The Theory of Invariants That Was Named After Its Shadows

The question deserves to be taken at its word, because it is a good one. If everything depends on the frame of reference, how can the speed of light be the same in all of them? Either something is absolute, in which case the slogan that everything is relative is false, or nothing is, in which case a constant at the foundation of the theory is a contradiction. The answer is not that the question is naive. The answer is that the theory has the wrong name, and the name has been doing philosophical damage for a century.

Here is the claim, stated as plainly as it can be. Special relativity does not say that everything is relative. It says that a short, specific list of things is relative — and says which — and that everything on a second list is not, and the second list is what the theory is actually about. The speed of light heads the second list. It is not an exception to relativity; it is the axiom from which the first list is derived. Get the two lists the right way round and the paradox turns inside out: the reason lengths and durations bend is precisely that c will not.

The Magnet and the Coil

Begin where Einstein began, which was not with light but with a magnet. In the 1860s Maxwell wrote down the equations of electricity and magnetism and found, folded inside them, a wave: a disturbance of the two fields that would travel through empty space at a speed fixed by two constants of the laboratory bench, the permittivity ε₀ and the permeability μ₀, one measured with charged plates, the other with current-carrying wires. The speed came out as 1/√(ε₀μ₀), about three hundred and ten thousand kilometres a second, which was, within the error bars of the day, the speed of light. Maxwell drew the obvious conclusion. Light was the wave.

Notice what is not in that formula. There is no velocity in it. Nothing in ε₀ or μ₀ says ‘as measured by someone at rest with respect to the ether’ or ‘with respect to the lamp’. The equations give a number, and the number does not know who is asking. Nineteenth-century physics took this for a defect. There must be a medium, the luminiferous ether, in whose rest frame the equations held; in every other frame light must go faster one way and slower the other, as sound does in a wind. In 1887 Michelson and Morley built an interferometer to find the wind. The Earth moves round the Sun at thirty kilometres a second, and the predicted shift of the fringes was forty times what the instrument could see. They found nothing. Not a small wind: none.11The experiment ran in July 1887 in the basement of a dormitory at the Case School of Applied Science in Cleveland, the interferometer floating on a pool of mercury so that it could be turned without strain. The apparatus could resolve a hundredth of a fringe; the predicted shift was about four tenths.

Einstein’s 1905 paper opens with the magnet rather than the interferometer. Move a magnet through a coil and a current flows, and the textbooks explained it by an electric field induced around the moving magnet. Hold the magnet still and move the coil instead, and the same current flows, but now the explanation was different: no electric field, only the magnetic force on charges carried through the field. Two explanations for one current, the choice turning on nothing but which object you had decided to call at rest. His remedy was to take Maxwell’s frameless c seriously instead of apologising for it. Two postulates: the laws of physics are the same in every inertial frame; and the speed of light in vacuum is the same in every inertial frame, whatever the motion of the source. Everything else in the theory is arithmetic performed on those two sentences.

The magnet and the coil get their answer as a by-product. What is electric and what is magnetic is a fact about the frame, not the world: the two fields are components of one object, the electromagnetic field tensor, as north and east are components of one displacement, and turning your frame trades one for the other. The current in the coil is invariant. The story about which field produced it is not.

One Rotation

Here is the arithmetic. If c is to be the same for you and for someone moving past you at speed v, the two of you cannot be reading the same clocks and rulers — something has to give, and what gives is the assumption that time is a single quantity everybody reads alike. The Lorentz transformation converts one frame’s coordinates into another’s: x′ = γ(x − vt) and t′ = γ(t − vx/c²), with γ = 1/√(1 − v²/c²). Read the second equation slowly, because everything strange lives in it. Your time depends on my position. Two events I call simultaneous, if they happen at different places, you will not.

In 1908 Hermann Minkowski saw what the transformation was. It is a rotation. Not a rotation in the ordinary plane, where what is preserved is x² + y², but in a plane of space and time where what is preserved is c²t² − x². That quantity is the spacetime interval, and it is the same in every frame the way the length of a stick is the same however you turn it. A frame is a choice of which direction to call time and which to call space, no more a fact about the world than which wall you call north. The stick is the fact.

“doomed to fade away into mere shadows”—Hermann Minkowski, on space by itself and time by itself, Cologne, 1908

The rule for adding velocities is a consequence, not an extra assumption. If you throw something at speed u inside a frame that moves at v relative to me, I do not see it at u + v. I see w = (u + v)/(1 + uv/c²). At everyday speeds the denominator is one to a dozen decimal places and Galileo’s sum holds. Set u = c and the formula returns w = c for any v whatever. Light from a rocket travelling at nine tenths of c does not reach me at 1.9c; it reaches me at c, and the rocket’s crew and I will disagree about its wavelength, its energy and the timing of its emission — about everything except its speed. c is constant across frames because the transformation between frames is the one that preserves it; asking why it does not add is asking why turning a stick does not change its length.

The invariance of c also divides the world around any event into three regions every frame agrees on: the future cone, the past cone and the elsewhere. The order of two events inside each other’s cones is invariant; cause precedes effect for everybody. The order of two events in each other’s elsewhere is frame-dependent, and harmlessly so, because nothing can travel between them in time to make one the cause of the other. This is why the entangled pairs of the electron’s complaint cannot carry a message: a correlation between two elsewhere events has no frame-independent ‘first’, so it cannot be a signal from either to the other.

The Two Ledgers

So here is the ledger the name conceals. In the column of things that depend on the frame: simultaneity; the length of a moving rod, contracted by γ; the duration of a moving process, stretched by γ; kinetic energy, zero in a body’s own frame and anything you like in another; the split of the electromagnetic field into E and B; velocity; momentum. In the other column: c; the interval; proper time, the time a clock accumulates along its own path, which is what your wristwatch actually reads; rest mass, since E² − p²c² = m²c⁴ wherever you stand; electric charge; the causal order of every pair of events that can affect each other; and the laws themselves, the same in every frame because that was the first postulate.

Look at how the columns pair off. Each relative quantity is the frame-dependent projection of an invariant. The relative things are not illusions: a metre stick moving past you really is shorter, by every measurement you can make, and the muon really does live longer. But they are projections, the way the length of a shadow really does change with the hour and really is not a property of the tree. Felix Klein saw this in 1910 and said the theory was better described as the theory of the invariants of the Lorentz group. He was right, and nobody listened, because the shadows had already been given top billing.

Is It Really the Case?

The honest answer has two parts, and popular accounts usually give only one. The first concerns what is actually measured. Every measurement of the speed of light ever made with a single clock is a measurement of the two-way speed: light goes out to a mirror and comes back, and one clock times the round trip. To measure the one-way speed you need a clock at each end, and the two must be synchronised; but synchronising them by sending a signal assumes you already know how long the signal took, which is the thing you set out to measure. In 1928 Hans Reichenbach made the loophole exact with a parameter ε. If a flash leaves me at t₁ and returns at t₃, the moment it reached the mirror is, by convention, t₁ + ε(t₃ − t₁). Einstein chose ε = ½, which makes the out and back speeds equal. Any ε strictly between zero and one is consistent with every experiment ever performed; it gives a one-way speed of c/2ε outward and c/2(1 − ε) back, and a round trip at exactly c.22Reichenbach’s Philosophy of Space and Time (in Further reading) sets out the parameter and the argument. The convention was also recognised, in different words, by Poincaré in 1898 and by Einstein himself, who in 1905 wrote that the equality of the out and back times was established ‘by definition’.

This is not a scandal, and it is not a crack through which the ether creeps back. The two-way speed is the fact; ε = ½ is the choice that makes the laws take their simplest, symmetric form, and physicists choose it the way cartographers choose a projection, knowing that they are choosing. So the scepticism is well placed as far as it goes. The constancy of the one-way speed cannot be shown. The constancy of the round-trip speed, its sameness in every direction and its independence of the source’s motion have been shown, to a precision hard to hold in the mind.

The second part concerns the unit, and here the sceptic has a real point. In 1983 the General Conference on Weights and Measures redefined the metre as the distance light travels in vacuum in 1/299,792,458 of a second. From that day c has been exactly 299,792,458 metres a second, not because it was measured but because it was defined. Nobody can now measure the speed of light in metres per second; what they measure is the length of their metre in light-time. Is c constant only by decree, then? Partly, and the two parts come apart cleanly. The number is a convention. It inherited a 1972 laser measurement at Boulder so precise that the old definition of the metre, a wavelength of krypton light, had become the larger source of uncertainty; the measurers had outrun the ruler, so the ruler was redefined in terms of what they measured.33The two events were the 1972 measurement by Evenson and colleagues at the National Bureau of Standards in Boulder, 299,792,456.2 ± 1.1 metres a second, by comparing the frequency and wavelength of a stabilised laser, and the 1975 recommendation of the value 299,792,458 by the 15th General Conference on Weights and Measures, which the 1983 definition of the metre then adopted unchanged.

The constancy, though, is not a convention, and could not be made one. The definition presupposes that light in vacuum has a single reproducible speed: in Paris and in Boulder, in the frame of a satellite, for red light and for blue. If it did not, a metre would be a different length in different laboratories and lengths measured by light would disagree with lengths measured by other means. They do not. The decree was possible only because the fact had been established. So: the value of c in metres per second is a choice; its being the same everywhere and for everyone is a discovery. The quantity no committee could fix, and the one that would move if the physics moved, is the dimensionless fine-structure constant, about 1/137.036, which carries no units to redefine and which the spectra of quasars and the natural fission reactor at Oklo in Gabon show to have shifted by no more than a few parts in ten million over two billion years.

And the modern tests are not subtle. Every GPS satellite carries atomic clocks that, twenty thousand kilometres up and moving at nearly four kilometres a second, run slow by about seven microseconds a day for their speed and fast by about forty-five for the weaker gravity, a net gain of thirty-eight microseconds. Uncorrected, that would put positions out by ten kilometres a day. In 1963 Frisch and Smith counted muons at the summit of Mount Washington and again at sea level, nineteen hundred metres lower. Muons live two microseconds at rest; without time dilation about twenty-seven an hour should have survived the descent. Five hundred and sixty-three an hour arrived at the top and four hundred and twelve at the bottom. In 1938 Ives and Stilwell caught a moving clock running slow in the Doppler shift of hydrogen ions; in 2014 the same experiment, on lithium ions circling a Darmstadt storage ring at a third of the speed of light, confirmed the dilation factor to two parts in a billion. And in 2015 a Michelson-Morley experiment with cryogenic sapphire oscillators bounded any dependence of the speed of light on direction at about a part in a million million million. The wind is not small. It is absent at the eighteenth decimal place.

Relativism and Relativity

Now the word, and the philosophy the word has bred. Einstein disliked the name. On 30 September 1921 — one hundred and five years ago to the day this essay is dated — he answered a letter from Ernst Zschimmer, who had suggested that the theory be called Invariantentheorie. Einstein agreed that ‘relativity’ was unfortunate and had given rise to philosophical misunderstanding, thought the proposed name truer to the theory’s method and aim, and judged that after sixteen years it was too late to change. He was right on all three counts.

The misunderstanding is a category error, and it is worth naming precisely. ‘Everything is relative’ in the cultural sense means there is no fact of the matter, that no vantage is better than another. Relativity in the physical sense says the opposite. It says there is a fact of the matter so exact that my measurements can be converted into yours, and yours into a third person’s, without loss, by a formula with one constant in it, known to nine figures. A theory whose equations translate between frames is the negation of a doctrine in which each frame has its own truth. If everything were relative there would be no Lorentz transformation, because there would be nothing for it to transform into. The relativist has borrowed the name of the theory whose content refutes him, the oldest confusion in the book.

“there are no facts, only interpretations”—Friedrich Nietzsche, notebooks, 1886-87

Nietzsche’s perspectivism is the serious version of the doctrine, and it deserves a serious answer. Yes: every measurement is made from somewhere. There is no view from nowhere, no frame at rest in a way the others are not; the ether was the last view from nowhere physics tried to keep. But relativity adds the thing perspectivism lacks. The perspectives are related by a law, and the law is not itself a perspective. What survives every turning of the view — the interval, the rest mass, the cone — is what deserves to be called real. A painter knows a version of this: the frame decides what is in the picture, but not what was in the room. Perspectivism is right that no picture is the room. It is wrong to conclude that there is no room.

The Block and the Now

There is one consequence that genuinely unsettles. In 1966 C. W. Rietdijk and in 1967 Hilary Putnam argued as follows. Whatever is simultaneous with me now is real; that is what ‘now’ means. But a stranger walking past me has a differently tilted plane of simultaneity, and for events far enough away her ‘now’ includes events in my future. If those events are real for her, and she is real for me, then they are real. Run the argument for every stranger at every speed and the whole of the future is already real, and the past too: a four-dimensional block in which nothing happens because everything already has. This is the picture an earlier essay took seriously and then found the flow again on the way back up; the narrower question is whether relativity forces it.

It does not, quite. In 1968 Howard Stein pointed out that the argument smuggles back in the very thing relativity removed. ‘Simultaneous with’ is not, in relativity, a relation between events; it is a relation between events and a frame, and it is not transitive across frames. The argument works only if ‘real for’ may hop from frame to frame as though there were a universal now underneath, which is the premise the theory denies. So relativity forbids a universal now without settling whether the future is real. What it leaves is the cone: for every observer the causal past is fixed and the causal future open, and no change of frame can rearrange your causes and your effects. That is thinner than a cosmic present. It is also, I think, all that the sense of an open future ever needed.

The Frame of the Frames

Which brings the question to its best form. Why constants at all? If the theory’s whole discovery is that no frame is privileged, what business does it have privileging a number? The answer is that c is not a number in the sense that a speed is a number. It is a conversion rate between two things once thought to be different kinds of thing, space and time, and it is exactly because it is the same from every frame that the frames can be compared. It plays the role ħ plays between energy and frequency and G plays between mass and the curvature of the geometry mass sits in. Planck saw this in 1899: from c, ħ and G alone one can build units of length, time and mass that owe nothing to any planet, metal bar or human convention. Since 2019 the International System has fixed ħ and the electron’s charge exactly, as it fixed c in 1983, turning the constants that do not depend on a frame into the frame all the others are measured against. G alone is still measured, to about twenty parts in a million, a scandal of a different kind.

The constants are not privileged frames. They are what a frame is made of: coordinates are numbers, numbers need a scale, and the scale has to be the same for everyone or the conversions would fail. The invariants are that scale. They are not one more perspective; they are the reason perspectives can be translated at all — the frame of the frames, if you like, though it is truer to say they are the part of the world that is not a frame. Another essay here calls the constants the places where it is possible to stand. In relativity that is literal: the interval is where you stand while everything else turns.

So consider what survived and what did not. Newton’s absolute space is gone; there is no fact about whether you are moving. Absolute time is gone; there is no fact about which distant events are happening now. The ether is gone, and with it the last hope of a rest frame for the universe. Those were the absolutes of the container. What arrived in their place were the absolutes of the relation: the interval between events, the speed that converts one kind of separation into the other, the cone that says which events can touch which, and the laws, which say the same thing in every language they are translated into. The absolute did not die in 1905. It moved — out of the stage and into the rules. That is a stranger kind of absolute than Newton’s and a more durable one, because it does not require anyone to find the stage.

The speed of light, then, is not an exception that relativity has to tolerate. It is what relativity is made of. The frames are relative to one another because c is not, as the shadows move because the tree stands still. ‘Everything is relative’ is what you say when you have read the first column of the ledger and not turned the page. The theory’s own name told you to stop there. Einstein knew it, and said so, and it was already too late.