Now is a choice of slicing

Of all the conclusions of special relativity, the one that does the most damage to ordinary thinking is not that clocks slow or that lengths shrink. It is that two things happening at the same time in different places is not a fact.

It depends on who is asking. Two observers moving past each other, both careful, both using good instruments, both correct, will disagree about whether two distant events happened together — and neither of them is making a mistake.

What “at the same time” has to mean #

The trouble starts with a question that seems too basic to matter: how would anybody establish that two distant events were simultaneous?

Not by seeing them together, because light takes time to arrive and the events are at different distances. What is needed is a pair of synchronised clocks, one at each place, and then a comparison of readings.

Which pushes the problem back a step: how are two distant clocks synchronised? The standard procedure — Einstein’s — is to send a light signal from one to the other and back. If it leaves at time $t_1$ and returns at $t_3$ by the first clock, and the second clock read $t_2$ when the signal arrived, the clocks are synchronised if

$$t_2 = \frac{t_1 + t_3}{2}.$$

The signal is assumed to take the same time each way. That assumption is what everything hinges on, and it is not something that can be tested without already having synchronised clocks — the procedure is a definition, and it is the only reasonable one available.

Now here is the difficulty. Two observers in relative motion both apply this procedure, both correctly, and they get different answers about which events are simultaneous. Not because either did the arithmetic wrong, but because the light signals travel at $c$ relative to each of them, and they are moving relative to each other.

The train, and why it is not a trick #

The standard illustration is worth its reputation.

A train moves along a platform. Lightning strikes both ends of the train at once — according to an observer on the platform, who has verified this by noting that the two flashes reach the platform’s midpoint together.

An observer at the middle of the train disagrees, and here is why. The train observer is moving toward the flash from the front and away from the flash from the rear. So the front flash reaches them first. Since light travels at $c$ in their frame too, and since they are equidistant from the two ends of their own train, they conclude that the front flash happened earlier.

Both accounts are complete and consistent. The platform observer says the strikes were simultaneous and the train observer moved into one of the signals. The train observer says the front strike came first and the flashes then travelled equal distances. Nothing distinguishes these as descriptions of reality, because there is no experiment that can pick one out — and the impossibility of detecting one’s own uniform motion is the principle doing the work, not a shortage of ingenuity.

The diagram makes it structural rather than anecdotal. An observer’s “now” is a slice through spacetime — the set of events they assign the same time coordinate. For a stationary observer the slice is horizontal. For a moving observer it is tilted, at an angle set by their speed, and the tilt is forced by the requirement that light travel at forty-five degrees for them too.

The size of the disagreement #

The tilt gives a formula. Two events separated by a distance $\Delta x$ and simultaneous in one frame are separated in time, in a frame moving at $v$, by

$$\Delta t' = \frac{\gamma v \Delta x}{c^2}.$$

Two features of it explain why nobody noticed for three centuries.

The disagreement is proportional to the separation. Two events at the same place are simultaneous for everybody; the effect needs distance to accumulate. This is why the relativity of simultaneity has no consequences whatsoever for anything happening in one room.

And it carries a factor of $c^2$ in the denominator, which is enormous. Two events a kilometre apart, judged by an observer moving at highway speed, differ in time by about $3\times10^{-13}$ seconds. The effect is real, and it is smaller than any clock could resolve for most of the history of clocks.

That last point is the essential one. The slice tilts toward the light line and never reaches it. So no observer’s “now” ever includes an event inside the light cone of another — the slicing can be tilted, and it cannot be tilted past the diagonal.

Causality survives, exactly #

The natural worry is that this breaks cause and effect. If observers disagree about which of two events came first, could one of them see an effect precede its cause?

No, and the reason is precise rather than reassuring. The order of two events can be reversed by a change of frame only if they are spacelike separated — outside each other’s light cones, so far apart in space relative to their separation in time that no signal could pass between them. And events that no signal can connect cannot be cause and effect.

So relativity permits disagreement about temporal order in exactly the cases where the order carries no causal meaning, and forbids it in every case where it does. That is not a patch; it is a structural consequence of the light cone being frame-independent, and it is the strongest internal evidence that the theory is consistent.

The pattern is a familiar one in physics: a quantity that seemed absolute turns out to be observer-dependent, and what survives is a different quantity that nobody had thought to look for. Simultaneity dissolves and the invariant interval takes its place, much as the separate conservation of mass and energy dissolves and a single conserved four-vector takes theirs. In each case the loss is of something intuitive and the gain is of something that every observer agrees on.

It also has a hard corollary. Any signal that travelled faster than light would connect spacelike-separated events, and some observer would see it arrive before it left. Faster-than-light signalling and backwards-in-time signalling are the same thing, not two different exotic possibilities, and the prohibition on the first is the prohibition on the second.

Why the two effects cannot be separated #

Simultaneity disagreement is not an extra oddity alongside time dilation. It is what makes time dilation consistent, and any account that drops it produces a contradiction within two sentences.

The symmetry is the problem. Each observer says the other’s clock runs slow, and both are right. That looks impossible until the measurement is described carefully: comparing a single moving clock against a stationary frame means reading it against one stationary clock at the start and a different one at the end. The moving observer agrees about each individual reading and denies that the two stationary clocks were ever synchronised.

Nothing is left over. Every apparent paradox in special relativity — the twins, the ladder in the barn, the pole-vaulter’s contradiction — dissolves the moment the simultaneity slices are drawn in, and none of them dissolves without.

Both the slowing of clocks and the tilt of the simultaneity slices are governed by the same factor, which is the hint that they are one phenomenon rather than two. A single function of speed appears in the rate, in the length and in the tilt; if these were separate effects there would be no reason for them to share it. They share it because they are three readings of one geometrical fact about how a boost acts.

The two numbers that bound the effect #

The formula’s two factors — separation and speed — set a range so wide that both ends of it are worth writing down, because the effect is simultaneously the most negligible and the most extravagant result in the theory.

At the small end: two events a metre apart, judged by someone walking past at a metre per second, differ in time by about $10^{-17}$ seconds. That is a hundredth of a femtosecond, which is shorter than the period of visible light by a factor of a hundred. No arrangement of ordinary objects produces a simultaneity disagreement that any instrument could see.

At the large end, with the same walking speed: two events in the Andromeda galaxy, 2.5 million light years away, judged by two people strolling in opposite directions at a metre and a half per second. The separation term is now $2.4\times10^{22}$ metres, and the disagreement comes to about four days. Two people passing on a pavement have “nows” in Andromeda that differ by the better part of a week — one of them slicing through events the other places firmly in the past.

Nothing physical distinguishes the two cases. The same formula, the same speeds, and a factor of $10^{22}$ in the only other quantity present. That is the honest summary of why the relativity of simultaneity was discovered so late and sounds so outrageous: the term that carries it is multiplied by a distance, and human experience contains no large ones.

What a shared present costs to build #

Since there is no universal now, any system that needs one has to construct it, choose it, and pay for it. Modern infrastructure needs one constantly, and the bill is itemised in nanoseconds.

The chosen slicing for terrestrial timekeeping is the one belonging to an observer rotating with the Earth, referred to the geoid. That is a convention, adopted because it is convenient, and it has to be maintained: every contributing atomic clock sits at a different altitude and a different rotational speed, and its rate is corrected to what it would read on the geoid before it is allowed to contribute. International Atomic Time is not a measurement of anything. It is an agreed slice, assembled from about 450 clocks that all disagree slightly and are all correct.

Signals travelling across a rotating Earth pay an explicit charge for the choice. Light sent eastward round the equator and light sent westward do not take the same time in the rotating frame, and the difference — the Sagnac term — comes to about 207 nanoseconds for a full circuit. Since one nanosecond of timing error is thirty centimetres of position error, satellite navigation cannot ignore it, and every GPS receiver applies the correction as a matter of course. The relativity of simultaneity arrives in consumer electronics as a term in a firmware routine.

The synchronisation convention itself is bought on credit throughout computing. Network time protocols time a round trip and split it in half, which is Einstein’s procedure with routers in place of mirrors — and it assumes, as he did, that the two directions take equal times. That assumption is false on almost any real network, since packets outbound and inbound frequently take different routes, and the resulting asymmetry is the dominant error in internet timekeeping. The most demanding installations measure the asymmetry of each fibre explicitly rather than assuming it away, which is as close as engineering comes to admitting that the one-way speed of light is a convention and then charging for it.

Where magnetism comes from #

The most consequential thing the tilted slice does is not to a clock. It is to a density, and the result is that one of the four forces turns out to be this page’s subject in disguise.

A current-carrying wire is electrically neutral: as many protons in the lattice as electrons drifting past them, so a charge sitting beside it feels no electric force. Set that charge moving parallel to the wire and it feels a force anyway — the magnetic one, which is why the experiment is done with a compass needle rather than an electroscope.

Now describe the same situation in the moving charge’s own frame. Here the lattice is moving, so the spacing between protons is contracted and the positive charge per metre is higher than it was. The electrons are moving too, at a different speed, so their spacing is contracted by a different factor and their negative charge per metre is lower. The wire that was neutral now carries a net positive charge per unit length, and the moving charge feels an ordinary electrostatic attraction — of exactly the size the magnetic force had in the other frame.

Charge per unit length is where the simultaneity enters, and it is worth being explicit about why. Counting how many charges lie in a metre of wire means marking the two ends of that metre at the same time, because the charges are moving. Two observers who slice differently mark different pairs of events, enclose different numbers of electrons, and get different densities — from the same wire, with nothing about it changed.

So magnetism is not a separate force that happens to accompany electricity. It is what the electric force of a moving distribution looks like when the observer’s “now” is tilted relative to the charges’, and the factor $v/c^2$ in the formula above is the factor in front of the whole magnetic interaction.

Lorentz had the formula and did not believe it #

The expression for the disagreement is older than the theory that explains it, which is a recurring shape in this collection and is unusually stark here.

Lorentz introduced a quantity he called local time in the 1890s — the time coordinate $t - vx/c^2$ — as a mathematical device for making Maxwell’s equations take the same form in a moving frame. It is precisely the tilt of the slice: a time offset proportional to position, with the same $v/c^2$ in front of it.

He did not think it was time. It was an auxiliary variable, a substitution that made the algebra come out, while the real time was the one belonging to the ether frame; the moving observer’s clocks were, on this account, simply wrong in a way that conspired to be undetectable.

Poincaré went a step further in 1900 and identified what the auxiliary variable actually was: it is what clocks in the moving frame read if they are synchronised by light signals in the way described at the top of this page, assuming the light takes equal times each way. That is the whole argument, and it was in print five years before 1905.

What was left was to stop calling one of the two times real. Einstein’s contribution at this point was not the formula, which existed, nor its interpretation, which Poincaré had given. It was the refusal to keep a preferred slice that no experiment could find — and everything on this page follows from taking the two observers as equally right rather than as one right and one usefully mistaken.

The universal now, and its absence #

The everyday picture of time is a moving present sweeping forward: a single global “now”, the same everywhere, dividing a fixed past from an open future.

Relativity does not contain this. There is no preferred slicing, so there is no fact about which events elsewhere in the universe are happening now. For events on Earth the ambiguity is nanoseconds. For an event in the Andromeda galaxy, the difference between two observers strolling in opposite directions on a pavement amounts to days.

That observation has been used to argue for a “block universe” — a spacetime that simply exists in its entirety, with the passage of time an artefact of how it is experienced from inside. It is a genuine philosophical position with genuine opponents, and it goes further than the physics strictly requires: the equations say there is no preferred foliation, not that the future is fixed. But the everyday notion of a universal present tense does not survive contact with the light cone, and something has to replace it.

What relativity offers instead is local and considerably better defined. Each observer has a proper time, read by a clock they carry, and it is unambiguous. Each event has a past and a future cone, and those are unambiguous. What has no observer-independent meaning is the comparison of times at separated places — and it turns out nothing depends on that comparison except intuition.

Where the model stops #

Flat spacetime. Everything above is special relativity. In general relativity the situation is worse and better at once: there is no global time coordinate at all in a general curved spacetime, and yet many spacetimes of physical interest — including the one describing the expanding universe — happen to permit a natural slicing. Cosmic time, on which the universe is 13.8 billion years old, is a choice singled out by the matter distribution rather than by the geometry. It is a preferred slicing in practice and not in principle.

Inertial observers. The tilted axes belong to an observer moving uniformly. An accelerating observer’s simultaneity slices rotate as they accelerate, and can sweep back and forth over distant regions in ways that look alarming and mean nothing — the slices are coordinates, and coordinates are free.

The synchronisation convention. The whole framework rests on defining light’s speed to be the same in both directions. This cannot be measured — measuring a one-way speed needs synchronised clocks, and synchronising clocks needs a one-way speed. Alternative conventions exist and give the same predictions for everything observable. The convention is chosen because it is the simplest, not because it has been verified.

The picture is two-dimensional. A simultaneity slice in real spacetime is a three-dimensional volume, not a line. The tilted line in the figure is the edge of a tilted hyperplane, and none of the pictures can show that — the same dimensional shortfall that makes a two-dimensional drawing of field lines misstate its own density law.

And a limit of the figures specifically: each draws two frames on one diagram, which requires choosing whose axes are perpendicular. That choice looks like a claim that one observer is the real one. It is not — the drawing simply has to start somewhere, and every statement in this essay could be drawn the other way round with the roles exchanged. The asymmetry is on the page, not in the world.

The ladder from here #

The next rung is length contraction, which is this disagreement applied to the two ends of one object and is arguably the clearest derivation of it. After it: the Lorentz transformation derived from synchronisation rather than assumed. The train argument drawn on a diagram instead of narrated. The ladder-and-barn paradox, which is simultaneity disagreement wearing a costume. The Andromeda argument and the block universe debate. The one-way speed of light and the conventionality thesis. Cosmic time and the preferred frame that cosmology has but relativity does not. And quantum entanglement, where correlations between spacelike-separated measurements coexist with the prohibition on signalling — a coexistence that is much stranger than either half.

Einstein’s 1905 paper opens not with light or with motion, but with two pages on how to set two clocks. Everything else in special relativity is downstream of that definition.