The ABC of Relativity — Chapter-by-Chapter Outline

Author: Bertrand Russell First published: 1925 (Kegan Paul / Allen & Unwin, London) Edition covered: The 1958 revised edition, edited by Felix Pirani (George Allen & Unwin), whose chapter structure is verified against the 1969 3rd revised edition (Allen & Unwin) and the 1985 4th revised edition (New American Library / Mentor). All editions have 15 chapters. The Pirani revision corrected errors, updated the physics, and — most visibly — retitled and rewrote Chapter 11: in the 1925 first edition it was "Is the Universe Finite?"; from 1958 onward it is "The Expanding Universe", reflecting Hubble's established recession of the spiral nebulae. Chapter 4's title is hyphenated as "Clocks and Foot-rules" in the revised editions ("Clocks and Foot Rules" in 1925); the other thirteen titles are unchanged across editions. Chapter content below follows the revised edition's structure, with the 1925 text's argument confirmed from the full first-edition text.

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Central thesis

Einstein's theory of relativity is not a mathematical mystery that only physicists can grasp, but a demand for a change in our imaginative picture of the world — a picture inherited from pre-human ancestors and learned in childhood. Russell's project is to rebuild that picture in plain language, without a single equation, so that a non-mathematical reader can see why the old ideas of space, time, motion, force, and matter had to be replaced, and what the new ideas mean.

The book's core claim is that physics must describe what belongs to physical occurrences themselves, and not what merely belongs to the observer's point of view. Einstein's achievement was to show that far more than we thought — including distance, duration, simultaneity, and mass — depends on the observer, and then to find what remains: an objective residue expressed in a four-dimensional structure called space-time. From this residue, gravitation emerges not as a force acting at a distance but as the geometry of space-time itself: bodies move along the "easiest" paths through a world that is not flat, and the laws of nature turn out to be far more abstract, and our knowledge of the intrinsic nature of things far more limited, than pre-relativity physics supposed.

If a man is twice as rich as another, this fact must appear equally whether you estimate the wealth of both in pounds or dollars or francs — the numbers change, but one number is always double the other. The theory of relativity is concerned to find what corresponds, in physics, to the "double" — the laws that are the same for all observers — and to discard everything that changes with the currency of the point of view.

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Chapter 1 — Touch and Sight: The Earth and the Heavens

Central question

Why is relativity so hard to understand, and why did physics need to abandon ideas that were built on the sense of touch?

Main argument

The imaginative difficulty Russell opens by stating the book's premise: everybody knows Einstein has done something astonishing, but very few people know exactly what. Popular accounts, he says, "generally cease to be intelligible just at the point where they begin to say something important." The difficulty is not mathematics but imagination: relativity demands a change in a picture of the world handed down from remote, perhaps pre-human, ancestors. The same sort of change was demanded by Copernicus; Einstein's ideas will seem easy to a generation that grows up with them.

The balloonist's world To show how much our physics rests on the accident of having senses of touch and sight together, Russell imagines a drugged man waking in a balloon on a dark night, able to see fireworks launched from ground, trains, and aeroplanes but unable to see the sources themselves. He would think nothing is permanent — only brief flashes of light. His geometry, physics, and metaphysics would differ entirely from ordinary mortals'. Yet he would understand Einstein more easily, being "free from a host of preconceptions."

King's Cross and Edinburgh On the earth's surface, most large objects stay put, and that luck — not logic — makes "place" seem definite. If houses moved like a swarm of bees and railways changed shape like avalanches, a journey from King's Cross to Edinburgh would have no meaning; you would ask the taxi-driver "Where is King's Cross this morning?" and the booking clerk would answer that Prince's Street has gone to Glasgow. The notion of "place" is a rough practical approximation that "cannot be made precise."

Einstein and Copernicus In astronomy, everything moves relative to everything else — the sun races toward Hercules faster than an express train, the "fixed" stars scurry "like a lot of frightened hens" — and there are no well-marked places. The question of which body is the "train" and which the "station" is one of convenience. Russell draws the provocative conclusion that Copernicus's dispute with his predecessors is "merely one of convenience": "the earth rotates once a day" and "the heavens revolve about the earth once a day" mean exactly the same thing, just as six feet means the same as two yards. All motion is relative; absolute motion is a fiction.

Force from touch The notion of "force" seemed intelligible because it was tied to muscle sensations and obvious pushing and pulling. But Newton's gravitation — a force between bodies ninety-three million miles apart — was mysterious even to Newton, who thought action at a distance impossible. Russell declares flatly: "The fact is that the whole conception of 'force' is a mistake." The sun exerts no force on the planets; in Einstein's law, the planet only pays attention to what it finds in its own neighborhood. Physics has advanced by finding sight less misleading than touch as a source of fundamental notions.

Key ideas

• Relativity's difficulty is imaginative, not mathematical: it requires replacing a childhood picture of the world.
• "Place" is not a logical necessity but a rough approximation dependent on the accidental stability of large earthly objects.
• All motion is relative; "the earth rotates once a day" and "the heavens revolve once a day" are the same statement in different words.
• The Copernican versus Ptolemaic dispute is a dispute about convenience, not about truth.
• "Force" is a concept derived from touch that physics must abandon; gravitation is a property of the neighborhood, not an influence from a distant body.
• Astronomy is physics stripped to sight alone, and it was there that the touch-based preconceptions broke down.

Key takeaway

Before any technicalities, the reader must accept that the world-picture built on touch and terrestrial stability is a set of useful fictions that relativity replaces.

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Chapter 2 — What Happens and What Is Observed

Central question

If different observers perceive the same occurrence differently, how can physics describe the occurrence itself rather than each observer's private view?

Main argument

"Everything is relative" is nonsense Russell immediately disposes of the popular slogan. "Everything is relative" is nonsense, "because, if everything were relative, there would be nothing for it to be relative to." The theory of relativity, he insists, "is wholly concerned to exclude what is relative and arrive at a statement of physical laws that shall in no way depend upon the circumstances of the observer." The name of the theory is unfortunate; it has led philosophers and uneducated people into confusions.

Three kinds of difference between observers When two observers perceive "one" occurrence, their perceptions differ in three ways: psychological (one man notices beauty, another waterfalls), physiological (long-sighted and short-sighted men), and physical. Only the physical kind matters — it survives when observers are replaced by cameras and phonographs. The nearer hearer hears louder and slightly earlier; two men see a falling tree from different angles. The physicist, like the plain man, believes his perceptions give knowledge of what really occurs — Einstein has made no difference in that respect — so physics must concern itself with the features an occurrence has in common for all equally good observers.

Light's finite speed changes the accounting Physical differences matter because light takes time to travel: 300,000 kilometers — about 186,000 miles — per second; eight minutes from the sun, three to a thousand years from the stars. On the earth's surface we treat seeing as instantaneous; in astronomy we cannot. The old physics allowed colors and noises to be subjective but kept shapes, positions, and time as objective properties. Russell argues this doctrine worked only while physics dealt with velocities far below light's. At the new speeds — electrons from radium at 170,000 miles a second — mass seems to increase with speed; a moving cube appears shorter in the direction of motion; and two accurate clocks that separate and reunite will not agree. "What we discover by means of clocks and foot rules, which used to be regarded as the acme of impersonal science, is really in part dependent upon our private circumstances."

The generating principle The single principle that generates the whole theory: physics must describe what a process has in common for all observers, so "the laws of phenomena should be the same whether the phenomena are described as they appear to one observer or as they appear to another." Russell's analogy: a man's being twice as rich as another must appear the same in any currency. Einstein found the mathematical instrument — the theory of tensors — for expressing laws in terms of the objective residue, and where his laws differed from the old ones, observation has favored him.

Key ideas

• The theory of relativity does not make everything relative; it identifies what is relative in order to exclude it from the laws of physics.
• Physical (not psychological or physiological) differences between observers' perceptions are the subject matter of the theory.
• Light's finite velocity means what an observer sees is a mix of the event and the point of view; at terrestrial speeds the mix was harmless, at astronomical and electronic speeds it is not.
• Clocks and foot rules do not reveal bare facts; they reveal relations between things and the measurer.
• A body's mass and length appear altered by its motion relative to the observer, and clocks in relative motion disagree.
• Physics must be expressed in laws that survive a change of reference body, exactly as a fortune's size survives a change of currency.

Key takeaway

Relativity is the systematic program of subtracting the observer's contribution from our measurements, and its surprising results are the cost of doing that subtraction honestly.

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Chapter 3 — The Velocity of Light

Central question

Why is the constancy of light's velocity — the same for all observers, however they move — the fact that forces the whole reconstruction of physics?

Main argument

The measured speed of light The finite velocity of light was first established by astronomy: eclipses of Jupiter's moons arrive a few minutes early when Jupiter is near, a few minutes late when remote, all accounted for by a constant speed of light in vacuo of almost exactly 300,000 kilometers a second. The same velocity belongs to wireless waves and X-rays; it is generally held to be the velocity with which gravitation itself is propagated.

The ether wind that was not there If light is waves, the waves were supposed to be waves in the ether, and their velocity ought to be relative to the ether. Since the ether offers no resistance to the planets, it presumably does not share the earth's motion — so the earth should experience an "ether wind," like a man walking against the wind on a windy day. Michelson and Morley tested this by sending light signals in two directions at right angles and reflecting them back: just as it takes longer to row upstream and back than across the stream and back, one beam should have lagged. No difference appeared, in 1881, again in 1887, and in every repetition. "The result was a surprise to them as to everybody else."

The Fitzgerald contraction and the White Knight Physicists first tried arbitrary hypotheses, above all the Fitzgerald contraction (developed by Lorentz): a body in motion shortens in the direction of motion by just enough to hide the missing effect — undetectable, because measuring rods share the shortening. Russell likens this to the White Knight's "plan to dye my whiskers green, and always use so large a fan that they could not be seen." The plan worked; Einstein later showed the contraction is not a physical fact but a result of conventions of measurement.

What the experiment really shows The plain meaning of Michelson-Morley and its successors: relatively to the earth, light travels at the same speed in all directions at all times of year, and any body, however it moves, remains at the center of the waves it sends out — as judged by observers moving with it. Russell's analogies: if sound behaved like light, a traveler on a train would hear an echo from the engine after the same interval whether at rest or rushing toward it; on an escalator moving at light-speed you would reach the top at the same moment whether you walked or stood still; a motor-car traveling at light-speed would be 186,000 miles from you after a second whether it met you or passed you; a fly touching a stagnant pool is always the center of its own ripples — while a physicist by the pool judges the center to be the point of contact. "This is exactly analogous to the Michelson-Morley experiment."

The only way out: clocks are affected by motion Such facts leave one exit: watches and clocks are affected by motion, fundamentally — not by flaws in construction. Two equally careful observers in relative motion may disagree about how long an hour is, and neither is wrong, "any more than you could if one used a clock showing Greenwich time and another a clock showing New York time."

Light as the unattainable limit No material body can ever travel as fast as light. Russell's moving-platform analogy: a series of platforms each moving four miles an hour faster than the previous one, with a man stepping from one to the next — you might expect speeds to add indefinitely, but they fall a little short, and no number of platforms ever reaches the velocity of light. "The discrepancy... makes the velocity of light an unattainable limit." (A 1925 footnote notes Dayton Miller's challenge to Michelson-Morley, then sub judice, discussed unfavorably by Eddington.)

Key ideas

• The speed of light is not merely large but equal for every observer, in every direction, regardless of the observer's motion.
• The ether wind expected by nineteenth-century physics was not found; the Michelson-Morley experiment is the pivotal fact.
• The Fitzgerald contraction "explained" the result only by a convention that could never be detected — a warning that measurements are conventions, not revelations.
• If light behaved like sound, echoes, escalators, and passing cars would behave normally; their behaving otherwise shows the depth of the break with common sense.
• The constancy of light's velocity forces the conclusion that time itself is affected by motion.
• Light's velocity is an absolute limit that no material body can reach, however long a force acts.

Key takeaway

The one experimental fact — light's velocity is the same for all observers — is impossible to reconcile with old ideas of space and time, and everything strange in relativity follows from accepting it.

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Chapter 4 — Clocks and Foot-rules

Central question

Can two events in different places be said to happen "at the same time," and what actually does a clock measure?

Main argument

Simultaneity is not what it seemed Until the special theory, nobody thought there was ambiguity in "two events in different places happened at the same time." It turned out to be a mistake: two distant events may appear simultaneous to one careful observer, while another equally careful observer, moving rapidly relative to the first, judges the first event to precede the second — and a third the reverse. "They would all be equally right." The time order of events is in part dependent on the observer.

The brigands and the millionaire's will The natural definition: events are simultaneous when seen simultaneously by a person halfway between them — two lightning flashes at Greenwich and Kew, seen from St. Paul's dome. But Russell makes the case vivid with a sound-based version: on a foggy night, two brigands shoot the guard and the engine driver of a train. An old gentleman exactly in the middle of the train hears both shots together; a station master halfway between the brigands hears the guard's shot first. An Australian millionaire uncle has left his fortune to whichever cousin survives. The House of Lords must decide; both witnesses are right. The train recedes from one shot and advances toward the other.

No parochial prejudices We earth-dwellers prefer an earth-based definition, but in theoretical physics "no such parochial prejudices are permissible." A physicist on a comet has as much right to his simultaneity as one on earth — the train is not "really" in motion any more than the earth; "there is no 'really' about it." Russell's rabbit-and-hippopotamus argument: each thinks its own size natural, but there is as little substance in arguing whether the earth or the train is "really" moving.

Proper time and the end of cosmic time The universal cosmic time taken for granted by historians (the eclipse of August 29, 776 B.C.) and astronomers is no longer admissible. Each body has its own proper time — the time order of events in its own neighborhood. Terrestrial time is just the proper time of large bodies on the earth; a Beta particle in a laboratory would want a different time, and from its point of view "it is we who suddenly grow thin or corpulent. The history of a physicist as observed by a Beta particle would resemble Gulliver's travels."

What a clock measures A clock, in relativity, is anything with a regular periodic performance: the earth rotating, an atom whose electrons orbit the nucleus. Each clock gives a correct measure of its own proper time — an important physical quantity — but not of events on bodies moving rapidly relative to it. Distances in space, like periods of time, are partly observer-dependent; the distance between two events replaces the distance between two bodies (the sun-dweller and the flea give opposite accounts of the London–Edinburgh journey).

The interval: the one objective quantity The book's central technical notion appears here. Between any two events, either a body could physically travel to be present at both (the interval is time-like) or it could not (the interval is space-like); when one event is the seeing of the other, the interval is zero. The interval is a physical fact about the two events, "not dependent upon the particular circumstances of the observer," and it is computed from an observer's own measures of distance and time by a simple geometric construction (a circle drawn with radius the distance light travels in the time) — the same result for every observer. This single quantity replaces the two "intervals" — one in space, one in time — of the older physics.

Key ideas

• Simultaneity between distant events is not a fact of nature but a convention tied to an observer's state of motion.
• The brigands story shows that two incompatible judgments of simultaneity can both be correct.
• "Proper time" is the time a body's own clocks measure; universal cosmic time is an illusion.
• A clock is any periodic process — an atom, a planet, a rotating earth — not just a manufactured instrument.
• Distance between events, like time between events, is observer-dependent; neither separately is objective.
• The interval between two events — time-like, space-like, or zero — is the one measurable relation that all observers agree on.

Key takeaway

What looked like the acme of impersonal science — measuring with clocks and foot rules — is shot through with the observer's motion; the objective residue is the single quantity called the interval.

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Chapter 5 — Space-Time

Central question

What does it really mean to replace "space and time" with "space-time"?

Main argument

Four quantities, not three plus one Everyone who has heard of relativity knows the phrase "space-time," but few non-mathematicians know what it means. To fix where and when an event occurs — an explosion on an airship — you need four quantities: latitude, longitude, height, and time. The old view treated the first three as position in space, the fourth as position in time, with the two reckonings wholly independent. Relativity changes this: the space and time reckonings are no longer independent. Alter the way of reckoning position in space and you may alter the time interval between events; alter the time reckoning and you may alter a spatial distance. Four quantities are still needed, but they can no longer be divided into three independent ones plus a fourth.

The Sirius example and "contemporary" events Russell pictures an event E happening to him while a flash goes out in all directions. Anything that happens after the light reaches it is definitely after E for everyone; anything he could have seen before E is definitely before. But events in the intervening period — a person in Sirius acting between his seeing E and E's happening — "is not definitely before or after E." Because light takes years to travel from Sirius, there is a period of twice as many years in Sirius that may be called "contemporary" with E. A. A. Robb's theory gives the philosophical point: one event is definitely before another only if it can influence it. Newspapers spread influence at twenty miles an hour, sound faster, wireless at the velocity of light — nothing faster is ever to be hoped for. So when a notable event happens on the sun, there are sixteen minutes on the earth that are neither before nor after it.

Measuring moving bodies: the Fitzgerald contraction explained In daily life we measure a body by applying a foot rule while it is at rest relative to us, so we always get the proper length. "Not being Joshua, we cannot make the sun stand still while we measure it." For bodies in rapid motion the results are strange: lengths in the direction of motion are shortened, at right angles unaffected, and everything is reciprocal — each observer sees the other's lengths shrunk. Russell shows geometrically that the ratio depends on the body's speed relative to light (the OP/MP figure). The contraction is no longer a hypothesis invented to rescue Michelson-Morley: "it now emerges naturally from the fact that the two observers do not make the same judgment of simultaneity. ... the trouble about time is at the bottom of the trouble about distance."

Why "space-time" is forced on us The old separation of space and time rested on the belief that simultaneity was unambiguous. Once simultaneity is relative, the description of "the state of the world at a given instant" is relative too. Physics must deal in events — things that happen at a given time — because "there is no such thing as the 'same' time for different observers, unless they are at rest relatively to each other." Four measurements fix the position of an event in space-time; "three measurements are not enough to fix any position. That is the essence of what is meant by the substitution of space-time for space and time."

Key ideas

• "Space-time" is not a literary flourish: the four numbers fixing an event can no longer be split into an independent space-part and time-part.
• There are events that are definitely before or after a given event, and others — over a wide "contemporary" band — that are neither.
• Before and after, in the definite sense, correspond to what can and cannot be influenced: nothing travels faster than light.
• Apparent length contraction is a consequence of disagreement about simultaneity, not a physical squeezing of bodies.
• All the strange ratios — lengths, times, masses — depend on one ratio: the body's speed compared to light's.
• Physics must describe events, not bodies-at-an-instant, because there is no universal "now."

Key takeaway

The replacement of space and time by space-time is not a slogan but the unavoidable consequence of taking the relativity of simultaneity seriously.

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Chapter 6 — The Special Theory of Relativity

Central question

How does the special theory of relativity account for the experimental facts of electromagnetism, and what exactly does it predict?

Main argument

The electromagnetic background The special theory arose from the facts of electromagnetism. Eighteenth-century electricity was dominated by the Newtonian analogy of inverse-square action at a distance, until Faraday's experiments showed the role of the intervening medium and Clerk Maxwell gave them mathematical form, proving that light is an electromagnetic phenomenon. Hertz then manufactured electromagnetic waves — the basis of wireless telegraphy. But just as the ether seemed secure, the discoveries of the electron, the proton, and quanta showed that nature is discontinuous, not the continuous world the ether program had hoped for. The problems solved by special relativity are typified by Michelson-Morley: Maxwell's equations were "more accurate than they should be" — valid for every observer regardless of motion through the (supposed) ether.

Galileo's precedent The same sort of puzzle was solved by Galileo at the start of modern physics: a weight dropped in the cabin of a steadily moving ship falls exactly as if the ship were at rest. Uniform motion in a straight line has no discoverable effects. Einstein's problem was harder: to show how electromagnetic phenomena could likewise be unaffected by uniform motion — and this could not be done by merely extending Galileo, because time itself had to be rethought.

The Lorentz transformation The quantitative solution is the Lorentz transformation — found as a formula by Lorentz, "interpreted and made intelligible by Einstein." It answers the question: given one observer's estimates of where and when an event occurred, what will another observer, moving uniformly relative to the first, judge? Russell gives the conditions the solution must satisfy — light's velocity the same for all observers, the laws of electromagnetism the same for all observers, and all effects reciprocal — then illustrates with the train traveling at three-fifths the velocity of light.

Trains, dinner plates, and fishing rods To the traveler, his own train is a hundred yards long; the people who glimpse it as it passes calculate it at eighty yards. Dinner plates, circular to the diners, look oval to the outsider. A man outside holds a fishing rod: upright it looks fifteen feet, along the railway it looks twelve. "All lengths in the direction of motion are diminished by twenty per cent, both for those who look into the train from outside and for those who look out of the train from inside."

Eddington's aviator and the cigars Time is stranger still. Russell quotes Eddington's aviator traveling at 161,000 miles a second: "If we observed the aviator carefully we should infer that he was unusually slow in his movements... His cigar lasts twice as long as one of ours." And reciprocity: "in the aviator's opinion it is we who are traveling at 161,000 miles a second past him... Our cigar lasts twice as long as his." Russell's gloss: "What a situation for envy! Each man thinks that the other's cigar lasts twice as long as his own. It may, however, be some consolation to reflect that the other man's visits to the dentist also last twice as long."

The two-fold correction and the new stars For events along the railway, the traveler makes a two-fold correction to the earth-dweller's dates: multiply the time by five-fourths, then subtract three-fourths of the time light took to arrive. The consequences are dramatic: if two new stars flare up, one ahead of the train and one behind, and the earth-dweller infers they exploded fifty and a hundred years ago respectively, the traveler "will exactly reverse these figures." Both argue correctly on correct data; both are right, "unless they imagine that the other must be wrong." Both nevertheless agree on the velocity of light, because distance estimates shift in exactly the same proportion as time estimates. "The theory of relativity is just as little startling as is compatible with the facts."

The interval as the reality Although distances and times vary between observers, the interval — the square of the distance between events minus the square of the distance light travels in the time between them — is the same for all. It is time-like, space-like, or zero as before. Russell: "we are tempted to say that it is the reality of which distances and periods of time are confused representations." A closing section, for readers with elementary geometry or algebra, derives the Lorentz transformation explicitly.

Key ideas

• Special relativity solves one definite problem: making the laws of physics — especially electromagnetism — identical for bodies in uniform relative motion.
• Maxwell's equations are true for any observer, and the Lorentz transformation is the exact change of measurements that makes this possible.
• Length contraction and time dilation are reciprocal: each observer sees the other's lengths shortened and clocks slowed.
• Simultaneity dissolves into the two-fold correction: forward events are ante-dated, backward events post-dated, by amounts proportional to light-travel time.
• Two observers can disagree completely about which of two explosions happened longer ago — and both be right.
• The interval, not distance or time, is the invariant physical reality underlying both.

Key takeaway

The special theory is the complete, coherent description of what a world looks like in which the velocity of light is the same for everyone — and its predictions are stranger than fiction but verified by experiment.

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Chapter 7 — Intervals in Space-Time

Central question

How does the special theory generalize into the general theory, and why must geometry itself be rebuilt to get there?

Main argument

From uniform to accelerated motion The special theory solved the problem of uniform motion in a straight line. But what of accelerated motion — the earth itself, "in an extended sense, a falling body," perpetually accelerated toward the sun? In a steadily moving train or lift nothing seems amiss; at starting and stopping you feel it in the pit of the stomach, and a weight dropped in an accelerating ship falls in a curve. Galileo and Newton treated accelerated motion as radically different from uniform — but that distinction "could only be maintained by regarding motion as absolute, not relative." The people on the ground have no sensations in the pit of the stomach when the lift starts. Einstein's general theory (1905 special, 1915 general — ten years of extraordinarily difficult work) removes the restriction that the observer must be unaccelerated, and in doing so produced the new law of gravitation.

From Pythagoras to Gauss: the genealogy of interval The notion of interval is a generalization of the theorem of Pythagoras, and Russell walks through the generalization in full. Pythagoras himself may never have existed; the theorem generalized the Egyptian rule that a 3-4-5 triangle is right-angled. Descartes built analytical geometry on it — in New York, you are told to go so many blocks east and so many north, and the theorem gives the direct distance. But on a sphere the theorem fails: a triangle made of the equator and two meridians has three right angles; walking 100 miles west, north, east, and south does not return you to your starting point; a ship from New York to Lisbon sails a great circle. These are intrinsic differences — discoverable from measurements on the surface alone. Gauss developed this into the theory of surfaces: whatever coordinates you use, a formula (a generalization of Pythagoras) gives small distances, and from it all intrinsic properties follow. Geodesics — the lines of shortest route on a surface — take the place of straight lines.

Riemann and non-Euclidean space Riemann's 1854 dissertation extended Gauss from surfaces to three-dimensional spaces: all the essential characteristics of a kind of space can be deduced from the formula for small distances. There is no a priori reason why space should be Euclidean; "whether this does happen or not, can only be discovered by actual measurements." Einstein's step is the final generalization: substitute the interval between events for the distance between points. In the general theory the interval is assumed only for neighboring events, in a general (Riemannian) form that reduces to the special-theory form at a great distance from matter. "What goes beyond these initial assumptions depends upon observation of the actual motion of bodies." With that, Russell announces: "We are now at last in a position to tackle Einstein's theory of gravitation."

Key ideas

• The general theory of relativity removes the special theory's restriction to uniform motion and took Einstein from 1905 to 1915 to complete.
• The interval is the latest descendant of the theorem of Pythagoras, via Descartes, Gauss, and Riemann.
• On a sphere, angles of triangles exceed two right angles, "straight" lines are great circles, and the square on the hypotenuse is less than the sum of the squares — all discoverable intrinsically.
• Non-Euclidean geometry is an empirical possibility, decidable only by measurement, not by reasoning.
• In the general theory the interval formula is assumed only for neighboring events and in full generality.
• Near matter, space need not be Euclidean; the local structure of space-time is something to be discovered from the motion of bodies.

Key takeaway

The interval between neighboring events is the generalization of Pythagoras on which the general theory is built, and its non-Euclidean character in the presence of matter is exactly what gravitation turns out to be.

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Chapter 8 — Einstein's Law of Gravitation

Central question

Why must Newton's law of gravitation be wrong, and what does Einstein put in its place?

Main argument

Newton's law cannot be quite right Newton's law says the force between two particles is proportional to the product of their masses and inversely proportional to the square of their distance — "the distance at a given time." But we have seen that "distance at a given moment" is a subjective conception: what one observer judges to be the same moment on earth and sun, another judges to be two different moments. A cosmic law cannot depend on so parochial a convention as Greenwich time; "this is as absurd as it would be if the question whether one man had murdered another were to depend upon whether they were described by their Christian names or their surnames."

Coordinates are conventions To build the new law we must not assume we know how to measure anything. Coordinates are merely a systematic way of cataloguing events — so long as they are continuous. Russell's illustrations: numbering American houses by blocks would have sudden jumps; so would dating events by the births of successive people called Smith. His favorite image is the live eel: "instead of using a steel measuring-rod to fix our co-ordinates, let us use a live eel, which is wriggling all the time" — the distance from tail to head counts as one unit whatever shape it takes. The eel is just as good as the steel rod, "not because eels are really rigid, but because steel rods really wriggle." To an observer in one possible state of motion the eel would appear rigid and the rod wriggle; "there is no saying that one observer is right and another wrong." Geometry ceases to exist as a separate science and becomes merged in physics.

The three postulates The law is found by combining this coordinate freedom with the known facts, and Russell states it as three postulates:

• Every body travels in a geodesic in space-time, except in so far as electromagnetic forces act upon it.
• A light ray travels so that the interval between two parts of it is zero.
• At a great distance from gravitating matter, coordinates can be chosen so that the interval takes the special-theory form; this is approximately true wherever gravitation is not very powerful.

Cosmic laziness A body left to itself chooses the route that makes the time between two stages of its journey as long as possible — by its own clocks. "If you leave London at 10 a.m. and arrive in Edinburgh at 6.30 p.m. Greenwich time, the more slowly you travel the longer you will take — if the time is judged by your watch." A ray of light doing the journey would judge it to take exactly no time. "Bodies left to themselves do their journeys as slowly as they can; it is a sort of law of cosmic laziness." Its mathematical expression is the geodesic.

The hill in space-time Near a piece of matter, space-time is like a hill that grows steeper toward the top, "like the neck of a champagne bottle," ending in a sheer precipice. A body coming near will not go straight over the top but will go round. "This is the essence of Einstein's view of gravitation. What a body does, it does because of the nature of space-time in its own neighborhood, not because of some mysterious force emanating from a distant body." The lantern analogy: on a dark night, men with lanterns cross a plain with a hill and a beacon on top; watched from a balloon, they seem to swerve as they approach the beacon — you might think the beacon repels them; at daylight you see the hill. "In this analogy the beacon corresponds to the sun... and the coming of daylight corresponds to the coming of Einstein." The sun is at the top of a hill, only the hill is in space-time, not in space. "Each little bit of matter is at the top of its own little hill, like the cock on his own dung-heap. What we call a big bit of matter is a bit which is at the top of a big hill. The hill is what we know about; the bit of matter at the top is assumed for convenience."

The abolition of action at a distance The law no longer involves action at a distance: "the sun exerts no force on the planets whatever. Just as geometry has become physics, so, in a sense, physics has become geometry."

Key ideas

• Newton's law uses "distance at a given time," a notion shown to be observer-dependent and therefore unfit for a cosmic law.
• Coordinates are pure conventions — a live eel is as good a unit as a steel rod — so laws must be expressed independently of them (tensors).
• The law of gravitation reduces to three postulates about geodesics, light rays, and the asymptotic special-theory form of the interval.
• Bodies left to themselves move as slowly as possible — the "law of cosmic laziness" — which is the geodesic principle.
• Gravitation is a hill in space-time: bodies deviate from Euclidean straightness because of the neighborhood they are in.
• Action at a distance is abolished: physics becomes geometry.

Key takeaway

Einstein's law of gravitation replaces a mysterious force across empty space with the simple statement that bodies follow the easiest paths through a curved space-time.

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Chapter 9 — Proofs of Einstein's Law of Gravitation

Central question

On what evidence — empirical and logical — is Einstein's law of gravitation accepted over Newton's?

Main argument

The perihelion of Mercury Einstein's law gives very nearly the same results as Newton's for the orbits of the planets — it must, since Newton's consequences are almost exactly verified. In 1915, when Einstein published, there was exactly one empirical fact in his favor: the motion of the perihelion of Mercury. Mercury's orbit is an ellipse with the sun at a focus, and from one perihelion (nearest approach) to the next the planet travels a little more than once round. The discrepancy is forty-two seconds of angle per century — less than half a second per year — after all perturbations by other planets are allowed for. It had puzzled astronomers; Einstein's theory accounted for it exactly, and for its (unobservably small) absence in the other planets.

The deflection of light, 1919 The second success was sensational. Orthodox opinion held that light in vacuo travels in straight lines and is unaffected by gravitation; even allowing it to be bent like a stream of particles gave half the deflection Einstein's law demanded. Einstein predicted that a star's ray passing very near the sun would be turned through just under one and three-quarters seconds. Such a star can only be seen during a total eclipse — and Eddington noted that the best day of the year for bright stars near the sun is May 29. "It happened by incredible good fortune that there was a total eclipse of the sun on May 29, 1919 — the first year after the armistice." Two British expeditions photographed the stars near the sun during the eclipse; the results confirmed Einstein's prediction, and doubters were convinced when their own observations in a subsequent eclipse gave the same result. The estimate is now universally accepted.

The displacement of spectral lines The third test is on the whole favorable but not decisive: the quantities are so small they are only just measurable. Russell explains the spectrum and the Doppler effect — the whistle of an approaching express seems shrill, receding, flat — and distinguishes the ordinary Doppler shift from Einstein's prediction: any periodic process in an atom on the sun, where gravitation is intense, must, measured by our clocks, run slightly slower than in a similar atom on earth, so its spectral lines ought to be displaced slightly toward the red. "It now seems highly probable that it exists." (In the revised editions this section is updated with the later confirmation and its quantitative details.)

The logical grounds But the considerations that originally led Einstein to his law were not these detailed results; they were more abstract. The relativity of motion is conclusive: "There is no physical occurrence which can be called 'absolute motion.'" Combined with the constancy of light's velocity, this yields the relativity of distances and times, and hence the collapse of "distance at a given time" — Newton's law is logically untenable. What should replace it? Here enters the equality of gravitational and inertial mass: in a given gravitational situation all bodies behave exactly alike — Galileo's discovery that a feather falls as fast as a lump of lead in a vacuum, Newton's that a comet at a given distance from the sun experiences exactly the same acceleration as a planet. "The way in which gravitation affects a body depends only upon where the body is, and in no degree upon the nature of the body. This suggests that the gravitational effect is a characteristic of the locality, which is what Einstein makes it."

Tensors and the choice of law Finally, if the law is to be a characteristic of a neighborhood, it must be expressed in a formula unchanged however we change coordinates — the business of the theory of tensors. Tensors show that one formula suggests itself as possibly the law of gravitation; examined, it gives the right results. "If Einstein's law had not been found to agree with experience, we could not have gone back to Newton's law. We should have been compelled by logic to seek some law expressed in terms of 'tensors'." Russell closes: of the method of tensors, "Einstein's law of gravitation is the most splendid example."

Key ideas

• Three empirical tests: the perihelion of Mercury (42 seconds per century), the eclipse deflection of starlight (1.75 seconds, confirmed May 29, 1919), and the gravitational red shift of solar spectral lines.
• The 1919 eclipse expeditions — two British parties, Eddington among them — made Einstein famous overnight by confirming the doubled deflection.
• Newton's law is logically untenable because it presupposes an observer-independent "distance at a given time."
• The equality of gravitational and inertial mass — feather and lump of lead, comet and planet — shows gravitation is a property of the locality, not of the body.
• Tensor methods guarantee laws are independent of the choice of coordinates; the law of gravitation is their most splendid example.

Key takeaway

Einstein's law of gravitation is accepted because it explains the one anomaly Newton could not (Mercury), predicts effects Newton's could not (bent light, red shift), and is forced by logic — not merely by experiment — once motion is recognized as relative.

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Chapter 10 — Mass, Momentum, Energy and Action

Central question

How do the fundamental quantities of classical mechanics — mass, momentum, energy, and action — change when space-time replaces space and time?

Main argument

Mass and weight For daily life mass is much the same as weight, but accurate measurement forces a distinction. Scales measure mass — the same pound of feathers and pound of lead everywhere; a spring balance measures weight, the force of the earth's attraction, which varies from place to place: more at the poles, less at the equator, less high in an aeroplane, less at the bottom of a coal mine (where part of the earth pulls upward). Theoretically, mass is defined by the force needed to produce a given acceleration: "It takes a more powerful engine to make a long train attain a speed of ten miles an hour... than it does to make a short train do so."

Mass is relative — and mass is energy Radio-active bodies emit beta-particles (electrons) with enormous velocities; their paths can be traced through water vapor, their masses compared by bending them with known electric and magnetic forces. The faster they travel, the greater their measured mass — in exactly the same proportion as lengths and times change. The remedy is the proper mass, measured by an observer sharing the body's motion. But even the proper mass is not strictly constant: "When a body absorbs energy — for example, by growing hotter — its 'proper mass' increases slightly," the increase being the energy divided by the square of the velocity of light. The most notable case: four hydrogen atoms combine into one helium atom, which has rather less than four times the mass of one hydrogen atom. Mass becomes absorbed into energy: "it represents, so to speak, the energy which the body expends internally, as opposed to that which it displays to the outer world."

Conservation of momentum and energy The three great conservation principles of classical mechanics fare differently. Conservation of momentum remains true: for different observers mass differs but so does velocity, and the two differences neutralize. From the space-time point of view, the measured mass is "the momentum in the time direction," naturally paired with momentum in space directions. Conservation of energy is subtler: potential energy cannot be adapted to relativity, so strict conservation cannot be maintained — but relativity finds a new formula for kinetic energy alone which expresses exactly the phenomena classical mechanics attributed to conservation. Russell quotes Eddington: "The relativity treatment adheres to the physical quantity and modifies the law; the classical treatment adheres to the law and modifies the physical quantity."

What "conservation" means In practice conservation means: a quantity in any region changes only by crossing the region's boundary — like population, which changes only by births, deaths, and migration. Russell's cautionary tale: a physiologist friend "once put four mice into a thermos. Some hours later, when he went to take them out, there were eleven of them. But... the mass of the eleven mice at the end of the time was no greater than the mass of the four at the beginning." Mass, unlike mice, does not fluctuate.

Energy is measured mass With the velocity of light as unit, the measured mass of a body is its proper mass divided by the square root of (1 − v²), while traditional kinetic energy is ½mv²; adding any constant is permissible since energy is a profit-and-loss account. The two expressions become indistinguishable at ordinary speeds and identical in exact form: "energy and measured mass become identified."

Action Action is energy multiplied by time — a measure of how much has been accomplished. The quantum is a small amount of action. In relativity, since a "region" becomes a volume lasting for a time, the density of matter in a region yields an amount of action — explaining why action should be fundamental. All the laws of dynamics have been gathered into the Principle of Least Action: a body passes from one state to another by a route involving less action than any slightly different route — "again a law of cosmic laziness." The quantum's being a unit of action suggests action is also fundamental empirically, though "at present there is no bridge connecting the quantum with the theory of relativity."

Key ideas

• Mass is not quantity of matter: measured mass is relative to the observer, and even proper mass varies with energy content.
• Mass and energy are the same thing; the change of mass with energy is measured by dividing the energy by the square of the velocity of light.
• Conservation of momentum survives relativity; conservation of energy survives only in a modified, less rigorous form.
• "Conservation" in practice means flow across boundaries, not a directly observed constancy of the whole world.
• Action — energy multiplied by time — becomes the fundamental quantity of relativity mechanics, and the quantum is a unit of it.
• The Principle of Least Action unifies all of dynamics and is another expression of cosmic laziness.

Key takeaway

The classical trio mass, momentum, and energy collapses into two: momentum and its time-component mass — that is, energy — and the deepest law of mechanics turns out to be the Principle of Least Action.

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Chapter 11 — The Expanding Universe

Central question

Is the universe finite, and what do the relativistic models — Einstein's and De Sitter's — say about its large-scale structure?

Main argument

Speculation versus acquired science Russell opens by drawing the line between results "pretty certainly in the line of advance" and "speculations which may or may not prove to be well founded." The finite-universe speculations belong to the second class — fascinating, but not to be confused with the solidity of what has gone before.

Why a finite universe is thinkable There are reasons for thinking the total amount of matter in the universe is limited — otherwise the gravitational effects of enormously distant matter would make our kind of world impossible — and if matter is finite it must be contained within a finite region, beyond which space would be "waste, like unfurnished rooms in a house too large for its inhabitants." Formerly this seemed futile because no one could conceive of an edge to space. Non-Euclidean geometry shows the alternative: the surface of a sphere has no boundary yet is not infinite. In a spherical universe, what we take for straight lines are like great circles — they ultimately return to their starting point — and light rays in empty space travel in what are really great circles. "The Euclidean straight line may remain as a beautiful dream, but not as a possibility in the actual world." This is not the local non-Euclidean character of the law of gravitation (hills and valleys); it is the possible curvature of the universe as a whole.

Einstein's and De Sitter's worlds Two finite universes have been constructed. In Einstein's, only space is queer: light travels round the whole universe in something like a thousand million years, and all rays from the sun meet again at the antipodes of their starting point and then at the starting point — like aeroplanes setting out from London in great circles in all directions, meeting at the antipodes and again in London. A person near the sun's antipodes sees what appears to be a body as bright as the sun was five hundred million years ago, and so on — "but all these suns are only ghosts; that is to say, you could pass through them without experiencing resistance, and they do not exert gravitation... It is rather disturbing to reflect that, if this theory is true, any number of the objects we see in the heavens may be merely ghosts." Russell notes the objection that absolute space and time re-enter "by another door," and that optical blurring would in any case spoil the images.

In De Sitter's world, time goes mad as well as space: distant clocks appear to go slower and slower, until at a quarter of the circumference they seem to have stopped altogether — a "lotus land, where nothing is ever done," always just beyond your ken, since light can never get across. Empirical evidence for or against is possible: if distant clocks are slow, distant atoms emit redder light. And indeed, "in a large majority of spiral nebulae there is a considerable displacement of spectral lines towards the red" — the most distant objects known, perhaps a million light-years away. The usual interpretation is a Doppler effect of recession, but De Sitter's theory predicts the reddening even for bodies at rest, and also predicts that distant bodies should recede, faster and faster, because gravitation cannot hold them. "However, it cannot be said that the argument, on either ground, is very strong" — Eddington lists forty-one nebulae, five with violet shifts.

The verdict Neither hypothesis is quite right — "it hardly seems probable that either is quite true. But it does seem probable that something more or less analogous is true." If the universe is finite, "it is theoretically conceivable that there should be a complete inventory of it." (In the revised editions this chapter is retitled "The Expanding Universe" and rewritten around Hubble's law: the recession of distant nebulae with velocity proportional to distance is now the established fact, and the chapter reports the expanding-universe models — with the history of the discovery and its confirmation — while keeping Russell's caution that such cosmological questions are not settled science.)

Key ideas

• The finite-universe question is a speculation built on relativity, not an acquired result; science aims at truth by successive approximations.
• A spherical universe has no edge and no center of the ordinary kind; "straight lines" close back on themselves like great circles.
• In Einstein's static universe, light returns to its source and every object appears multiplied as ghost images at the antipodes.
• In De Sitter's world, distant clocks seem to stop, a "lotus land" beyond the reach of light; red shifts and recession follow.
• The red shift of the spiral nebulae is real but was, in 1925, weak evidence — forty-one nebulae, five with violet shifts.
• Something like a finite, non-static universe is probable, though neither simple model is exactly true.

Key takeaway

Relativity raises the possibility that the universe is finite and non-static — a question whose answer, the recession of the nebulae, was already beginning to show in the red shifts when Russell wrote, and which the revised edition reports as the expanding universe.

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Chapter 12 — Conventions and Natural Laws

Central question

How much of what physics calls "laws of nature" is really convention — disguised choices about how we measure — and what genuinely remains?

Main argument

Disputes about words versus disputes about facts The hardest thing in controversy is to distinguish disputes about words from disputes about facts — and physics is no exception. The seventeenth century's great debate about what "force" is was really a debate about how to define the word. The method of tensors exists to eliminate what is purely verbal in physical laws. Russell's example: a man punting walks along the boat but keeps a constant position relative to the river bed; the Lilliputians might debate endlessly whether he is walking or standing still — "the debate would be as to words, not as to facts."

Convention survives in laws Even after we choose measures, an element of convention survives in the laws. The Fitzgerald contraction illustrates this: if lengths pointing north were half those pointing east, and this were true of all bodies, a fishing rod turned from west to north would still measure fifteen feet — your foot rule shrinks too — and you would only detect the change by something like Michelson-Morley, after which you still must choose whether to blame a change of length or a change of the velocity of light. "In any way you choose to adopt, there will be an element of convention." Even the electron's size and shape are partly conventional: any hypothesis affecting all electrons equally is undetectable and therefore "no change at all." Russell candidly reports that Eddington concludes the symmetry of an electron's forces is a matter of convention — "The argument is difficult and I have not fully understood it; but I feel some hesitation in accepting it as valid."

World-building Eddington calls the advanced part of relativity "world-building": the economical architect constructs the physical world from the smallest possible amount of raw material. The raw material is events; from them, mathematics builds an expression with the conservation properties of momentum and energy — and "it seems not surprising that we should come to believe in 'bodies.' These are really mere mathematical constructions out of events." The indestructibility of matter, once an empirical law, "ceases to be a proposition of physics, and becomes instead a proposition of linguistics and psychology": "Matter" is the name of a mathematical expression; our senses are such that we notice it.

What is left of physics The reader may ask what is left. Russell divides physics into three departments. First, what is included in relativity: the four-dimensional order of events and the measurable interval between neighboring events — beyond that, little. Second, what cannot yet be brought within relativity: the existence of electrons and protons, the structure of the atom, the theory of quanta — relativity "cannot give any reason why matter should exist in little lumps," and its outlook of continuity is precisely why. Third, geography (Russell's term, including history): everything that, as crude fact, distinguishes one part of space-time from another — where the sun is, where the earth is, that there are light waves here and matter there. "No amount of physical laws will enable us to infer a physical fact unless we know other facts as data for our inference." Relativity deals with structure; geography deals with material. And as to the intrinsic nature of events outside our own lives: "physics tells us the pattern of them, but is quite unable to tell us what they are like in themselves."

Key ideas

• Much apparent "law" is convention: coordinate choices, unit choices, and undetectable hypotheses all masquerade as facts.
• A hypothesis that changes everything measured equally — including the measuring instruments — is not a hypothesis at all.
• "Bodies" are mathematical constructions out of events, built for their permanence; even indestructibility becomes linguistics and psychology.
• Physics splits into: relativity (order and interval), non-relativistic physics (atoms, quanta, discontinuity), and geography (crude fact, including history).
• Relativity's assumption of continuity is why it cannot yet account for quanta, electrons, and protons.
• We can know the structure of events but not their intrinsic nature; our perceptual apparatus is selective.

Key takeaway

Under scrutiny, most of what looked like natural law dissolves into convention and the selectiveness of our senses; what remains is structure — the interval and the order of events — plus the brute facts of geography.

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Chapter 13 — The Abolition of "Force"

Central question

If there is no force in relativity, what does the Newtonian idea of "force" actually amount to — and can we learn to do without it?

Main argument

Where force came from In the Newtonian system, bodies under no forces move in straight lines with uniform velocity; deviations are ascribed to "force." Forces like ropes and collisions seemed intelligible from muscle sensations, but gravitation did not: "It seems odd that the earth can float in the void: the natural thing to suppose is that it must fall. That is why it has to be supported on an elephant, and the elephant on a tortoise, according to some early speculators." The whole notion of force arose, Russell argues, from the determination to preserve Euclidean geometry at all costs: if we insist our space is Euclidean when it is not, we must invent causes for bodies not moving in what we insist are straight lines. He quotes Eddington: a physicist who clings to the wrong geometry "will attribute the discrepancy of the observations to some influence which is present and affects the behavior of his test-bodies... a 'field of force'... A field of force represents the discrepancy between the natural geometry of a co-ordinate system and the abstract geometry arbitrarily ascribed to it."

Force gone, freedom found In Einstein's world there are no straight lines, only "straightest lines," or geodesics, involving time as well as space. A poet might say water runs down hill because it is attracted to the sea; a physicist says it moves "as it does, at each point, because of the nature of the ground at that point, without regard to what lies ahead of it." The planets move round the sun because that is the easiest thing to do — in the technical sense of least action — "because of the nature of the region in which they are, not because of an influence emanating from the sun." Russell draws the wider moral with characteristic wit: learning to conceive the world without force "would alter not only their physical imagination, but probably also their morals and politics." In Newton's solar system the sun seems a monarch whose behests the planets obey; "in Einstein's world there is more individualism and less government." The word "dynamic," he complains, has come to mean "energetic and forceful"; properly it should describe "the people in hot climates who sit under banana trees waiting for the fruit to drop into their mouths." If nature as portrayed by Einstein is our model, "it would seem that the anarchists will have the best of the argument": "The physical universe is orderly, not because there is a central government, but because every body minds its own business." No two particles of matter ever come into contact; a man charged with knocking another down "would be scientifically correct in pleading that he had never touched him. What happened was that there was a hill in space-time in the region of the other man's nose, and it fell down the hill."

The top-hats and the mirror To see what "reality" means without force, Russell examines reflections: twenty or thirty top-hats in facing mirrors all tumble when the real one is knocked off, and we suppose no force is needed for them. He shows the image is not "imaginary" — real light waves reach your eye — but is optically derivative: its rays do not spread from where it seems to be, and changes in it center about the real object, not about a point behind the mirror. General principle: most events belong to groups connected with a small region of space-time — a center. "Changes that happen to an object are those that affect the whole group of events which center about the object." The candle before a mirror: shut your eyes and its appearance to you changes, but its center is your eyes; blow it out and its appearance everywhere changes — then you say the change happened to the candle. This gives "a perfectly verifiable meaning to the statement that the image is 'only' a reflection," and lets us regard the heavenly bodies, seen but not touched, as more real than images.

Cause and effect as shorthand What is an "effect"? The electric light changes everything in the room; the tiger loosed in a Bank Holiday crowd makes everyone move — the tiger's action "is of the nature of a repulsive force," but the people flee because waves reach their eyes and ears, and if the same waves came without the tiger they would flee just as fast. The sun's gravitation differs only in being attractive: "the sun acquires its apparent power through the fact that there are modifications of space-time all round the sun." To say the sun "causes" them "adds nothing to our knowledge." What we know is "what kind of space-time is the presence of gravitating matter": when space-time is non-Euclidean in a region, growing more so toward a center, according to a law, "we describe this state of affairs briefly by saying that there is gravitating matter at the center." Cause and effect, of which force is a particular case, "is thus merely a convenient shorthand for certain purposes; it does not represent anything that is genuinely to be found in the physical world."

Key ideas

• "Force" was invented to patch the errors of insisting on Euclidean geometry; a field of force is the discrepancy between natural and ascribed geometry.
• In relativity, bodies follow geodesics because of the neighborhood they are in — the "law of cosmic laziness" — not because anything pulls them.
• The abolition of force changes the moral imagination: the universe is orderly because every body minds its own business, not because of central government.
• "Reality" can be defined verifiably through groups of connected events centered in a region of space-time: images are less real than their objects because their changes center elsewhere.
• Effects — like a tiger's on a crowd — are transmitted locally by waves and modifications of the neighborhood, not by remote influence.
• Cause and effect language is convenient shorthand that adds nothing to the formula and misleads when taken literally.

Key takeaway

Force is not a feature of the world but a relic of a mistaken geometry, and once it is abolished the language of cause and effect loses its claim to describe anything genuinely in the physical world.

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Chapter 14 — What Is Matter?

Central question

If physics is about events and intervals, what is "matter" — is there anything at all behind the word?

Main argument

Two traditional answers, both obsolete The question "What is matter?" is of the kind asked by metaphysicians "and answered in vast books of incredible obscurity"; Russell asks it as a person who wants the moral of modern physics. Two traditional conceptions: the atomists, who thought matter consisted of tiny indivisible lumps that hit and bounced; and the plenum-theorists, like Descartes, who held matter exists everywhere and attributed the planets' motions to vortices in the ether. Modern physics proved the atomic structure of gross matter while leaving the ether in its curious status — necessary, some said, for the propagation of light, but serving no other purpose. Russell's verdict: relativity demands the abandonment of the old conception of "matter," which is "infected by the metaphysics associated with 'substance.'"

Matter as a series of events In the old view, a piece of matter survived through all time while never being in more than one place at a time — a picture bound to the complete separation of space and time. Substitute space-time and the natural constituents are events: "An event does not persist and move, like the traditional piece of matter; it merely exists for its little moment and then ceases. A piece of matter will thus be resolved into a series of events." Each particle is "its history, not some metaphysical entity to which the events happen."

The group of events about a center From the previous chapter we know how a group of connected events is ranged about a center in space-time — like widening ripples on a pool when a fly has touched it. Common sense invents a hypothetical occurrence at the center and calls it the cause; and because one wave is followed by closely similar waves (the sun does not change its appearance suddenly), it invents a hypothetical particle occupying a string of neighboring centers, and says all the hypothetical occurrences are part of one history. "It is only by this double use of hypothesis, perfectly unnecessary in each case, that we arrive at anything that can be called 'matter' in the old sense of the word." The economical alternative: "an electron at a given moment is the various disturbances in the surrounding medium which, in ordinary language, would be said to be 'caused' by it" — taken, not at our moment, but traveling outward from the electron with the velocity of light. The closely similar set of disturbances, with nearly the same center, found slightly earlier or later, "will be defined as being the electron at a slightly earlier or slightly later moment." All the laws of physics are preserved without inferred entities.

The table that common sense sees "Common sense imagines that when it sees a table it sees a table. This is a gross delusion." What happens: light waves reach the eyes, cause occurrences in the retina, the optic nerve, the brain; any one of these happening without its preliminaries would cause the sensation "seeing the table" even if there were no table. Touch is an electric disturbance in the electrons and protons of the fingertips, produced by the proximity of those in the table; "if the same disturbance in our finger tips arose in any other way, we should have the same sensations, in spite of there being no table." Testimony is second-hand — a witness may not testify on hearsay. "For all these reasons, when we say that a man 'sees a table,' we use a highly abbreviated form of expression."

The electron and "empty space" Everything that occurs elsewhere owing to an electron can in theory be explored experimentally — "but what occurs within the electron (if anything occurs there) it is absolutely impossible to know." An electron is known by its "effects," but the word belongs to an obsolete view of causation: all we have a right to say is that groups of occurrences happen together in neighboring parts of space-time, a connection that "works equally backwards and forwards" — the future determines the past in the same sense as the past the future; the asymmetry "is only due to our ignorance," and there might be beings who remember the future. "The moral of this is that, if an electron is only known by its 'effects,' there is no reason to suppose that anything exists except the 'effects.'" What is called empty space is regions where disturbances propagate freely; each disturbance has a center, and near the center the law of propagation ceases to hold — "this region within which the law does not hold is called 'matter'; it will be an electron or proton according to circumstances."

Formulas and interpretations The whole latitude is characteristic of mathematical physics: we can be far more certain that our formulas are approximately correct than that any particular interpretation of them is. "A definite and conclusive answer to our question is not possible just because a variety of answers are compatible with the truth of mathematical physics." Russell's choice: define matter so that there must be such a thing if the formulas of physics are true — a minimum definition. Matter so defined is, as he says in the final chapter, like Isabella's gruel in Jane Austen, "thin, but not too thin."

Key ideas

• The traditional alternatives — indivisible atoms or a universal plenum — both presuppose "substance" and are obsolete.
• Matter is resolved into a series of events; a particle is its history, not an entity to which events happen.
• "Cause" and "particle" are unnecessary hypotheses invented to occupy the centers of groups of connected events.
• An electron at a moment may be defined as the disturbances spreading outward from its center at the velocity of light.
• Seeing a table is a highly abbreviated description of a chain of light waves, nerve impulses, and brain events; matter itself is not observed.
• Time's direction is an accident of ignorance; connection between events works equally backwards and forwards.
• The formulas of physics permit many interpretations; the definition of matter should be the minimum that guarantees matter exists if the formulas are true.

Key takeaway

Matter is not a substance behind phenomena but the name we give to groups of events arranged about a center — and physics cannot tell us, and does not need to know, what (if anything) lies at the center.

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Chapter 15 — Philosophical Consequences

Central question

What does relativity really change in philosophy — and what does it leave untouched?

Main argument

Neither idealism nor Kant The philosophical consequences of relativity "are neither so great nor so startling as is sometimes thought." It throws very little light on time-honored controversies such as realism versus idealism. Some think it supports Kant's view that space and time are subjective "forms of intuition" — but they have been misled by the word "observer": the observer in relativity "is just as likely to be a photographic plate or a clock" as a mind. The subjectivity involved "is a physical subjectivity, which would exist equally if there were no such things as minds or senses in the world." And it is strictly limited: the theory "gives a technique for distinguishing what is relative from what belongs to a physical occurrence in its own right." If it supports Kant about space and time, it refutes him about space-time; in Russell's view neither statement is correct, and philosophers may stick to their previous views.

What the great laws collapse into Physics tells us much less about the physical world than we thought. Almost all the "great principles" of traditional physics turn out to be either truisms — like the "great law" that there are always three feet to a yard — or false. Conservation of mass illustrates both: measured mass is the same thing as energy, conserved by a law that is itself a truism of measurement; proper mass is very nearly constant but not quite — "if you have four one-pound weights... they weigh rather less" — and four hydrogen atoms put together to make one helium atom weigh measurably less. "Broadly speaking, traditional physics has collapsed into two portions, truisms and geography."

A world of events The world relativity presents to the imagination is "not so much a world of 'things' in 'motion' as a world of events." Electrons and protons persist, but "are really to be conceived as strings of connected events, like the successive notes of a song." Between neighboring events there is the interval — "the physical reality of which lapse of time and distance in space are two more or less confused representations"; between distant events, the geodesic is the route that makes the sum of intervals along it greater than by any other route — the route a body chooses if left to itself. Everything is step-by-step: Euclid's straight lines become light rays, which are not quite straight near heavy bodies; the sum of the angles of a triangle is two right angles only in remote regions. "We, who cannot leave the earth, are incapable of reaching a place where Euclid is true."

The decay of reasoning's pretensions As reasoning improves, its claims to prove facts grow less. "Logic used to be thought to teach us how to draw inferences; now, it teaches us rather how not to draw inferences." Animals and children are prone to inference — a horse is surprised beyond measure at an unusual turning; the "great principles" of the uniformity of nature and universal causation are attempts to bolster up "our belief that what has often happened before will happen again, which is no better founded than the horse's belief that you will take the turning you usually take." Causation, in the old sense, no longer has a place in theoretical physics.

The choice of clocks The collapse of one all-embracing time must in the long run affect our views of cause and effect, evolution, and more. Whether there is progress in the universe "may depend upon our choice of a measure of time": choose one equally good clock and the universe progresses as fast as the most optimistic American thinks; choose another and it goes from bad to worse as fast as the most melancholy Slav could imagine. "Thus optimism and pessimism are neither true nor false, but depend upon the choice of clocks." The poet's "One far-off divine event / To which the whole creation moves" is spoiled: if the event is far off and the creation moves quickly, "some parts will judge that the event has already happened, while others will judge that it is still in the future." The second line ought to be "To which some parts of the creation move, while others move away from it." "But this won't do. I suggest that an emotion which can be destroyed by a little mathematics is neither very genuine nor very valuable."

Structure, not music What we know about the physical world is much more abstract than was supposed: we know the laws of occurrences — "just so much as can be expressed in mathematical formulae — but of their nature we know nothing." The analogy: between orchestral music as played and as printed in the score there is a resemblance in structure, so that one can be inferred from the other. A person stone deaf from birth, living among musical people, could understand that the scores represent something "quite different from themselves in intrinsic quality, though similar in structure" — the value of the music would be unimaginable, but all its mathematical characteristics inferable. "Now our knowledge of nature is something like this. We can read the scores... But we have not the advantages which he derived from association with musical people. We cannot know whether the music represented by the scores is beautiful or hideous; perhaps, in the last analysis, we cannot be quite sure that the scores represent anything but themselves." The book closes by conceding that even relativity may not be the end of the stripping-away of imagination: "far as we have traveled in the direction of abstraction, it may be that we shall have to travel further still." Abstraction, difficult as it is, "is the source of practical power" — the financier, whose dealings are more abstract than any other practical man's, is also more powerful.

Key ideas

• Relativity favors neither realism nor idealism; the "observer" is a physical instrument, and the subjectivity is physical and strictly limited.
• Most "great laws" of physics are truisms of measurement or false; what remains divides into truisms and geography.
• The physical world is a world of events; persistence is a string of connected events like the successive notes of a song.
• The interval is the reality of which time and distance are confused representations; geodesics replace straight lines.
• Logic now teaches how not to draw inferences; causation in the old sense disappears from physics.
• Progress, optimism, and pessimism depend on the choice of clocks; an emotion destroyed by a little mathematics is not very genuine.
• Physics reveals structure, not intrinsic nature — the score, never the music — and may be capable of still further abstraction.

Key takeaway

Relativity's philosophical upshot is deflationary: it subtracts our imaginative additions from physics, leaving a world of events and structure of which we know the pattern — the score — and, in the last analysis, nothing else.

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The book's overall argument

1. Chapter 1 (Touch and Sight: The Earth and the Heavens) — establishes that relativity demands an imaginative reconstruction of the world-picture, and that touch-based notions of "place," "motion," and "force" are conventions, all motion being relative.
2. Chapter 2 (What Happens and What Is Observed) — establishes that physics must state laws in terms of what is common to all observers, subtracting the observer's physical circumstances from measurements.
3. Chapter 3 (The Velocity of Light) — establishes the pivotal fact that light's velocity is the same for all observers, an unattainable limit, which the old physics cannot accommodate.
4. Chapter 4 (Clocks and Foot-rules) — establishes that simultaneity between distant events is relative to the observer, that proper time replaces universal time, and that the interval between events is the one objective quantity.
5. Chapter 5 (Space-Time) — establishes that space and time can no longer be separated, that "before" and "after" mean influence, and that four coordinates fix an event in space-time.
6. Chapter 6 (The Special Theory of Relativity) — establishes the quantitative content of the special theory: the Lorentz transformation, reciprocal length contraction and time dilation, and the invariant interval.
7. Chapter 7 (Intervals in Space-Time) — establishes the genealogy of the interval from Pythagoras through Descartes, Gauss, and Riemann, preparing the generalized interval of the general theory and its non-Euclidean geometry.
8. Chapter 8 (Einstein's Law of Gravitation) — establishes the new law: bodies follow geodesics in a space-time whose local structure near matter is a "hill," so gravitation is geometry, not force.
9. Chapter 9 (Proofs of Einstein's Law of Gravitation) — establishes the empirical (Mercury's perihelion, the 1919 eclipse, the red shift) and logical (relativity of motion, equality of gravitational and inertial mass, tensors) grounds for accepting the law.
10. Chapter 10 (Mass, Momentum, Energy and Action) — establishes that mass is relative and convertible with energy, that momentum and energy conservation survive in modified form, and that action is the fundamental quantity.
11. Chapter 11 (The Expanding Universe) — establishes the speculative reach of relativity: finite and non-static universes, the ghost images and lotus lands of Einstein's and De Sitter's models, and the red shifts of the nebulae pointing to expansion.
12. Chapter 12 (Conventions and Natural Laws) — establishes how much of physics is convention and how much is genuine: order, interval, and "geography," with the atomic and quantum discontinuities still outside relativity.
13. Chapter 13 (The Abolition of "Force") — establishes that force was invented to preserve Euclidean geometry and that cause and effect is convenient shorthand, not something found in the physical world.
14. Chapter 14 (What Is Matter?) — establishes that matter is a series of events grouped about a center, a minimum-definition construction that guarantees matter exists if the formulas of physics are true.
15. Chapter 15 (Philosophical Consequences) — establishes the deflationary philosophy of the whole: physics yields structure, not substance; the great laws collapse into truisms and geography; and knowledge is the score, never the music.

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Common misunderstandings

Misunderstanding: "Relativity proves that everything is relative." Russell disposes of this in Chapter 2: if everything were relative there would be nothing for it to be relative to. The theory is wholly concerned to exclude what is relative and find laws that hold for all observers — it is a search for the invariant, not a celebration of the arbitrary.

Misunderstanding: "Einstein showed Copernicus was wrong and the earth is really at rest." Chapter 1 argues the opposite move: all motion is relative, so "the earth rotates" and "the heavens revolve" mean the same thing. Copernicus's system is preferred as more convenient, not as more true.

Misunderstanding: "Relativity says space and time are subjective in the Kantian, mind-dependent sense." Chapter 15: the "observer" is as likely to be a photographic plate or a clock as a mind; the subjectivity is physical — it would exist if there were no minds at all — and it is strictly limited, since the theory identifies what remains objective.

Misunderstanding: "The Fitzgerald contraction is a real physical squeezing of bodies." Chapter 5: the contraction emerges from disagreement about simultaneity; each observer measures the other's lengths as shorter. It is a convention of measurement, not a stress on matter — which is why it cannot be detected by a co-moving measuring rod.

Misunderstanding: "Light travels at 186,000 miles per second and everything else must simply accept that limit as a rule." The constancy is an experimental fact from which the whole theory follows; the "limit" on material speeds is a consequence of the geometry of space-time, not an imposed decree. Russell's moving-platform analogy shows why velocities do not simply add.

Misunderstanding: "In relativity, mass increases with speed, so mass is not conserved." Chapter 10: measured mass does increase with velocity, but proper mass is invariant; and mass and energy are one quantity, so the conservation laws survive — modified, not abolished. Even proper mass changes slightly with energy content (helium weighs less than four hydrogens).

Misunderstanding: "Gravitation is a force that pulls things, and Einstein's theory bends light by a stronger pull." Chapters 8 and 13: there is no force at all; the sun exerts none on the planets. Bodies follow geodesics through space-time that is non-Euclidean near matter, and light follows zero-interval geodesics. A "field of force" is the discrepancy between the natural geometry and the geometry we arbitrarily ascribe.

Misunderstanding: "The expanding universe means the galaxies are flying apart through a fixed space." Chapter 11 (revised): what was observed is the recession of the nebulae — red shifts — interpreted within relativistic models of space-time itself; whether the universe is finite or infinite remains an open question that Russell treats as speculation, not acquired science.

Misunderstanding: "Relativity tells us what things are really like." The opposite, per Chapters 12–15: it tells us the structure of events — the pattern — while the intrinsic nature of what underlies the pattern is precisely what we cannot know.

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Central paradox / key insight

The book's most counterintuitive idea is the relativity of simultaneity: there is no such thing as "now" for the universe. Two events in distant places can be simultaneous for one observer, and for another — equally careful, equally right — the first may precede the second or follow it. All the other paradoxes of the book are corollaries: lengths shrink, clocks slow, masses grow, because they are all defined in terms of simultaneity, and simultaneity is not a fact of nature but a convention tied to a state of motion. The objective residue of all this dissolving measurement is the interval — the one quantity all observers agree on — and from that single invariant, the whole theory, up to and including the geometry of gravitation, is rebuilt.

We, who live on the earth, would naturally, in such a case, prefer the view of simultaneity obtained from a person at rest on the earth to the view of a person traveling in a train. But in theoretical physics no such parochial prejudices are permissible. ... We cannot therefore say unambiguously that two events in distant places are simultaneous. Such a statement only acquires a definite meaning in relation to a definite observer.

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Important concepts

Relativity of simultaneity The dependence of "at the same time" on the observer's motion: two distant events simultaneous for one observer need not be for another, and no one is wrong. The foundation of everything else in the theory (Chapter 4).

Proper time The time measured by a body's own clocks — the time order of events in its own neighborhood. Each body has its own; universal cosmic time is an illusion. Terrestrial time is just the proper time of the earth (Chapters 4, 6).

Interval The one measurable relation between two events that is the same for all observers, replacing the separate spatial and temporal separations of the old physics. It is time-like, space-like, or zero (for events connected by light). "The reality of which distances and periods of time are confused representations" (Chapters 4, 6, 7).

Space-time The four-dimensional continuum of events: four quantities are needed to fix an event, and they cannot be divided into three independent spatial ones plus an independent temporal one. Space and time reckonings interlock (Chapter 5).

Event Something that happens at a given time and place. Physics deals in events rather than bodies-at-an-instant because there is no universal "now." Matter itself is a series of events (Chapters 5, 14).

Proper length / proper mass Length as measured by an observer at rest relative to the body; mass as measured by an observer sharing the body's motion. The invariants behind the observer-dependent measured values (Chapters 5, 10).

Lorentz transformation The exact rule connecting one observer's estimates of distances and times with another's when the two are in uniform relative motion; the mathematical heart of the special theory (Chapter 6).

Geodesic The "straightest" line in a curved space — on the earth, a great circle; in space-time, the route that makes the total interval between two events greater than by any alternative route. Bodies left to themselves move in geodesics; light rays are zero-interval geodesics (Chapters 7, 8).

Cosmic laziness Russell's name for the geodesic principle: bodies left to themselves do their journeys as slowly as possible — the route that maximizes their own measured time (Chapters 8, 10).

The "hill" in space-time Russell's image for gravitation: near matter, space-time is non-Euclidean, like a hill growing steeper toward its summit; bodies deviate from Euclidean straightness because of the neighborhood they are in. No force acts (Chapter 8).

Equality of gravitational and inertial mass The empirical fact that all bodies behave alike in a given gravitational situation (feather and lump of lead; comet and planet), showing that gravitation is a property of the locality, not of the body (Chapter 9).

Tensor The mathematical technique for writing laws in a form independent of the choice of coordinates — the instrument by which Einstein eliminated the observer's contribution and found the law of gravitation (Chapters 2, 9, 12).

Action Energy multiplied by time — a measure of how much has been accomplished; the quantum is a unit of action. The Principle of Least Action unifies dynamics: bodies choose routes of least action, "again a law of cosmic laziness" (Chapter 10).

The expanding universe The revised Chapter 11's subject: the red shifts of the spiral nebulae interpreted as recession, against the background of Einstein's and De Sitter's finite-universe models. Russell treats the finiteness question as speculation, not settled science.

Event-particle / the center of a group of events The reconstruction of matter: a particle is the string of events grouped about a center in space-time; what (if anything) is at the center is unknowable and unnecessary (Chapters 13, 14).

Geography (in Russell's extended sense) All the brute facts that distinguish one part of space-time from another — where the sun is, where the earth is, what waves pass through a region. Facts that no amount of law can replace; they must be given as data (Chapter 12).

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References and Web Links

Primary text: the 1925 first edition (public domain scans)

• Internet Archive, full text of the 1925 first edition: https://archive.org/details/abcofrelativity0000russ
• Internet Archive, second public scan of the 1925 first edition: https://archive.org/details/abcofrelativity0000bert_l1g3
• HathiTrust, full view of the 1925 first edition (University of Michigan): https://babel.hathitrust.org/cgi/pt?id=mdp.39015066436612

The Pirani-revised editions (records and scans)

• Internet Archive, 1958 revised edition edited by Felix Pirani (George Allen & Unwin): https://archive.org/details/abcofrelativityr00russ
• Internet Archive, 1969 Mentor edition (New American Library): https://archive.org/details/abcofrelativity00russ
• Internet Archive, 1985 4th revised edition (New American Library): https://archive.org/details/abcofrelativity00russ_0
• HathiTrust catalog record, 1969 3rd revised edition: https://catalog.hathitrust.org/Record/001477648
• HathiTrust catalog record, 1985 4th revised edition: https://catalog.hathitrust.org/Record/007068971
• WorldCat record for the 1969 3rd revised edition: https://search.worldcat.org/title/28858

Catalog records and editions

• HathiTrust catalog record for the 1925 first edition: https://catalog.hathitrust.org/Record/001477646
• Open Library work page for the 1969 edition: https://openlibrary.org/works/OL1088543W
• Open Library work page for the 1958 revised edition: https://openlibrary.org/works/OL1088582W

Current in-print edition and publisher

• Routledge Classics edition (2009), publisher page with table of contents: https://www.routledge.com/The-ABC-of-Relativity/Russell-Pirani/p/book/9780415473828
• Waterstones listing for the Routledge Classics edition: https://www.waterstones.com/book/the-abc-of-relativity/bertrand-russell/9780415473828

Audiobook of the 1925 text

• LibriVox recording of The ABC of Relativity: https://librivox.org/the-abc-of-relativity-by-bertrand-russell/
• Internet Archive copy of the LibriVox recording (chapters as individual files): https://archive.org/details/abcofrelativity_2204_librivox

Background and further reading (supplements, not substitutes for the book)

• Eddington, Space, Time and Gravitation — the book Russell repeatedly quotes and recommends; Internet Archive scan: https://archive.org/details/spacetimegravita00eddi
• Wikipedia, Theory of relativity: https://en.wikipedia.org/wiki/Theory_of_relativity
• Wikipedia, Special relativity: https://en.wikipedia.org/wiki/Special_relativity
• Wikipedia, General relativity: https://en.wikipedia.org/wiki/General_relativity
• Wikipedia, Michelson–Morley experiment: https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment
• Wikipedia, Geodesic (geometry underlying Chapter 7): https://en.wikipedia.org/wiki/Geodesic
• Wikipedia, Bertrand Russell: https://en.wikipedia.org/wiki/Bertrand_Russell
• Wikipedia, Felix Pirani (editor of the revised editions): https://en.wikipedia.org/wiki/Felix_Pirani