At a glance
This sim draws two events on the spacetime diagram of frame S and shows the times a second frame S′, moving along x, assigns to them. It is about what the two frames agree on and what they do not.
What you'll learn
- How to read a time in S′ off the tilted S′ axes of the diagram.
- How the times S′ assigns to the two events depend on the boost v/c and on their separation Δx.
- How the order of the two events in S′ depends on the direction of the boost.
- What the interval Δs² is, and how it sorts pairs of events.
Frame S (pale axes) and frame S′ (teal axes). A and B are level: the same ct in S. Not to scale.
What's on screen
- Event A (blue) and event B (rose), each labelled with (x, ct). Drag either one sideways.
- Pale axes and grid: frame S, with x across and ct up in the same unit.
- Teal axes ct′ and x′: frame S′, with squares at whole units and a faint dashed grid.
- Dashed yellow lines: light rays at 45° through the origin.
- Dashed lines from A and B to the ct′ axis. The label there is that event's ct′.
- Corner box: v/c and Δct′. Readings panel: the full numbers for both frames.
See itShow the diagram with S′ at its opening v/c = 0.60. Watch for: A and B level with each other, and the teal S′ axes leaning toward the yellow light line.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| v/c (S′ relative to S) | The speed of S′ along x, as a fraction of c. Positive means S′ moves toward +x. Moving it pauses Play. | −0.90 to +0.90 opens at +0.60 | |
| Drag A or B | Moves an event sideways. Its ct stays at 3, so only Δx = xB − xA changes. | x from −4.7 to 4.7 opens at A −1.50, B 2.53 | |
| S grid (Display layers) | Lines of constant x and constant ct, for frame S. | on / off opens on | |
| ct′ readouts | The ct′ label and dot where each event's dashed line meets the ct′ axis. | on / off opens on | |
| S′ grid | Lines of constant ct′ and constant x′, for frame S′. | on / off opens on | |
| Light cone | The two 45° light lines through the origin. | on / off opens on | |
| Readings | γ, the times and separations in both frames, and Δs² in S and in S′. | panel opens collapsed | |
| ▶ Play / ⏸ Pause | Sweeps v/c back and forth between −0.90 and +0.90, or freezes it. | top bar | |
| Speed | How fast the sweep runs. It reaches the same limits at any setting. | 0.25×–4× | |
| ↻ Reset | Restores the events and display layers to their opening state, and pauses the sweep. | top bar |
Fixed: c = 1, so x and ct share one unit, and the sweep runs at 0.35 in v/c per second at 1×.
The S′ axes lean toward the light lines
S′ moves along x at speed v = βc. Its time axis ct′ is the path of its own origin, the line x = β ct, and its space axis x′ is the line ct = βx.
β is the speed of S′ measured in S, as a fraction of c.
- Light must travel at c in S′ as well. A light ray at 45° stays at 45°, so the ct′ and x′ axes lean toward the same light line, one from each side.
- This is not a Euclidean rotation. A rotation would turn both axes the same way and keep them at 90° on the page. Here the axes close together like scissor blades.
- Each axis tilts by arctan β from its S counterpart: 31° at β = 0.60, 42° at β = 0.90. For |β| < 1 it never reaches the light line.
See itSet v/c to 0.90. Watch for: the teal ct′ and x′ axes leaning close to the same light line from opposite sides, without touching it.
A time in S′ is read along a line of constant ct′
An event at (x, ct) in S has coordinates (x′, ct′) in S′. With c = 1, the sim computes:
The boost from S to S′. It is derived in Lecture 10; here, read what it does to the axes.
- ct′ is a clock reading, not an arrival time. It is what a clock at rest in S′, sitting at the event and synchronised with the other clocks of S′, reads there. Light-travel delays are already removed.
- Events with the same ct′ lie on a line parallel to the x′ axis. The sim draws that line from each event to the ct′ axis and prints the value there.
The two axes, and the line of constant ct′ through any event.
The label sits at height γ·ct′ on S's scale, because the tick marks on the S′ axes are spaced wider than those on S. Read the number, not the height.
See itFollow each dashed line from its event to the ct′ axis. Watch for: the blue and rose lines running parallel to the teal x′ axis, and the ct′ values printed where they meet the ct′ axis.
Equal times in S become unequal in S′
A and B share one ct in S, so Δct = 0. Put that into the boost and one term drops out:
Δct′ = ct′B − ct′A and Δx = xB − xA, in units where c = 1.
v/c
Δx
The v/c slider here is the sim's slider. Δx changes when you drag A or B.
- Δct′ is proportional to Δx. It needs both a boost (β ≠ 0) and a separation (Δx ≠ 0); lacking either, it is zero.
- It grows with β through γβ, which is 0.31, 0.75 and 2.06 at β = 0.30, 0.60 and 0.90. Doubling β more than doubles it.
The same two events at β = 0 and at the opening β = 0.60. A is at x = −1.50 and B at x = 2.53, both at ct = 3.
See itAt v/c = 0.60, bring B toward A: to half, then a tenth of its Δx. Watch for: Δct′ falling in step with Δx, to half and then a tenth of its first value.
Reversing the boost reverses the order
Δct′ = −γβΔx changes sign when β does, with the same size. The event further along the boost direction is the earlier one in S′.
Same events, opposite boosts. The dashed lines are lines of constant ct′ through each event.
- At β = +0.60, S′ moves toward +x and B is further along: ct′B = 1.85 comes before ct′A = 4.88, so B is first. At β = −0.60, A is first.
- In S′ the clocks of S move toward −x, so the clock at A leads. A moving array's leading clock reads behind its trailing ones, so A's reading of 3 happens later in S′.
- Only β = 0 or Δx = 0 gives a tie.
See itRun v/c = +0.60, then v/c = −0.60, on the same two events. Watch for: Δct′ changing sign, and the card naming the first event each time.
Only some pairs can swap order
This sim uses Δs² = −Δct² + Δx² (with c = 1), so a spacelike pair has Δs² > 0: no signal, even one at c, can join its two events. Δs² has the same value in S and in S′.
The interval is the same number in both frames. The sim prints both in Readings.
- For a spacelike pair, a boost can change the order (section 06). For a timelike pair (Δs² < 0) or a null pair (Δs² = 0) it cannot: then |Δx| ≤ |Δct|, so βΔx can never outweigh Δct, and Δct′ = γ(Δct − βΔx) keeps the sign of Δct.
- In this sim A and B always share ct, so Δs² = +Δx² and the pair is spacelike for any Δx ≠ 0. The sim shows the order swapping, but cannot show a pair that keeps its order.
The lines of constant ct′ through A for every boost the sim allows (|β| ≤ 0.9). They stay inside the spacelike region.
The yellow lines the sim draws are the light cone of the origin, not of A or B. "Spacelike" here describes the pair A and B.
See itPress Play to sweep v/c while Readings is open. Watch for: Δct′ swinging through zero and changing sign, while both Δs² values hold still.
Where it matters
- Why signals cannot exceed c. A signal faster than light would join a spacelike pair, and in some frame it would arrive before it was sent. This is the standard argument for c as the limit on signals.
- Synchronising clocks in a moving frame. Clocks synchronised in S, as read from S′, are offset by vΔx/c² at low speed. For a jet at 250 m/s and two points 100 m apart that is 0.28 ps: real, and far too small to see.
- GPS. Receivers work in an Earth-fixed frame that rotates, and GPS timing includes the Sagnac correction, an offset of the same form: velocity times distance over c². The rotating Earth is not an inertial frame, so the sim's two frames are only an analogy here.
Go deeper: the offset in ordinary units
- Putting c back, two events simultaneous in S and Δx apart are Δt′ = −γvΔx/c² apart in S′.
- γ is close to 1 until v is a sizeable fraction of c (it is 1.25 at 0.6c), so at everyday speeds the offset is just vΔx/c².
Summary
In one line. Two events at the same time in S but at different places get different times in S′, by Δct′ = −γβΔx. For a spacelike pair the order depends on the frame, and Δs² does not.
Key points
- Δct′ = −γβΔx is zero only if β = 0 or Δx = 0.
- The event further along the boost direction is first in S′. Reversing the boost reverses the order.
- Δs² = −Δct² + Δx² is the same in S and S′. Only spacelike pairs (Δs² > 0) can swap order.
Common mix-ups
- "They only look non-simultaneous because light from A and B reaches S′ at different times." A time coordinate is a synchronised clock reading, so the gap survives every delay correction.
- "There is one real 'now', and the frames just measure it badly." Each frame's lines of constant ct′ slice spacetime equally validly.
- "If the order can flip, cause and effect are in danger." Only spacelike pairs can flip. Pairs a signal can join keep their order in every frame.
Things to try on your own
- Drag B onto A so that Δx = 0, then sweep v/c with Play and read Δct′.
- Compare Δct′ at v/c = 0.30 and 0.60 for the same Δx, and explain the ratio with γβ.
- Open Readings, drag A or B, and check that Δs² in S and in S′ still agree.