in a coordinate system rigidly attached to the carriage the stone describes a straight line, and in a coordinate system rigidly attached to the ground it describes a parabola. The lesson, in Einstein's own words, is that "there is no such thing as a path in itself, only a path relative to a particular reference body." Said another way, a trajectory is not a property of the stone alone; it is a relation between the stone and whatever rigid body you choose to measure against. Your GPS receiver reports a track across the Earth's surface, but the same journey, measured in a coordinate system fixed to the orbiting satellites, would be an entirely different curve; neither is the true one and both are correct.

A list of places is still only half a description of motion. To describe the fall completely we must also say, for every point on the path, at what time the stone was there. And that word "time" must be defined in such a way that time values are, at least in principle, things one can observe, results of a measurement, not something taken on faith. On the footing of classical mechanics, Einstein meets this demand simply. Take two clocks built exactly alike; the person at the carriage window holds one, the person on the footpath holds the other. Each of them notes, at every tick of the clock in their own hand, the point of their own reference body with which the stone coincides at that instant. The result is two tables, one pairing ticks with points of the carriage, the other pairing ticks with points of the ground, and each table is a full description of the motion relative to its body.

Einstein then sets aside, for now, the small error that creeps in because the observer sees the stone by light, and light takes time to travel from the stone to the eye. He also mentions a second difficulty lurking here, without naming it yet, and promises that both will be treated at length later. That promise is the doorway to everything that follows in the book, and it is far from academic. Every GPS satellite carries an atomic clock and tags its signals with the time of sending; your receiver works out its position precisely from how long those signals took to arrive, so the finite speed of light is not a nuisance to be ignored but the whole method. And the satellite clocks must be corrected for the fact that they do not keep the same time as clocks on the ground, which is exactly the kind of difficulty Einstein is hinting at.

The footnotes gathered at the end of this section tidy up loose ends from the earlier parts. A straight line, too, is given a natural object: three points of a rigid body lie on a straight line when the middle one is placed so that the sum of its distances to the two outer ones is as small as possible. The counting of unit rods was assumed to come out whole, and the remedy is a divided scale, which brings no new principle. Spatial coincidence needs no deeper analysis, since in real cases people hardly ever disagree about whether it holds. And the last note warns that only general relativity, in the second half of the book, will force these ideas to be refined and changed.

With this the first section of the course closes, and so does the ground-laying part of Einstein's book. We have learned to say what a geometrical statement means, how a length is measured, how a place is named, and now how a motion is described: always against a named rigid body, and always with clocks.

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