Special Relativity

For two centuries, Newtonian mechanics rested on three assumptions. Space was absolute, an immutable stage upon which events unfolded. Time flowed universally, identical for everyone, everywhere in the universe. Gravity acted instantaneously between bodies, without any medium to transmit it. As the previous chapter showed, none of the three went unchallenged: Mercury's perihelion refused to fit, and the ether that was supposed to carry light kept requiring new assumptions to survive contact with experiment.

The instantaneous-gravity assumption is the one that sits worst with electromagnetism. If the Sun were to suddenly disappear, when would we know? Newton's law contains no delay: the Earth would leave its orbit at that same instant. Maxwell's theory, by contrast, carries electromagnetic effects at a finite speed.

In 1905 Albert Einstein, then a technical expert third class at the Bern patent office, published "On the Electrodynamics of Moving Bodies". He was not alone in working on the problem. Hendrik Lorentz and Henri Poincaré had already written down most of the equations. What Einstein changed was what the equations were taken to mean.

This chapter follows that reinterpretation to its conclusion: spacetime, a single entity in which space and time mix under a change of velocity. It is the structure General Relativity will later bend.

Newtonian physics rests on three structural assumptions:

• Absolute and universal time: there exists a single "now" shared by the entire universe. A clock on Earth and one on Mars tick in unison.
• Rigid and independent space: space is an immutable container, the same for all observers, regardless of their state of motion.
• Instantaneous action at a distance: the gravitational force propagates instantaneously. If the Sun were to disappear, the Earth would "know" immediately.

The tension is between two well-tested theories: Newtonian gravity has no propagation delay at all, while in electromagnetism every effect takes time to travel. Resolving it starts with light.

In 1864 James Clerk Maxwell unified electricity and magnetism into a single system of equations. (The calculation showing that the wave speed matched the measured speed of light had appeared two years earlier, in On Physical Lines of Force.) From the equations emerged a wave propagating at:

The speed of light was written into the laws of electromagnetism. But the equations did not say speed relative to what? In classical physics, if you walk forward at 5 km/h on a train moving at 100 km/h, the ground sees you move at 105 km/h. Applied to light, the same rule predicts that an observer chasing a light beam should measure less than c. The natural fix was to name a medium — the ether — and say that c is the speed relative to it.

As the previous chapter described, the Michelson-Morley experiment of 1887 looked for the Earth's motion through that ether and found nothing. The result did not by itself establish that c is the same for every observer: George FitzGerald (1889) and Hendrik Lorentz (1892) accounted for the null result within an ether theory, by proposing that moving bodies contract along their direction of travel by exactly the factor needed to hide the effect. That is a viable explanation, and it kept the ether.

There was a third option, and it needed no ether at all. Suppose light leaves its source at c relative to that source, the way a ball thrown from a moving train leaves the hand at its own speed plus the train's. Then Michelson and Morley were bound to find nothing: their lamp, their mirrors and their interferometer all moved together, so the light was always at c relative to the apparatus. This is an emission theory, set out in detail by Walther Ritz in 1908. Einstein had worked along the same lines himself before 1905 — he says so in a letter of 1912, and an unpublished manuscript from the same year, edited by John Stachel, sets out the version he had entertained and dropped.

Two things told against it. The first came from a question about light in moving matter. Light slows down in water: instead of c it travels at c/n, where n is the refractive index, about 1.33 for water. Now set the water flowing. Does the light get carried along with it?

Each theory has to answer, and the answers differ. In a stationary ether, light is a disturbance of the ether and the water is only an obstacle it passes through: the flow adds nothing, and the light travels at c/n whatever the water does. An emission theory has no ether at all — matter is the only thing there is to refer a speed to — so the light inside the water can only be moving at c/n relative to the water, and the water's velocity should add in full. Two clean predictions: nothing, or everything.

In 1851 Hippolyte Fizeau measured it. He split a beam, sent the two halves down a pair of tubes with water running one way in one and the other way in the other, and recombined them to see whether the flow had shifted the interference fringes. It had — but by less than full carrying would require. He repeated the experiment in 1859 with the same outcome:

So the water drags the light, but only partly — and by an amount that depends on the refractive index, which is a property of how the water responds to light rather than of how it moves. Lorentz's electron theory later derived exactly this coefficient. An emission theory has no room for it: it predicted the full amount.

The second objection was about waves. If the speed of light depends on the emitter, then the wavelength λ = c/ν no longer follows from the frequency alone: the same frequency, sent by sources in different states of motion, arrives with different wavelengths. Optics had spent a century measuring interference and diffraction, and those measurements assume that a given colour of light has a definite wavelength.

Fizeau's result stayed with Einstein. The historian Gerald Holton drew a distinction worth keeping here: when Einstein was asked about Michelson and Morley his answers varied, but when he raised the origins of relativity himself he named the water tubes and stellar aberration.

A third objection has a story attached to it. In 1946, aged sixty-seven, Einstein wrote a short intellectual autobiography for a volume of essays about his work. In it he describes a thought he had had at sixteen, at school in Aarau: what would you see if you ran alongside a light beam at c? You would see an electromagnetic field standing still — the crests and troughs of the wave sitting beside you, going nowhere. He records that this struck him as impossible, both from experience and on the basis of Maxwell's equations.

The second half of that is worth checking, because it says something about what a light wave is in this theory. In Maxwell's equations neither field stands on its own: a magnetic field changing in time is what gives the electric field its curl, and an electric field changing in time is what gives the magnetic field its curl. Freeze the pattern — which is what running alongside it at c would mean — and both time derivatives vanish, taking both curls with them. What is left is a uniform field, the same value everywhere, with no crests and no troughs. So a light wave here is not a shape moving through space; it is two fields regenerating each other, and the regeneration runs only while both are changing.

What the argument reaches, though, is narrower than it first appears. Lorentz's theory never claimed Maxwell's equations hold in every frame — it claimed they hold in the ether's frame, and under the Galilean transformation they change form in any other. An observer travelling at c is in one of those other frames, so the theory was not asserting the thing the frozen field contradicts. Against an emission theory the objection does land: there the same optical laws are meant to hold in every inertial frame, which makes the frozen state an allowed solution of the theory's own equations in the observer's frame. John Norton has argued on these grounds that what the thought experiment tells against is the emission theory rather than the ether.

The recollection itself needs the same care. Einstein set it down fifty years after the fact, and there he presents the thought as containing the beginnings of special relativity. An earlier telling of the same episode, recorded by his biographer Carl Seelig in 1956, ends in puzzlement about a wave field with no time dependence and makes no connection to relativity or to any principle. Norton reads the 1946 account as a reconstruction; Balaban has argued instead that the young Einstein was already trying to make a problem out of the ether frame. The physics of the frozen wave holds either way. How much of the older man's framing belongs to the sixteen-year-old is not settled.

Einstein took a different route. In his later recollections he gave the Michelson-Morley result little weight in his own thinking; what bothered him were asymmetries in how Maxwell's theory described a magnet moving near a coil versus a coil moving near a magnet — the same observable current, two unrelated explanations. His 1905 paper opens on that asymmetry, not on the ether-drift experiments.

The first thing to go is the velocity-addition rule we started with. It is replaced by a rule in which c is a ceiling rather than one speed among others. For an object moving at u in a frame that itself moves at v:

At everyday speeds uv/c² is negligible and the formula returns the familiar sum: the walker on the train really does move at 105 km/h, to within about one part in 10¹⁷. Set u = c, however, and the result is c for any v whatsoever. The classical rule was not wrong so much as a low-speed approximation, and the new rule makes the second postulate self-consistent rather than paradoxical.

Consistency with the second postulate is incompatible with a universal time. To see why, Einstein first had to say what it means for two distant clocks to read the same time at all.

Section 1 of Einstein's 1905 paper does not begin with light beams and trains. It begins with two clocks, A and B, sitting at rest some distance apart, and a question: what would it even mean to say they agree?

Comparing them by eye fails, because seeing B from A takes time. Carrying one clock over to the other fails too, as the theory will shortly explain. Einstein's answer is a procedure. Send a flash from A at time t₁ by A's clock; let it reflect off B and return to A at t₂. Then B's clock is defined to read, at the moment of reflection:

The assumption is in the factor one-half: it takes the light to have gone out in the same time it took coming back — and that assumption cannot be tested without a second synchronised clock at B, which is the thing being set up. Any measurement of the one-way speed of light runs into the same circle. What experiments actually measure is the round-trip average. The one-half is therefore a stipulation, chosen because it is simple and consistent, not a fact read off from nature.

Not an Idea Out of Nowhere

Coordinating distant clocks by signal exchange was, by 1905, routine engineering. Through the second half of the nineteenth century, surveyors had been fixing longitudes by telegraphing time signals between observatories and halving the round-trip to correct for transmission delay — the same arithmetic as Einstein's definition, done with wires instead of light beams. Cities were being wired for pneumatic and electric clock networks, in which a master clock pushed a signal out to slave dials. Bern, where Einstein worked, was one of them.

Nor was he the first to draw a philosophical conclusion from it. In La mesure du temps (1898), Henri Poincaré argued that we have no direct intuition of the simultaneity of distant events, and that the rules we use to judge it are conventions adopted for convenience. Poincaré was then closely involved with the Bureau des Longitudes, whose business was precisely the telegraphic coordination of distant clocks. The historian Peter Galison has argued at length that the abstract claim and the practical procedure belong to a single story rather than to separate ones.

By 1904 Lorentz had published the transformations that carry his name, including the quantity he called local time — the time coordinate of a moving frame. He treated it as a mathematical convenience with no physical meaning; the ether frame still had the real time. Poincaré went further, giving local time an interpretation in terms of clocks synchronised by light signals, but he too kept a true time behind the apparent one.

Einstein's step was to drop the distinction. There is no true time hiding behind the procedure; the procedure is all there is, and it gives different answers in different frames. The equations were largely already on the page. What changed was the claim that they describe time itself rather than an artefact of measurement.

The standard illustration is a train. Two lightning bolts strike the two ends of a moving carriage. One observer stands on the embankment, level with the midpoint of the carriage at the moment the bolts strike; another sits at the middle of the carriage itself. Each uses the same criterion: if the two flashes reach me at the same moment, and I am equidistant from where they struck, the strikes were simultaneous.

One common way of putting this does not work. It would be easy to say that the passenger "runs into" the front flash, so of course it arrives early — but that is velocity addition applied to light, which the second postulate forbids. Running into a light beam does not make it arrive faster. The description in terms of the passenger moving belongs to the embankment observer, who is entitled to it because in that frame the passenger really is moving. In the passenger's own frame there is no motion to appeal to, and the unequal arrival times can only be explained by unequal emission times.

This is not a delay in perception, and it does not go away once each observer corrects for light-travel time — the corrections are exactly what is being carried out above. There is no universal "now". Given two events far enough apart, whether one precedes the other can depend on who is asking.

If simultaneity is relative, so must be the rate of time. Imagine a clock made of light: a photon bouncing between two mirrors. For a stationary observer, the photon travels a vertical path. But if the clock is moving, the photon must travel a longer diagonal path.

Since the speed of light is constant, the photon takes longer to complete a cycle for the external observer. The moving clock runs slower.

The relation is also symmetric, which is easy to misread. If you and I fly past each other at constant velocity, I measure your clock running slow and you measure mine running slow — with the same γ. There is no contradiction, because we are comparing different pairs of events, and by the previous section we do not even agree on which distant events are simultaneous. Neither of us is right; the question "whose clock is really slower" has no frame-independent answer.

Length contraction is the spatial counterpart of time dilation. A moving object shortens in the direction of motion:

Length contraction follows directly from the previous two sections, and the link is worth making explicit. To measure the length of a moving rod, you must locate both its ends at the same moment — otherwise the rod moves between the two readings and the answer is meaningless. But "the same moment" is exactly the notion that turned out to be frame-dependent. Two observers measuring the same rod are marking different pairs of events, so it is no surprise that they get different lengths. L₀ here is the proper length: the length measured in the frame where the rod is at rest, the counterpart of proper time.

Space and time are no longer separate entities. To see how tightly they are bound, we need to put them into a single structure: spacetime.

Three months after the electrodynamics paper, Einstein published a three-page follow-up in the same journal, asking whether the inertia of a body depends on its energy content. The argument considers a body that emits two equal pulses of light in opposite directions, so that it does not recoil, and works out the bookkeeping in two frames. The accounts only balance if the body has lost mass equal to the emitted energy divided by c²:

Read carefully, this says that mass is a form of energy rather than a separate substance — and that the conversion factor is enormous, since c² ≈ 9 × 10¹⁶ m²/s². A gram of anything corresponds to about 9 × 10¹³ joules. The equation does not say the conversion is easy: nothing here explains how to release that energy, and most processes release a tiny fraction of it. Chemical burning converts on the order of one part in a billion of the fuel's mass; nuclear fusion in the Sun's core reaches roughly 0.7%.

That combination is not an accident of algebra. Just as t and x mix under a change of velocity while ds² stays fixed, energy and momentum mix while the combination E² − (pc)² stays fixed, and its fixed value is (mc²)². The mass m is therefore the frame-independent part of the pair. Energy and momentum turn out to be components of a single four-dimensional object, in the same way that time and space are — which is the subject of the next section, and the reason General Relativity ends up treating energy, not mass alone, as the source of gravity.

In 1908, mathematician Hermann Minkowski reformulated Special Relativity in geometric terms. No longer separate "space" and "time", but a single four-dimensional entity: spacetime.

The Invariant Spacetime Interval

In Euclidean space, the distance between two points is invariant:

In Minkowski spacetime, there is an analogue: the spacetime interval:

Classification of Events

The interval classifies the causal relationship between events:

• Timelike (ds² > 0): the events can be causally connected. One can influence the other.
• Spacelike (ds² < 0): the events are causally disconnected. No signal can connect them.
• Lightlike (ds² = 0): connected by a ray of light — the boundary between the two cases above.

For timelike separation, ds is directly meaningful: ds/c is the proper time a clock would record travelling between the two events. For spacelike separation ds² is negative, so one quotes the proper distance √(−ds²) instead — the separation measured in the frame where the two events are simultaneous.

The Light Cone

Drawn on a spacetime diagram, that three-way classification takes a definite shape. Take any event and consider all the lightlike directions leading away from it: they trace out a cone opening into the future and a mirror cone opening into the past. This is the light cone of the event.

• Events inside the future cone are the ones this event can influence; events inside the past cone are the ones that could have influenced it.
• Events outside both cones are spacelike-separated — no causal link is possible in either direction, and observers in different frames can disagree about which happened first. The train's two lightning bolts are exactly this case.

The cone structure is what keeps the relativity of simultaneity from reordering cause and effect. Observers may reorder spacelike-separated events, but never timelike-separated ones: no change of frame can put an effect before its cause. Every massive object's worldline stays inside its own light cone. This is the structure that later chapters bend — a black hole's horizon is the surface where the cones have tipped over far enough that every future-directed path leads inward.

Read in sequence, the results so far look like a list of losses: no absolute simultaneity, no universal time, no frame-independent length. But each of these comes with a replacement that does hold in every frame.

• The speed of light is the same in every inertial frame — the assumption the whole theory is built on.
• The spacetime interval between two events is the same for everyone, even though its separate time and space parts are not.
• Proper time along a given worldline is the same for everyone. Observers disagree about how fast your clock runs; they agree on what it reads when you get there.
• Causal order between timelike-separated events is the same for everyone. Effects never precede causes in any frame.
• Rest mass is the same for everyone, being what is left of E² − (pc)² once the frame-dependent parts are removed.
• The form of the laws is the same in every inertial frame — the first postulate, which is the reason any of the above is worth stating.

The same expansion also shows why the relativistic energy relation is not a break with the old mechanics. Expanding E = γmc² for small v gives mc² + ½mv² + …: the rest energy, then the familiar Newtonian kinetic energy as the first correction to it.

Free Fall and Acceleration

The step from Special to General Relativity rests on the observation made by Galileo: all bodies fall the same way. Mass does not matter, composition does not matter. A feather and a hammer, in the absence of air, fall at the same rate.

Why This Implies Curvature

A tempting shortcut runs: acceleration bends worldlines, gravity is equivalent to acceleration, so gravity bends spacetime. The shortcut does not work. Accelerated worldlines are curved in perfectly flat Minkowski spacetime — a rocket firing its engines traces a curved path through a spacetime with no curvature at all. A bent worldline says nothing about the geometry it is drawn in.

The actual argument turns on the limits of the elevator. Inside a small enough box, free fall erases gravity exactly as the equivalence principle says. Now enlarge the box. Release two balls side by side, far apart, while falling toward the Earth: each falls toward the planet's centre, so their paths converge. Release one above the other: the lower falls into a slightly stronger field and the two drift apart. An observer in a large enough falling elevator sees the balls accelerate relative to one another, despite being in free fall.

These residual relative accelerations are tidal effects — the same ones that raise tides in the oceans. They cannot be transformed away by any choice of frame, because they measure the difference between neighbouring free-fall paths. A single accelerated frame can cancel a uniform field; nothing can cancel a field that varies from place to place.

That is the signature of curvature. On a flat surface, two initially parallel straight lines stay parallel; on a sphere, two travellers heading due north from the equator converge without either turning. Gravity behaves like the second case: initially parallel free-fall paths converge or diverge on their own. What gravity contributes beyond a choice of frame is therefore the tidal part — the piece that free fall does not remove.

What Stops Working

Something built earlier in this chapter has already given out, and it is worth naming before the next chapter picks it up. Simultaneity was defined by a procedure: a flash leaves A at t₁, reflects off B, returns to A at t₂, and B is set to t₁ + ½(t₂ − t₁). Run that procedure across a lattice of clocks at rest in an inertial frame and the settings hold. The clocks share a common rate, so one synchronisation lasts.

Accelerate the lattice and it stops lasting. In a cabin accelerating at a, a clock at the ceiling and a clock at the floor run at rates differing by roughly ah/c², so a lattice set once drifts apart on its own. No gravity is involved — this is flat spacetime and Special Relativity throughout. The equivalence principle then carries the result across: if the accelerating cabin and the cabin on the ground cannot be told apart from inside, the same rate difference has to appear in a gravitational field. That is the route by which Einstein obtained gravitational time dilation in 1907, eight years before he had field equations to derive it from.

So the single time coordinate the synchronisation procedure delivers belongs to inertial frames in particular, not to relativity in general. Outside them it survives over a patch — one small enough that the rate differences across it stay below what the clocks inside it can resolve. The GPS figures given earlier are that arithmetic run on a real system: the gravitational term alone is +45.9 μs/day, which is why satellite oscillators are offset before launch instead of being reset from the ground.

Acceleration on its own does not call for new geometry — the accelerating cabin sits in flat spacetime, and its clock split is a Special Relativity result. What gravity adds is the tidal part described above, which no choice of frame removes. Anything that describes it has to do two things at once: work in patches small enough for Special Relativity to hold, and supply a rule for how the geometry changes from one patch to the next.

Spacetime is now in place, along with a reason to think gravity curves it. What is still missing is the quantitative link: how much curvature a given distribution of matter and energy produces, and how bodies move once it is there. That took Einstein another eight years, and it is the subject of the next chapter.