At a glance
Two observers in relative motion can disagree about which of two events happened first. This sim places two events on a spacetime diagram and shows which pairs can change order in a second frame.
What you'll learn
- How the distance and time between two events decide whether light from one can reach the other.
- How the order of a pair in a moving frame S′ depends on the speed v/c of that frame.
- How the interval Δs² is built from cΔt and Δx, and what its sign tells you.
- How to read Δx, cΔt and the orderings off the Readings panel.
Frame S: ct up, x across. E1 and E2 are the two events.
What's on screen
- E1 (teal dot) and E2 (rose dot): the two events. Labels give (ct, x) in frame S.
- Light lines (dashed yellow): the two 45° lines through E1, the paths of light. The faint shading is the region above E1 between them.
- S grid and axes (grey): frame S. The S′ axes (teal, with a dashed teal grid) appear when you switch S′ on.
- Readings panel: Δx, cΔt, Δs², and the orderings (the primed ones once S′ is on).
See itPut E1 at the origin and E2 at x = 3, ct = 1. Watch for: the two dots on the diagram, and Δx and cΔt in the Readings panel matching E2's position.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Drag E1, E2 | Moves an event on the diagram. Positions snap to 0.25. | x and ct from −5.5 to +5.5 opens: E1 (0, 0), E2 at x = 3, ct = 1 | |
| Causality preset | Puts E1 at the origin and E2 at a fixed place: Timelike at x = 1, ct = 3; Spacelike at x = 3, ct = 1; Light-like at x = 2.5, ct = 2.5. | 3 buttons none highlighted at open | |
| S grid | Shows or hides the grey grid of frame S. | on / off opens on | |
| S′ axes and grid | Draws the axes and dashed grid of a second frame S′, adds the cΔt′ and Ordering in S′ rows to Readings, and reveals the v/c slider. Switching it on from v/c = 0 sets v/c = +0.60. | on / off opens off | |
| v/c: S′ relative to S | The velocity of S′ along x, as a fraction of c. A positive value means S′ moves toward +x. It tilts the S′ axes. | −0.90 to +0.90, step 0.01 opens at 0; shown once S′ is on | |
| Readings | Δx, cΔt and Δs² for the pair, a Classification badge, and the Ordering in S. With S′ on it adds cΔt′ and Ordering in S′. All but the primed rows are frame-S values. | panel | |
| ↻ Reset | Puts the events back to their opening places, switches S′ off and the S grid on. | top bar |
Fixed: ct and x share one scale, so light runs at 45°. Δs² = −(cΔt)² + (Δx)², measured from E1 to E2. Nothing in this diagram moves on its own, so Play/Pause and Speed have no effect here.
What E1 can influence
A signal leaving E1 travels at most at the speed of light, so it can reach E2 only if E2 lies in the shaded wedge above E1, or on its edge. In frame S, with Δx and cΔt measured from E1 to E2:
The reach condition.
The sim packs this into one number, the interval, and names its sign:
The interval, as the sim prints it.
- Δs² < 0, timelike: E2 is inside the wedge above E1 (or its mirror below). Something slower than light can link the events.
- Δs² = 0, light-like: E2 is on a light line. Only light can link them.
- Δs² > 0, spacelike: E2 is outside the wedge. No signal can link them.
With Δs² = −(cΔt)² + (Δx)², timelike pairs are negative and spacelike pairs positive.
See itPlace E2 at x = 3, ct = 1. Watch for: E2 outside the dashed light lines, and Δs² = +8.00 in the Readings panel.
How a second frame reorders events
Frame S′ moves along x at v relative to S. It gives the same two events new coordinates:
The boost from S to S′, with γ = 1/√(1 − (v/c)²).
- The sign of cΔt′ is the order in S′: positive means E1 first, negative means E2 first, zero means simultaneous.
- The teal x′ axis is the line ct = (v/c)·x. Events on a line parallel to it, such as the one through E1, are simultaneous in S′.
- So E2 comes before E1 in S′ when E2 lies below the x′ line through E1.
- The interval is the same in both frames: −(cΔt′)² + (Δx′)² = −(cΔt)² + (Δx)². The Readings panel shows cΔt′ but not Δx′.
See itSwitch on S′ at v/c = +0.60 with E2 at x = 3, ct = 1. Watch for: E2 below the teal x′ line, and Ordering in S′ reading E2 first.
Where the order flips
For a pair outside the wedge, cΔt′ passes through zero at one frame speed:
- For the opening pair, cΔt = 1 and Δx = 3, so the flip is at v/c = 1/3. At that speed S′ calls the events simultaneous. Faster, S′ puts E2 first.
- The threshold is a property of the pair, and it is the speed of a frame. No signal travels between the events.
- The slider stops at ±0.90. A spacelike pair with |cΔt/Δx| between 0.90 and 1 flips beyond the slider’s range.
v/c
This slider moves the sim's v/c slider too, and follows E1 and E2.
See itSet v/c to the flip speed for E2 at x = 3, ct = 1. Watch for: Ordering in S′ reading simultaneous, and swapping to E1 first or E2 first when you nudge the slider.
Why a pair inside the wedge keeps its order
Because Δs² is the same in every frame, (cΔt′)² = −Δs² + (Δx′)². For a timelike pair Δs² < 0, so cΔt′ can never reach zero.
The smallest cΔt′ is √(−Δs²), the time between the events in the frame where they happen at the same place.
- For the timelike preset (cΔt = 3, Δx = 1), Δs² = −8. Sweeping v/c from −0.90 to +0.90, cΔt′ runs +8.95, down to +2.83 (= √8) at v/c = +0.33, then up to +4.82.
- For a light-like pair, |cΔt′| = |Δx′|, so cΔt′ shrinks toward zero only as v/c → 1. The slider stops at 0.90, where the light-like preset gives cΔt′ = +0.57.
See itSweep v/c from −0.90 to +0.90 on the timelike preset. Watch for: cΔt′ sliding down to +2.83 and back up, and Ordering in S′ always reading E1 first.
What a faster-than-light signal would break
The sim draws events, not signals. This section is the argument for why the wedge matters.
One faster-than-light signal A → B, drawn in S and in S′ at v/c = +0.60.
- Suppose a signal could go from E1 to E2 outside the wedge. In S it leaves E1 and arrives at E2 later.
- In S′ at v/c = +0.60 the same two events are in the opposite order: the signal would arrive before it left.
- With a fast enough reply sent from S′, a message could even return before it was sent. Every frame agrees on the order of events inside a wedge, so signals slower than light cannot do this.
See itTake the same pair in S′ at v/c = +0.60. Watch for: E2 below the x′ line, so S′ reads E2 first while S reads E1 first.
Where it matters
- Tests of quantum entanglement. In the 2015 Delft experiment, two electron spins 1.3 km apart were measured so that the two measurements were spacelike separated. No signal at light speed could pass from one to the other.
- Gravitational-wave detectors. LIGO's Hanford and Livingston sites are about 3,000 km apart, so light needs about 10 ms between them. Detections closer together than that cannot be cause and effect, and the searches use this window to match triggers.
- Quantum field theory. Measurements at spacelike separation commute, so one cannot change the outcome of the other.
Go deeper: what faster than light would cost a particle
A particle with v > c would need pc > E, which makes m² negative.
- For a particle with mass, E = γmc² grows without bound as v → c, so no finite energy reaches c.
- Particles with m² < 0, called tachyons, have been proposed. None has been observed.
Summary
In one line. Δs² = −(cΔt)² + (Δx)² is the same in every frame, and only pairs outside the wedge (Δs² > 0) can have their order reversed by a change of frame.
Key points
- E1 can influence E2 only if |Δx| ≤ cΔt: E2 inside or on the wedge.
- Outside the wedge, cΔt′ changes sign at v/c = cΔt/Δx, which is below 1.
- Inside the wedge, cΔt′ is never smaller than √(−Δs²), so the order holds in every frame.
- Pairs that can swap order are exactly the pairs no signal can link.
Common mix-ups
- "If S′ reads E2 first, E2 caused E1." For a pair outside the wedge, neither event can cause the other.
- "The order only differs because light takes time to arrive." The frames compare coordinate times, not when light arrives.
- "A fast enough frame reverses any pair." A frame is slower than light, so only pairs with |cΔt/Δx| < 1 have a flip speed.
Things to try on your own
- Switch on S′ at v/c = +0.60 and drag E2 along ct = 1 to find where Ordering in S′ stops reading E2 first. Compare with x = cΔt/(v/c) ≈ 1.67.
- Move E2 to x = 5, ct = 4.75. Sweep the slider across its range and find out whether the order ever changes.