At a glance
This sim puts two events on a spacetime diagram and reads their separation in frame S and, once you switch it on, in a moving frame S′. Comparing the two frames shows what observers can and cannot disagree about.
What you'll learn
- How to read two events as the separations cΔt and Δx.
- How the light lines through E1 relate to every other event on the diagram.
- How the interval Δs² is built, and how its sign classifies a pair.
- How the readings in S and in a moving frame S′ compare, including the order of the two events.
Two events, E1 and E2, and the light lines through E1. ct runs up and x across.
What's on screen
- E1 (teal): the first event, labelled with its (ct, x).
- E2 (rose): the second event.
- Grey axes and grid: frame S. Both axes share one scale, so light runs at 45°.
- Dashed yellow lines through E1, shaded between the upper pair: the paths of light.
- Teal tilted axes and dashed grid: frame S′, when switched on.
- Orange dashed lines from each event to x = 0, labelled with its reception ct (Light signals switched on).
- Readouts: the Frame S panel, and the Frame S′ panel when S′ is on.
See itPut E1 and E2 at the opening pair. Watch for: E1 and E2 level on one ct line, with the Frame S panel reading cΔt = +0.00 and Δx = +4.00.
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. | ct and x within ±5.5, snapping to 0.25 opens at E1 (1, −1), E2 (1, 3) | |
| Event coordinates | Lets you type ct and x for either event, entered in frame S or in frame S′. | ±8 in S | |
| S grid | Shows or hides frame S's grid lines. The axes stay. | on / off opens on | |
| S′ axes and grid | Adds frame S′: its tilted ct′ and x′ axes, a dashed grid, the v/c slider and the Frame S′ panel. | on / off opens off | |
| v/c: S′ relative to S | The speed of S′ along x, as a fraction of light speed. | −0.90 to +0.90 step 0.01, shows +0.60 when S′ is first switched on | |
| Light signals and reception times | Draws a light signal from each event to an observer at x = 0, and labels each with its reception ct. | on / off opens off | |
| Frame S panel | Δx, cΔt, Δs² and a Classification label for the pair, all measured in S. | panel | |
| Frame S′ panel | The same readings measured in S′. It appears when S′ is on. | panel | |
| ▶ Play / ⏸ Pause | Runs or freezes the moving pulse on each light signal. Nothing else moves on its own. | top bar | |
| Speed | Plays the pulses faster or slower. The arrival times do not change. | 0.25×–4× opens at 1× | |
| ↻ Reset | Puts E1 and E2 back at their opening positions and switches S′ and Light signals off. | top bar |
Fixed: the Classification reads null whenever |Δs²| is below 0.035.
Events and the light lines through E1
A point on the diagram is an event: where and when something happens, written (ct, x). The Frame S panel reports the separation between E1 and E2:
Both measured in frame S. Equal ct means simultaneous in S.
- Light from E1 covers Δx = ±cΔt, a 45° line, because both axes use the same scale. The dashed yellow lines through E1 are the paths of light through E1, and the shading lies between the upper pair.
- A signal leaving E1 at speed c or less covers the distance |Δx| in the time Δt only if |Δx| ≤ cΔt. That puts E2 inside the shaded wedge or on its edge, and light itself arrives on the edge.
- An E2 outside the wedge cannot be reached from E1 by anything. The light lines belong to E1, so they move when E1 moves.
The condition for E1 to reach E2.
See itPut E1 at the origin and E2 inside the wedge. Watch for: E2 sitting between E1's two upper light lines, with cΔt larger than Δx in the Frame S panel.
Seeing is later than happening
With Light signals on, an observer sits at x = 0. Light from an event at (ct, x) runs along a 45° line to the ct axis, which is the observer's worldline. The sim's reception rule:
Arrival ct at x = 0 for the opening pair.
- The arrival ct is the event's own ct plus the light's travel time |x|. The two are equal only for an event at x = 0.
- Equal ct in S means the events happen together. They are seen together only if their |x| is also equal.
See itTurn on light signals for the opening pair. Watch for: the two orange reception labels at different ct, although E1 and E2 start at the same ct.
The interval sorts pairs into three kinds
The sim combines the two separations into one number:
The Δs² row of the Frame S panel.
- Δs² < 0, timelike. E2 is inside E1's light lines, in its future or its past. √(−Δs²) is cΔτ, where τ is the proper time: what an inertial clock present at both events records.
- Δs² = 0, null. E2 is on one of E1's light lines. Only light connects the pair.
- Δs² > 0, spacelike. E2 is outside E1's light lines. √Δs² is the proper distance, the separation in the frame where the two events are simultaneous.
E1 at the origin. Only the future wedge is shaded, as on the sim's canvas.
With Δs² = −(cΔt)² + (Δx)², timelike pairs are negative and spacelike pairs positive.
See itMove E2 to three places in turn. Watch for: Δs² changing from negative to zero to positive as E2 goes from inside the light lines, to one of them, to outside, with the Classification following.
The same pair in a moving frame
Frame S′ moves along x at speed v relative to S. Its readings follow from S's, with γ = 1/√(1 − (v/c)²):
The boost the Frame S′ panel applies.
v/c
This slider is the sim's own v/c control. The numbers follow E1 and E2.
- cΔt′ and Δx′ change with v/c. Δs² does not: −(cΔt′)² + (Δx′)² = −(cΔt)² + (Δx)² in every frame, which is why it is called the invariant interval.
- On the diagram, the ct′ and x′ axes tilt toward the light line by the same angle. They stop being perpendicular, so this is not a Euclidean rotation. The light lines do not move, so a pair keeps its kind in every frame.
E1 at the origin, E2 at ct 3, x 1, and v/c = +0.60. The dashed line holds every event that S′ calls simultaneous with E2.
See itPut E2 at ct 3, x 1 with S′ on at v/c = +0.60, then drag the v/c slider. Watch for: Δs² reading −8.00 in both panels, and staying there as cΔt′ and Δx′ change.
Order can change only for spacelike pairs
cΔt′ is zero when v/c = cΔt/Δx. A frame has |v/c| < 1, so it can reach that value only if |cΔt| < |Δx|, which is a spacelike pair. Past that value cΔt′ has the opposite sign, and S′ sees the two events in the opposite order.
| pair | kind | reverses at v/c | what other frames see |
|---|---|---|---|
| cΔt = 1, Δx = 3 | spacelike | +0.33 | reversed order above +0.33 |
| cΔt = 3, Δx = 1 | timelike | never (needs 3) | the same order in every frame |
Timelike and null pairs keep their order in every frame. No signal, light included, can connect a spacelike pair, so a reversed order never puts an effect before its cause.
The sim's v/c slider stops at ±0.9, so a spacelike pair with |cΔt/Δx| above 0.9 cannot be seen to reverse on it.
See itSweep v/c from −0.9 to +0.9 for each pair in the table. Watch for: cΔt′ turning negative for the spacelike pair and staying positive for the timelike one.
Where it matters
- Looking out is looking back. Sunlight is about 8.3 minutes old when it reaches you, Proxima Centauri's light is 4.25 years old, and Andromeda's is about 2.5 million years old. Each is the reception rule with x as the distance.
- Commands to a Mars rover. Light takes about 3 to 22 minutes one way, depending on where the planets are. Nothing sent now can affect what the rover does before then.
- GPS. A receiver gets its position from the arrival times of signals from several satellites. Each signal takes 67 to 86 ms, depending on how high the satellite is, and a 1 ns timing error is about 30 cm.
Go deeper: why Δs² is invariant, and the smallest cΔt′
- With β = v/c, −(cΔt′)² + (Δx′)² = γ²[−(cΔt − βΔx)² + (Δx − βcΔt)²] = γ²(1 − β²)[−(cΔt)² + (Δx)²] = −(cΔt)² + (Δx)². The cross terms cancel.
- For a timelike pair (cΔt′)² = −Δs² + (Δx′)², so cΔt′ is smallest, equal to √(−Δs²), in the frame where Δx′ = 0. For E2 at ct 3, x 1 that is v/c = 1/3, where cΔt′ = √8 ≈ 2.83.
Summary
In one line. Δs² = −(cΔt)² + (Δx)² is the same in every frame, its sign says whether E2 lies inside, on or outside E1's light lines, and only spacelike pairs can change order.
Key points
- Light from an event at (ct, x) reaches x = 0 at ct + |x|, later than the event itself.
- A boost changes cΔt and Δx, leaves Δs² and the light lines alone.
- Timelike and null pairs keep their order in every frame. Spacelike pairs can change it.
Common mix-ups
- "Two events with the same ct are seen together." Same ct means they happen together in S. They are seen together only if |x| is also the same.
- "A boost shrinks Δs²." A boost changes cΔt and Δx. Δs² stays the same.
- "If the order can reverse, an effect could come before its cause." Only spacelike pairs reverse, and no signal can connect them.
Things to try on your own
- Type E1 = (0, 0) and E2 = (2, 5) in Event coordinates, switch S′ on, and find the v/c where cΔt′ reads +0.00. Compare Δx′ there with √Δs².
- Put E2 on a light line through E1, switch S′ on and sweep v/c. Note what Δs² and the Classification do.
- Put E2 at ct 3, x 1 and sweep v/c to find the smallest cΔt′. Compare it with √(−Δs²).