Illustrated Physics
physics, drawn
Relativity

Now is a choice of slicing

Two events happening at the same time is not a fact about the events. It is a fact about who is asking, and different observers slice spacetime at different angles.
12 min read 6 figures Where the model stopsWhat stays the same

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.

Simultaneity at β = 0.5Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.xctsame time, for one observersame time, for the otherABneither slicing is the right one: that is the content of relativity
Fig. 1 Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

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 t1t_1 and returns at t3t_3 by the first clock, and the second clock read t2t_2 when the signal arrived, the clocks are synchronised if

t2=t1+t32.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 cc 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 cc 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.

A spacetime diagram at β = 0.5Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.xctlightct′x′here, nowfuturepastunreachableγ = 1.155 — the axes close on the light line together
Fig. 2 A spacetime diagram with a moving observer’s axes drawn in. Their space axis — the set of events they call “now” — is tilted, and that tilt is the whole of the disagreement.

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.

A spacetime diagram at β = 0.75Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.75 of the speed of light.xctlightct′x′here, nowfuturepastunreachableγ = 1.512 — the axes close on the light line together
Fig. 3 A faster observer. The slice has tilted further, so the set of events they call simultaneous with here-and-now has swung further from the horizontal.

The size of the disagreement

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

Δt′=γvΔxc2.\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 c2c^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 3×10−133\times10^{-13} seconds. The effect is real, and it is smaller than any clock could resolve for most of the history of clocks.

Simultaneity at β = 0.8Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.xctsame time, for one observersame time, for the otherABneither slicing is the right one: that is the content of relativity
Fig. 4 The same two events, sliced by a much faster observer. The tilt is steeper and the time difference they assign is larger — and at no speed does the tilt reach forty-five degrees.

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.

A light clock at β = 0.6The same clock at rest and moving. Light covers the hypotenuse rather than the height, and since its speed is the same for both observers, the moving clock must take longer to tick.at restlight goes straight upmovinglight travels 1.25× as farγ = 1.250the ratio is forced by one constant speed
Fig. 5 A light clock at rest and moving. Comparing the moving clock against stationary ones requires two stationary clocks at different places — and whether those two are synchronised is exactly what observers in relative motion disagree about.

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.

The Lorentz factor against speedHow much clocks slow and lengths shrink, plotted against speed as a fraction of light. At a tenth of light speed the effect is half a percent; it only becomes dramatic in the last stretch.00.20.40.60.80246speed (fraction of light)1.011.151.672.293.20everyday speeds live here, indistinguishable from 1
Fig. 6 The Lorentz factor against speed. Both the slowing of clocks and the tilt of the simultaneity slices are governed by it, which is a hint that they are one phenomenon rather than two.

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

Later rungs: the Lorentz transformation derived from synchronisation rather than assumed. The train argument drawn on a diagram instead of narrated. Length contraction as a consequence of slicing a worldsheet at an angle — which is arguably the clearest derivation of it. 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.

Where this goes next

xctlightct′x′here, nowfuturepastunreachableγ = 1.155 — the axes close on the light line together Two axes, one speed, and a diagram that does the arguing Put position across and time up, insist that light travels at forty-five degrees for everyone, and nearly every result in special relativity becomes something to read off rather than derive. at restlight goes straight upmovinglight travels 1.25× as farγ = 1.250the ratio is forced by one constant speed The clock that has to slow, and why no clock can refuse One constant speed and one right-angled triangle force a moving clock to tick slower. The argument is Pythagoras, which is what makes it inescapable rather than merely surprising. +every line leaves: net flux counts the chargethe shape of the surface never enters the answer Counting what comes out, and never looking inside Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside.