Relativity: The Special and the General Theory

Overview

Relativity: The Special and the General Theory is about a hundred and thirty pages in thirty-odd short sections, written in German in 1916 and translated into English in 1920. Einstein says in the preface that it is aimed at readers who are interested in the theory from a general scientific and philosophical point of view but who are not conversant with the mathematics, and that it demands of the reader a fair amount of patience and force of will.

Both halves of that are honest. There is very little mathematics, nothing beyond school algebra and the Pythagorean theorem, and the appendices where the real work is can be skipped. But the ideas are genuinely difficult, not because they are technical but because they require giving up things that feel like facts about the world rather than assumptions.

The date is worth noting. He had published general relativity in its final form in November 1915 and it had not yet been confirmed; the eclipse expedition that measured the bending of starlight and made him famous was in 1919. So this is written by a man confident in a theory the world had not yet accepted.

The method throughout is the thought experiment, and this is the best demonstration anywhere of what that technique can do.

The argument

Part I: special relativity. Einstein begins with the principle of relativity in its old Galilean form, which nobody disputed: if you are in a closed cabin moving smoothly, no mechanical experiment will tell you whether you are moving. He then adds the fact that had been causing trouble for a generation: light in empty space always travels at the same speed, whatever the motion of its source or its observer.

These two look incompatible. If a beam of light passes a train and the train chases it, the train should measure a slower speed, and it does not. Rather than abandon either principle, Einstein asks what would have to be true for both to hold, and follows it.

The answer arrives in the section on simultaneity, and it is the heart of the book. Two lightning bolts strike a railway embankment at points A and B. A man standing midway between them on the embankment sees the flashes arrive together and says they were simultaneous. A passenger on a train moving from A toward B is carried toward the light from B and away from the light from A, and sees B’s flash first, and says B happened earlier. Neither is mistaken. Simultaneity is not a property of the events; it depends on the reference body, and there is no privileged one.

Everything else follows. Moving clocks run slow, moving rods contract in the direction of motion, and the Lorentz transformation replaces the old rules for changing between frames. Einstein is careful to point out that these are not illusions or measurement errors; there is no fact of the matter about a clock’s rate independent of a frame.

He then shows that space and time have to be taken together as a four-dimensional continuum, an idea he credits to Minkowski, and derives the equivalence of mass and energy, presenting the famous relation as a consequence rather than as a separate discovery.

Part II: general relativity. Special relativity only covers observers in uniform motion, which leaves out acceleration and gravity. The route in is the equivalence principle, which Einstein called the happiest thought of his life: a man falling freely does not feel his own weight, and cannot, by any experiment in his immediate surroundings, distinguish free fall from the absence of gravity. Equally, a man in a sealed chest being accelerated in empty space cannot tell his situation from standing still in a gravitational field.

If gravity and acceleration are locally indistinguishable, then anything true of accelerated frames is true of gravity. Einstein works through a rotating disc, where the circumference contracts and the radius does not, so that the ratio of circumference to diameter is no longer pi. Geometry in an accelerated frame is not Euclidean. Therefore geometry in a gravitational field is not Euclidean either, and gravitation is not a force acting across space but a feature of the geometry of space and time, curved by the presence of matter.

He then gives the three predictions that could be tested: the anomalous shift in Mercury’s orbit, which the theory accounted for exactly and which had been an unexplained discrepancy for seventy years; the bending of starlight passing the sun, which Eddington’s expedition confirmed three years after this book appeared; and the slowing of clocks in a gravitational field, not confirmed until much later.

Part III: cosmology. The shortest section, on whether the universe is finite or infinite, in which Einstein argues that general relativity permits a universe that is finite and yet unbounded, like the surface of a sphere in a higher dimension. He is writing before Hubble, so the universe here is static, and the fudge factor he introduced to keep it that way was later described by him as a mistake.

How to read it

Take Part I slowly and do not proceed past the section on simultaneity until the train and the embankment are clear. Draw it. That section is the whole conceptual difficulty of special relativity, and everything afterwards is consequence.

Skip the appendices on a first reading. They contain the derivations and are for readers who want them; nothing in the argument depends on following them.

Be honest about which sentences you have understood. The prose is deceptively easy and it is possible to read a paragraph on the relativity of simultaneity, feel that it went down smoothly, and hold no actual belief at the end of it. The test is whether you can explain the train example to someone else.

Notice what Einstein is unwilling to give up. He keeps two principles and sacrifices absolute time, which everyone had taken to be a condition of thinking at all. That order of priority, holding to principles and letting intuitions go, is the methodological lesson.

Read it after some Newton, even a summary. Relativity is a modification of a specific picture, and knowing what absolute space and absolute time were supposed to be makes their removal intelligible.

Questions it raises

• If simultaneity depends on the observer, what is left of the idea of the present moment?
• Einstein keeps two principles and abandons absolute time. What makes a principle worth keeping when it conflicts with something that feels self-evident?
• Gravity becomes geometry rather than force. Is that an explanation of gravity or a redescription of it?
• The theory made three predictions and they were confirmed. How much should successful prediction settle?
• A falling man feels no gravity, and this is presented as the key to everything. Why should a local indistinguishability tell you about the structure of the universe?
• Einstein introduced a constant to keep the universe static and later called it a blunder. What does that episode show about how theory and expectation interact?