Relativity That ClicksEpisode 5 / Simultaneity Is Up to the Observer ── but Causality Is Invariant

The most counterintuitive heart of relativity ── and precisely the continuation of the view that "ct = const is now"

Simultaneity Is Up to the Observer ── but Causality Is Invariant "Now" = the surface where ct is constant. But how you slice it tilts from observer to observer. So "simultaneous" shifts.
Even so, the order of cause and effect alone is invariant for everyone ── the watchman is "nothing exceeds c."

Tools you'll need: the spacetime interval s² and light cone from Episode 3, the "ct is the surface of now" from Episode 1 This episode's core: simultaneity = gauge, causality = invariant

Up to now, both time dilation and $E=mc^2$ we've accepted fairly readily. Here we touch the heart of what is most counterintuitive in relativity ── the very concept of "simultaneous" differs from observer to observer. Two events that happened at the same instant for one person appear to another as "one of them first." This is neither an illusion nor a measurement error; it is the true face of spacetime. In fact, this is the direct continuation of the view that "relativity sees the world at $ct=$ constant" ── because "now" is the surface where $ct$ is constant (the surface of simultaneity), and how that surface is sliced tilts from observer to observer, simultaneity shifts. But rest easy. What shifts stops at "simultaneity"; the order of cause and effect can never be flipped for any observer. The watchman for that is exactly what we've been repeating since Episode 1: "nothing exceeds $c$." With the light cone, we cleanly sort simultaneity (stage machinery) from causality (invariant).

01Is "simultaneous" obvious?

In Newton's world, time was a single flow shared by the whole universe. "This present moment" was the same everywhere in the cosmos, and whether two events were "simultaneous" had one answer for everyone. It seems all too obvious. But relativity denies this "absolute now." The trigger, again, is the single fact that the speed of light is the same for everyone. When a moving person defines "simultaneous for me" using light, it ends up shifted from a stationary person's "simultaneous."

02ct=const is "now" ── and how it's sliced tilts by observer

On a spacetime diagram (vertical $ct$, horizontal $x$), what is "now"? For a given observer it is every place that exists at the same instant = the horizontal surface where $ct$ is constant. This is the "space of now." Time advancing means this $ct=$ constant surface sliding upward. Up to here, it's the same as Newton. The decisive difference is ──

The surface of "now" (the simultaneity surface) tilts from observer to observer

$$\text{The stationary person's now: }\ ct=\text{const (horizontal)}\qquad\text{The moving person's now: }\ ct=\beta x+\text{const (tilted)}$$

For an observer moving at speed $\beta$, the "surface of simultaneity" becomes, on the spacetime diagram, a line tilted by $\beta$ (just as one's own world line tilts from the $ct$ axis, the simultaneity surface tilts as its mirror image ── opening like a pair of scissors). So "which event is now" ── how the ct=const surface is sliced ── differs from observer to observer.

Why does it tilt? Just the intuition. A moving observer decides "I emit light left and right at the same time, and it arrives at both ends at the same time = simultaneous." But from the stationary person's view, the observer is moving, so the rear end approaches the light and gets hit first, while the front end runs away and gets hit later ── the two points the moving person calls "simultaneous" are not simultaneous for the stationary person. Because the speed of light is the same for both, it cannot be otherwise. Simultaneity was a convention (a gauge) that depends on the procedure of synchronizing clocks with light.

03So simultaneity shifts ── but order changes only for "spacelike"

When the simultaneity surface tilts, "which of two events came first" can change with the observer. But ── it does not change for every event. The tilt you can give the simultaneity surface is $\beta$, so it is always flatter than 45° (a light ray). Therefore ──

Events whose order changes vs. events whose order doesn't

•If two events are spacelike ($s^2<0$, outside the light cone) ── tilt the simultaneity surface and their before/after swaps. "Simultaneous," too, is up to the observer.
•If two events are timelike / lightlike ($s^2\ge0$, inside/on the light cone) ── for any $\beta$, the order is invariant. First is always first.

The sign of the Episode 3 $s^2$ does the work here. Events outside the light cone (spacelike) cannot even be linked by light = they cannot influence each other, so if their order swaps, nobody is troubled. Meanwhile, events inside the light cone (timelike) can be linked at less than light speed = they can be cause and effect, so their order had better be preserved ── and it is, properly.

04Let's play with it ── tilt the "now" line and the order swaps

The spacetime diagram below. Place event A at the origin and another event B. With buttons you can switch B between "spacelike (outside the light cone)" and "timelike (inside)." The slider is the observer's speed $\beta$ ── this tilts your "now" line (the simultaneity surface $ct=\beta x$).

When B is spacelike: move $\beta$ and the "now" line straddles B, so a B that was after A swaps to before ── simultaneity and order are both up to the observer. When B is timelike (inside the light cone): no matter how much you tilt the line (at $\beta<1$ you can't exceed 45°), B can never be straddled ── the order A→B is invariant for everyone. The order of events that could be causal is preserved.

Figure: spacetime diagram. Origin A and event B, with the light cone (45°). The slider tilts the observer's "now" line (simultaneity surface ct=βx). If B is spacelike (outside the cone) the order swaps; if timelike (inside the cone) it never changes.

observer's speed β (= tilt of the "now" line) 0.00

the "now" line (simultaneity surface ct=βx) light cone (45°) events A and B

05Causality is invariant ── the watchman is "nothing exceeds c"

To sum up, spacetime has a two-layer structure. "Now" (how the $ct=$ constant simultaneity surface is sliced) is each observer's convention = stage machinery, and it tilts. But the light cone (the sign of $s^2$) is invariant, and it protects the order of cause and effect. From cause to effect, the path always runs inside the light cone (timelike) ── so for any observer, the cause comes first. What makes this watchman possible is that nothing can send a signal faster than $c$. If there were a faster-than-$c$ signal (a tachyon), events outside the light cone could be linked causally, and to some observer the effect would appear before the cause ── a time paradox. The $c$ limit is not merely a speed limit; it was the keystone that protects the causality of the universe.

A handshake with "Fields That Click" and "Cosmology That Clicks" When "Fields That Click" Episode 1 said "the news (the change in a force) travels at c," that was precisely the flip side of this causality requirement ── influence propagates only inside the light cone. And the view that "$ct=$ constant is now, and how it's sliced is a gauge" is the same skeleton as the backbone of your "Cosmology That Clicks," $c\cdot t=\text{constant}$. There, the time evolution of the universe was read as the choice of the $c\cdot t$ surface (= the gauge of the time coordinate), and it held that "if the dimensionless $\alpha$ is invariant, the physics is the same" ── the choice of simultaneity surface is stage machinery; what is invariant is the causal structure and dimensionless quantities. Relativity and cosmology connect through the same single principle.

◇ ◇ ◇

The honest line ── the reach of "simultaneity is a gauge"

The relativity of simultaneity (the simultaneity surface tilting by $\beta$), the fact that spacelike-separated ($s^2<0$) events can have their before/after swapped depending on the observer, that for timelike/lightlike separation ($s^2\ge0$) the causal order is invariant for all observers, that a superluminal signal would break causality ── all of these are established physics. The mutuality of Episode 2's "each clock sees the other's running slow" is also fully explained by this difference in the tilt of the simultaneity surface.

"Simultaneity is a gauge (a convention)" is a statement about synchronizing clocks at different places. The proper time $\tau$ read by the same clock at the same place (Episode 3) is invariant, and that is no convention. Also, the horizontal "$ct=$ constant is now" surface is the natural slice an inertial observer chooses in flat spacetime; for accelerating observers or under gravity (curved spacetime, Episode 6), the surface of "now" curves in more complicated ways ── the freedom of the simultaneity surface is part of the coordinate (gauge) freedom we'll handle in Episode 7. Tachyons are a theoretical possibility and have not been found experimentally.

Practice problems (solvable with this episode's ideas)

What kind of line is a stationary observer's surface of "now" on a spacetime diagram? And for an observer moving at β?
See the answer
Stationary: horizontal (ct = const). Moving: a line of slope β (ct = βx + const). Just as one's world line tilts, the simultaneity surface tilts as its mirror image.
In what case does the order of two events swap depending on the observer? Answer in terms of s².
See the answer
Only for spacelike separation ($s^2<0$, outside the light cone). For timelike/lightlike ($s^2\ge0$, inside/on), the order is invariant for every observer.
Why is it no trouble if the order of spacelike-separated events changes?
See the answer
They cannot even be linked by light (they cannot send signals to each other), so they cannot have a causal relation. Since they are not cause and effect, it makes no contradiction whichever comes first.
If a faster-than-c signal could be used, what would happen? What is the true role of the c limit?
See the answer
Events outside the light cone could be linked causally, and to some observer the effect would appear before the cause = causality (a time paradox) breaks. The c limit is not a speed limit but the keystone that protects the causality of the universe.

Episode 5 SummarySimultaneity is a gauge, causality is invariant ── the light cone sorts them

"Now" is the simultaneity surface where $ct=$ constant. In Newton it is horizontal (absolute), but in relativity it tilts by $\beta$ from observer to observer (a consequence of the constancy of light speed) ── so "simultaneous" is each observer's convention (a gauge). But the surface can only be tilted flatter than 45°, so the order swaps only for spacelike-separated events ($s^2<0$, outside the light cone). For timelike/lightlike ($s^2\ge0$, inside/on), the order is invariant for all observers.

From cause to effect the path always runs inside the light cone, so the order of causality is protected ── the watchman for that is "nothing exceeds $c$." The $c$ limit is not a speed limit but the keystone that protects causality. The difference in the tilt of the simultaneity surface is also the true nature of Episode 2's mutuality, "each sees the other's clock running slow." Simultaneity (how it's sliced) is stage machinery; the light cone (the causal structure) is the invariant ── and this "how ct = const is sliced = a gauge" viewpoint is the same skeleton as your "Cosmology That Clicks," $c\cdot t=$constant.

Coming next ── Episode 6 "Gravity is the curvature of spacetime ── why c is the limit." Up to here, flat spacetime without gravity (special relativity). Next, at last, to gravity. Inside an elevator, gravity and acceleration cannot be told apart (the equivalence principle). From here opens general relativity, in which gravity is not a force but the curvature of spacetime itself, and objects are merely going "straight" through curved spacetime. And the answer to the homework we've been holding since Episode 1 ── ⟨why is c the absolute limit⟩ ── also lies in the very structure of spacetime. We'll shake hands with "Force That Clicks" Episode 5, "Gravity Is Geometry."

This document is Episode 5 of the "Relativity That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The relativity of simultaneity (the simultaneity surface $ct=\beta x+\text{const}$ tilting per inertial frame), the fact that for spacelike-separated ($s^2<0$) event pairs the time order is observer-dependent while for timelike/lightlike separation ($s^2\ge0$) the causal order is invariant, that causality breaks if superluminal signals exist, and that the mutuality of time dilation is explained by the difference in simultaneity surfaces ── all of these are established standard physics. That "simultaneity is a gauge (a convention)" is a statement about synchronizing clocks at distant places while the local proper time $\tau$ is invariant, that the horizontal simultaneity surface is the natural choice of an inertial observer in flat spacetime while in accelerating frames or curved spacetime (Episodes 6 and 7) the simultaneity surface curves, and that tachyons are a theoretical possibility not yet detected, are all noted in the "honest line" section. The figure is a schematic of the tilt of the simultaneity surface and the classification of events by the light cone in 1+1-dimensional spacetime. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are frozen and hidden). Neighboring episodes: Episode 4 E=mc² / Table of Contents / sister series Cosmology That Clicks.

Print / Save as PDF: ⌘+P (Ctrl+P on Windows). On screen, tilt the "now" line with β and check whether the order of B (spacelike/timelike) swaps. Click "See the answer" to open a solution.