At a glance
A spacetime diagram plots time upward and space sideways, so one moment at one place becomes a dot and a whole history becomes a line. This sim lets you place two events, draw worldlines, and read what frame S assigns to the pair.
What you'll learn
- How an event and a worldline are drawn on a diagram of ct against x.
- How the slope of a worldline relates to the speed it stands for.
- How the readings Δx, cΔt and Δs² follow from where E1 and E2 sit.
- How the sign of Δs² relates to the way the two events are joined.
Time upward (as ct), space sideways (x), and two events with the dashed 45° lines through the origin.
What's on screen
- E1 and E2 (teal, rose): the events you drag, labelled (ct, x).
- Dashed yellow diagonals: 45° lines through the origin; with c = 1, the paths of light.
- Dotted teal lines (Detail 2): from each event to both axes.
- Amber legs (Detail 3): the two parts of the separation.
- Worldlines you draw, each labelled with its v/c.
- Frame S panel: Δx, cΔt and Δs², signed.
See itStart from the opening picture. Watch for: E1 (teal) and E2 (rose), each labelled with its (ct, x), and the dashed yellow lines crossing at the origin.
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 quarter units. | ct, x: −5.5 to 5.5 opens at E1 (ct 2, x −2), E2 (ct 2, x 2) | |
| Detail | 1 shows the plain diagram. 2 adds dotted lines from each event to both axes. 3 adds the separation triangle for the pair. | 1 / 2 / 3 opens at 1 | |
| S grid | Shows or hides the unit grid of frame S. | on / off opens on | |
| Now ct sweep | Draws a green dashed line of constant ct that climbs up the diagram, labelled with its ct. | on / off opens off | |
| ✎ Add worldline | Arms the tool: click two points to draw a line through them, labelled with its v/c. It disarms after each line. | any two points | |
| Frame S | Reads out Δx, cΔt and Δs² for the pair (E2 minus E1), signed, to two decimals. | panel | |
| ⏸ Pause / ▶ Play | Stops or starts the Now sweep. Nothing else on the diagram moves by itself. | top bar | |
| Speed | Scales how fast the Now sweep climbs. | 0.25×–4× opens at 1× | |
| ↻ Reset | Puts E1 and E2 back where they started, clears your worldlines, and returns Detail, S grid and the sweep to their opening settings. | top bar |
Fixed: c = 1, so ct and x are in the same units and light is exactly 45°. Only frame S is drawn.
What a dot is, and what its numbers mean
A dot is an event: one place at one moment, such as a flash or a collision. It is not an object and not a path.
- Both axes are lengths. Multiplying time by c turns it into a distance, so ct and x share one unit. The sim sets c = 1: one unit of ct is the time light takes to cross one unit of x.
- A frame is a labelling rule. It gives every event a pair (ct, x). Frame S is the one drawn here, and its axes stay fixed. Another frame would label the same event with different numbers; the labels belong to the frame, the event does not.
- Equal ct, equal x. Two events with the same ct are at the same moment in frame S. Two with the same x are at the same place in frame S.
One second of ct is one light-second of distance, about 3.0 × 10⁸ m.
See itTurn on Detail 2 for the current pair. Watch for: dotted teal lines from each dot to the ct axis and the x axis, ending at the two numbers in its label.
A worldline is a history
A worldline is the whole history of one object drawn as one line: all its events in order. Nothing on the diagram moves; the picture shows the motion.
- At rest is vertical. An object that stays put still advances in ct, so its worldline runs straight up.
- Motion tilts it. If x advances with ct, the line leans. The sim labels it v/c = Δx / Δ(ct), signed: positive leans right.
- Steeper means slower. The slope drawn is Δ(ct)/Δx = 1/(v/c), so a nearly vertical line is a slow object.
- The 45° lines are light. They carry v/c = ±1. A line flatter than 45° would read |v/c| > 1. The tool lets you draw one, but no particle or signal has such a worldline.
At rest (teal), moving at v/c = +0.50 (blue), light (dashed). Shaded: slopes nothing can have.
See itDraw three worldlines with different tilts. Watch for: the vertical line labelled v/c = +0.00, the leaning ones labelled larger values, and all of them steeper than the dashed 45° lines.
Separating two events
In frame S, the pair E1, E2 is separated by a time part and a space part, both signed (E2 minus E1). Together they give the interval Δs².
In Δs² the time part carries the minus sign.
x₁
ct₁
x₂
ct₂
These sliders are E1 and E2: they move the events in the sim too.
- Δs² is neither a distance nor a time. It is the one combination of the two that every inertial frame agrees on, which is why it has a name.
- Detail 3 draws it: the vertical leg is cΔt, the horizontal leg is Δx.
See itTurn on Detail 3 for a sample pair. Watch for: the amber legs labelled Δx and cΔt, and the same two numbers, with Δs² = −5, in the card.
The sign of Δs² sorts every pair
- Δs² < 0, timelike. The time part wins; the joining line is steeper than 45°. Something slower than light can be at both events, so the earlier can cause the later. A negative value is a result, not an error.
- Δs² = 0, null. The joining line is exactly 45°: light from the earlier event arrives at the later one.
- Δs² > 0, spacelike. The space part wins; the line is flatter than 45°. No signal joins the pair, and some frame sees the two events at the same time.
- Frames split the interval differently between time and space, but they agree on Δs², so they agree on its sign.
Timelike pairs give Δs² < 0, null pairs 0, spacelike pairs Δs² > 0.
See itMove E2 through three positions around E1. Watch for: Δs² reading negative, then +0.00, then positive.
Where it matters
- A solar flare and its light arriving at Earth. The Sun is 1 AU, about 499 light-seconds (8 min 19 s), away. The pair has Δx = cΔt = 499 light-seconds, so Δs² = 0: null, and sunlight's worldline is the 45° line itself.
- "Now" here and "now" at Proxima Centauri. With Δx = 4.25 ly and cΔt = 0 in frame S, Δs² ≈ +18 ly²: spacelike. No signal can connect those two events.
- Everyday life. A handclap and a blink 1 ms later, 1 m apart, have cΔt ≈ 300 km against Δx = 1 m: very timelike. Light crosses 1 m in about 3.3 ns, which is why the 45° limit is never noticed at a desk.
Two real pairs of events, drawn like the sim's diagram.
Go deeper: what Δs² measures
- For a timelike pair, √(−Δs²) is c times the time recorded by a clock that moves at constant velocity and is present at both events: the proper time (Lecture 9).
- For a spacelike pair, √Δs² is the distance between the events in the frame where they are simultaneous.
Summary
In one line. A frame labels each event with (ct, x). The interval Δs² = −(cΔt)² + (Δx)² is the same in every inertial frame, and its sign says what can join the two events: slower-than-light travel, light, or nothing.
Key points
- A dot is an event; its coordinates are labels from a frame.
- A worldline is a history: vertical at rest, tilting with speed.
- With c = 1, light draws the 45° lines and v/c = Δx / Δ(ct).
- In this sim Δs² is negative for timelike pairs, zero for null, positive for spacelike.
Common mix-ups
- "An object at rest is a point." It still advances in ct, so its worldline is vertical.
- "Steeper means faster." The slope drawn is 1/(v/c), so steeper means slower.
- "Equal height means simultaneous for everyone." Equal ct means simultaneous in frame S; another frame can disagree for events at different x.
Things to try on your own
- Turn on Detail 3 and drag E2 until the pair lies along a dashed diagonal. Read Δs².
- Set Δx = 3 and cΔt = 1, then swap the two values. Compare the signs of Δs².
- Draw a steep and a flat worldline with ✎ Add worldline and compare their v/c labels.